Euler Systems

Karl Rubin

Author address:

Department of Mathematics Stanford University Stanford, CA 94305-2125 USA

E-mail address: rubin@math.stanford.edu

Contents

Acknowledgments xi

Introduction 3 Notation 6

Chapter 1. Galois Cohomology of p-adic Representations 9 1.1. p-adic Representations 9 1.2. Galois Cohomology 11 1.3. Local Cohomology Groups 12 1.4. Local Duality 18 1.5. Global Cohomology Groups 21 1.6. Examples of Selmer Groups 23 1.7. Global Duality 28

Chapter 2. Euler Systems: Definition and Main Results 33 2.1. Euler Systems 33 2.2. Results over K 36 2.3. Results over K_{∞} 40 2.4. Twisting by Characters of Finite Order 43

Chapter 3. Examples and Applications 47 3.1. Preliminaries 47 3.2. Cyclotomic Units 48 3.3. Elliptic Units 55 3.4. Stickelberger Elements 55 3.5. Elliptic Curves 63 3.6. Symmetric Square of an Elliptic Curve 73

Chapter 4. Derived Cohomology Classes 75 4.1. Setup 75 4.2. The Universal Euler System 78 4.3. Properties of the Universal Euler System 80 4.4. Kolyvagin’s Derivative Construction 83 4.5. Local Properties of the Derivative Classes 90 4.6. Local Behavior at Primes Not Dividing p𝔯 92 4.7. Local Behavior at Primes Dividing 𝔯 98 4.8. The Congruence 102

Chapter 5. Bounding the Selmer Group 105 5.1. Preliminaries 105 5.2. Bounding the Order of the Selmer Group 106 5.3. Bounding the Exponent of the Selmer Group 114

Chapter 6. Twisting 119 6.1. Twisting Representations 119 6.2. Twisting Cohomology Groups 121 6.3. Twisting Euler Systems 122 6.4. Twisting Theorems 125 6.5. Examples and Applications 125

Chapter 7. Iwasawa Theory 129 7.1. Overview 129 7.2. Galois Groups and the Evaluation Map 135 7.3. Proof of Theorem 2.3.2 141 7.4. The Kernel and Cokernel of the Restriction Map 145 7.5. Galois Equivariance of the Evaluation Maps 147 7.6. Proof of Proposition 7.1.7 151 7.7. Proof of Proposition 7.1.9 154

Chapter 8. Euler Systems and p-adic L-functions 163 8.1. The Setting 163 8.2. Perrin-Riou’s p-adic L-function and Related Conjectures 166 8.3. Connection with Euler Systems when d_{−} = 1 168 8.4. Example: Cyclotomic Units 171 8.5. Connection with Euler Systems when d_{−} > 1 173

Chapter 9. Variants 175 9.1. Rigidity 175 9.2. Finite Primes Splitting Completely in K_{∞}/K 178 9.3. Euler Systems of Finite Depth 179 9.4. Anticyclotomic Euler Systems 180 9.5. Additional Local Conditions 183 9.6. Varying the Euler Factors 185

Appendix A. Linear Algebra 189 A.1. Herbrand Quotients 189 A.2. p-adic Representations 191

Appendix B. Continuous Cohomology and Inverse Limits 195 B.1. Preliminaries 195 B.2. Continuous Cohomology 195 B.3. Inverse Limits 198 B.4. Induced Modules 201 B.5. Semilocal Galois Cohomology 202

Appendix C. Cohomology of p-adic Analytic Groups 205 C.1. Irreducible Actions of Compact Groups 205 C.2. Application to Galois Representations 207

Appendix D. p-adic Calculations in Cyclotomic Fields 211 D.1. Local Units in Cyclotomic Fields 211 D.2. Cyclotomic Units 216

Bibliography 219

Index of Symbols 223

Subject Index 227

Acknowledgments

This book is an outgrowth of the Hermann Weyl lectures I gave at the Institute for Advanced Study in October, 1995. Some of the work and writing was done while I was in residence at the Institute for Advanced Study and the Institut des Hautes Etudes Scientifiques. I would like to thank both the IAS and the IHES for their hospitality and financial support, and the NSF for additional financial support.

I am indebted to many people for numerous helpful conversations during the period that this book was in preparation, especially Avner Ash, Ralph Greenberg, Barry Mazur, Bernadette Perrin-Riou, Alice Silverberg, and Warren Sinnott. I am also very grateful to Christophe Cornut, Alice Silverberg, and Tom Weston for their careful reading of the manuscript and their comments, and to the audiences of graduate courses I gave at Ohio State University and Stanford University for their patience as I was developing this material. Finally, special thanks go to Victor Kolyvagin and Francisco Thaine for their pioneering work.

Karl Rubin December, 1999

Introduction

History. In 1986, Francisco Thaine [Th] introduced a remarkable method for bounding ideal class groups of real abelian extensions of Q. Namely, if F is such a field, he used cyclotomic units in fields F (μ_{ℓ}), for a large class of rational primes ℓ, to construct explicitly a large collection of principal ideals of F. His construction produced enough principal ideals to bound the exponent of the different Galois-eigencomponents of the ideal class group of F, in terms of the cyclotomic units of F. Although Thaine’s results were already known as a corollary of Iwasawa’s “main conjecture”, proved by Mazur and Wiles [MW], Thaine’s proof was very much simpler. The author [Ru1] was able to apply Thaine’s method essentially unchanged to bound ideal class groups of abelian extensions of imaginary quadratic fields in terms of elliptic units, with important consequences for the arithmetic of elliptic curves with complex multiplication.

Shortly after this, Victor Kolyvagin [Ko1] discovered (independently of Thaine) a similar remarkable method, in his case to bound the Selmer group of an elliptic curve. Suppose E is a modular^{1} elliptic curve over Q, with sign +1 in the functional equation of its L-function. Kolyvagin’s method used Heegner points on E over anticyclotomic extensions of prime conductor of an imaginary quadratic field K (in place of cyclotomic units in abelian extensions of Q) to construct cohomology classes over K (in place of principal ideals). He used these cohomology classes, along with duality theorems from Galois cohomology, to bound the exponent of the Selmer group of E over Q. The overall structure of his proof was very similar to that of Thaine.

Inspired by Thaine’s work and his own, Kolyvagin then made another fundamental advance. In his paper [Ko2] he introduced what he called “Euler systems.” In Thaine’s setting (the Euler system of cyclotomic units)

1C. Breuil, B. Conrad, F. Diamond, and R. Taylor have recently announced that they have succeeded in extending the methods of Wiles to prove that every elliptic curve over Q is modular. Given this result, one can remove the assumption that E is modular, both here and throughout the discussion of elliptic curves in §3.5.

Kolyvagin showed how to use cyclotomic units in fields F (μ_{r}), for a large class of integers r (no longer just primes), to bound the orders of the different Galois-eigencomponents of the ideal class group of F, rather than just their exponents. Similarly, by using a larger collection of Heegner points in the situation described above, Kolyvagin was able to give a bound for the order of the Selmer group of E. Thanks to the theorem of Gross and Zagier [GZ], which links Heegner points with the L-function of E, Kolyvagin’s bound is closely related to the order predicted by the Birch and Swinnerton-Dyer conjecture.

This book. This book describes a general theory of Euler systems for p-adic representations. We start with a finite-dimensional p-adic representation T of the Galois group of a number field K. (Thaine’s situation is the case where K = Q and T is \underleftarrow{lim} μ_{p^{n}} twisted by an even Dirichlet character; in Kolyvagin’s case T is the Tate module of a modular elliptic curve.) We define an Euler system for T to be a collection of cohomology classes c_{F} ∈ H^{1}(F, T), for a family of abelian extensions F of K, with a relation between c_{F′} and c_{F} whenever F ⊂ F′.

Our main results show how the existence of an Euler system leads to bounds on the sizes of Selmer groups attached to the Galois module Hom(T, μ_{p^{∞}}), bounds which depend only on the given Euler system. The proofs of these theorems in our general setting parallel closely (with some additional complications) Kolyvagin’s original proof. Results similar to ours have recently been obtained independently by Kato [Ka2] and Perrin-Riou [PR5].

What we do not do here is construct new Euler systems. This is the deepest and most difficult part of the theory. Since Kolyvagin’s introduction of the concept of an Euler system there have been very few new Euler systems found, but each has been extremely important. Kato [Ka3] has constructed a new Euler system for a modular elliptic curve over Q, very different from Kolyvagin’s system of Heegner points (see §3.5). Flach [Fl] has used a collection of cohomology classes (but not a complete Euler system) to bound the exponent but not the order of the Selmer group of the symmetric square of a modular elliptic curve.

One common feature of all the Euler systems mentioned above is that they are closely related to special values of L-functions. An important benefit of this connection is that the bounds on Selmer groups that come out of the Euler system machinery are then linked to L-values. Such bounds provide evidence for the Bloch-Kato conjectures [BK], which predict the orders of these Selmer groups in terms of L-values.

Our definition of Euler system says nothing about L-values. If there is an Euler system for T then there is a whole family of them (for example, the collection of Euler system cohomology classes is a Z_{p}-module, as well as a Gal(K̄/K)-module). If one multiplies an Euler system by p, one gets a new Euler system but a worse bound on the associated Selmer groups. The philosophy underlying this book, although it is explicitly discussed only in Chapter 8, is that under certain circumstances, not only should there exist an Euler system for T, but there should exist a “best possible” Euler system, which will be related to (and contain all the information in) the p-adic L-function attached to T.

A remark about generality. It is difficult to formulate the “most general” definition of an Euler system, and we do not attempt to do this here. The difficulty is partly due to the fact that the number of examples on which to base a generalization is quite small. In the end, we choose a definition which does not cover the case of Kolyvagin’s Heegner points, because to use a more inclusive definition would introduce too many difficulties. (In Chapter 9 we discuss possible modifications of our definition, including one which does include the case of Heegner points.) On the other hand, we do allow the base field K to be an arbitrary number field, instead of requiring K = Q. Although this adds a layer of notation to all proofs, it does not significantly increase the difficulty. A reader wishing to restrict to the simplest (and most interesting) case K = Q should feel free to do so.

Organization. In Chapter 1 we introduce the local and global cohomology groups, and state the duality theorems, which will be required to state and prove our main results. Chapter 2 contains the definition of an Euler system, followed by the statements of our main theorems bounding the Selmer group of Hom(T, μ_{p^{∞}}) over the base field K (§2.2) and over Z_{p}^{d}-extensions K_{∞} of K (§2.3).

Chapter 3 contains some concrete applications of the theorems of Chapter 2. We apply those theorems to three different Euler systems. The first is constructed from cyclotomic units, and is used to study ideal class groups of real abelian fields (§3.2). The second is constructed from Stickelberger elements, and is used to study the minus part of ideal class groups of abelian fields (§3.4). The third is constructed by Kato from Beilinson elements in the K-theory of modular curves, and is used to study the Selmer groups of modular elliptic curves (§3.5).

The proofs of the theorems of Chapter 2 are given in Chapters 4 through 7. In Chapter 4 we give Kolyvagin’s “derivative” construction, taking the Euler system cohomology classes defined over abelian extensions of K and using them to produce cohomology classes over K itself. We then analyze the localizations of these derived classes, information which is crucial to the proofs of our main theorems. In Chapter 5 we bound the Selmer group over K by using the derived classes of Chapter 4 and global duality. Bounding the Selmer group over K_{∞} is similar but more difficult; this is accomplished in Chapter 7 after a digression in Chapter 6 which is used to reduce the proof to a simpler setting.

In Chapter 8 we discuss the conjectural connection between Euler systems and p-adic L-functions. This connection relies heavily on conjectures of Perrin-Riou [PR4]. Assuming a strong version of Perrin-Riou’s conjectures, and subject to some hypotheses on the representation T, we show that there is an Euler system for T which is closely related to the p-adic L-function.

Chapter 9 discusses possible variants of our definition of Euler systems. Finally, there is some material which is used in the text, but which is

outside our main themes. Rather than interrupt the exposition with this material, we include it in four appendices (A–D).

Notation

If F is a field, F̄ will denote a fixed separable closure of F and

G_{F} = Gal(F̄/F).

(All fields we deal with will be perfect, so we may as well assume that F̄ is an algebraic closure of F.) Also, F^{ab} will denote the maximal abelian extension of F, and if F is a local field F^{ur} will denote the maximal unramified extension of F. If F is a global field and Σ is a set of places of F, then F_{Σ} will denote the maximal extension of F which is unramified outside Σ. If K ⊂ F is an extension of fields, we will write K ⊂_{f} F to indicate that [F : K] is finite.

If F is a field and B is a G_{F} -module, F (B) will denote the fixed field of the kernel of the map G_{F} → Aut(B), i.e., the smallest extension of F whose absolute Galois group acts trivially on B.

If O is a ring and B is an O-module then Ann_{O}(B) ⊂ O will denote the annihilator of B in O. If M ∈ O then B_{M} will denote the kernel of multiplication by M on B, and similarly if M is an ideal of O. If B is a free O-module, τ is an O-linear endomorphism of B, and x is an indeterminate, we will write

P (τ|B; x) = det(1 − τx|B) ∈ O[x],

the determinant of 1 − τx acting on B. Caution: If σ is, for example, a Galois automorphism which acts on B, then P (τ|B; σ) means P (τ|B; x) evaluated at x = σ, and not det(1 − τσ|B).

The Galois module of n-th roots of unity will be denoted by μ_{n}. If p is a fixed rational prime and F is a field of characteristic different from p, the cyclotomic character

ε_{cyc} : G_{F} ⟶ Z_{p}^{×}

is the character giving the action of G_{F} on μ_{p^{∞}}, and the Teichmüller character ω : G_{F} → (Z_{p}^{×})_{tors} is the character giving the action of G_{F} on μ_{p} (if p is odd) or on μ_{4} (if p = 2). Hence ω has order at most p − 1 or 2, respectively (with equality if F = Q) and ⟨ε⟩ = ω^{−1}ε_{cyc} takes values in 1 + pZ_{p} (resp. 1 + 4Z_{2}).

If B is an abelian group, B_{div} will denote the maximal divisible subgroup of B. If p is a fixed rational prime, we define the p-adic completion of B to be the double dual

Bˆ = Hom(Hom(B, Q_{p}/Z_{p}), Q_{p}/Z_{p})

(where Hom always denotes continuous homomorphisms if the groups involved comes with natural topologies). For example, if B is a Z_{p}-module then Bˆ = B; if B is a finitely generated abelian group then Bˆ = B ⊗_{Z} Z_{p}. In general, Bˆ is a Z_{p}-module and there is a canonical map from B to Bˆ . If τ is an endomorphism of B then we will often write B^{τ=0} for the kernel of τ, write B^{τ=1} for the subgroup fixed by τ, etc.

If {A_{i} : i ∈ I} is an inverse system, we will denote by {a_{i}}_{i} (or sometimes simply \{a_{i}\}) an element of \underleftarrow{lim} A_{i}.

Finally, if S is a set, then |S| will denote the cardinality of S. Most of these notations will be recalled when they first occur.

CHAPTER 1 Galois Cohomology of p-adic Representations

In this chapter we introduce our basic objects of study: p-adic Galois representations, their cohomology groups, and especially Selmer groups.

We begin by recalling basic facts about cohomology groups associated to p-adic representations, material which is mostly well-known but included here for completeness.

A Selmer group is a subgroup of a global cohomology group determined by “local conditions”. In §1.3 we discuss these local conditions, which are defined in terms of special subgroups of the local cohomology groups. In §1.4 we state without proof the results we need concerning the Tate pairing on local cohomology groups, and we study how our special subgroups behave with respect to this pairing.

In §1.5 and §1.6 we define Selmer groups and give the basic examples of ideal class groups and Selmer groups of elliptic curves and abelian varieties. Then in §1.7, using Poitou-Tate global duality and the local orthogonality results from §1.4, we derive our main tool (Theorem 1.7.3) for bounding the size of Selmer groups.

1.1. p-adic Representations

Definition 1.1.1. Suppose K is a field, p is a rational prime, and O is the ring of integers of a finite extension Φ of Q_{p}. A p-adic representation of G_{K} = Gal(K̄/K), with coefficients in O, is a free O-module T of finite rank with a continuous O-linear action of G_{K}.

Let D denote the divisible module Φ/O. Attached to a p-adic representation T we define

V = T ⊗_{O} Φ,
W = V/T = T ⊗_{O} D,
W_{M} = M^{−1}T/T ⊂ W for nonzero M ∈ O,

so W_{M} is the M-torsion in W. Note that T determines V and W, and W determines T = \underleftarrow{lim} W_{M} and V, but in general there may be different O-modules T giving rise to the same vector space V.

10 1. GALOIS COHOMOLOGY OF p-ADIC REPRESENTATIONS

Example 1.1.2. Suppose ρ : G_{K} → O^{×} is a character (continuous, but not necessarily of finite order). Then we can take T = O_{ρ}, where O_{ρ} is a free rank-one O-module on which G_{K} acts via ρ. Clearly every one-dimensional representation arises in this way. When ρ is the trivial character we get T ≅ O, and when O = Z_{p} and ρ is the cyclotomic character

ε_{cyc} : G_{K} ⟶ Aut(μ_{p^{∞}}) ⟶^{∼} Z_{p}^{×}

we get

T ≅ Z_{p}(1) = \underleftarrow{lim}_{n} μ_{p^{n}},
V ≅ Q_{p}(1) = Q_{p} ⊗_{Z_{p}} \underleftarrow{lim}_{n} μ_{p^{n}},
W ≅ (Q_{p}/Z_{p})(1) = μ_{p^{∞}}.

For general O we also write O(1) = O ⊗ Z_{p}(1), write Φ(1) = Φ ⊗ Q_{p}(1), and write D(1) = D ⊗ Z_{p}(1).

Definition 1.1.3. If T is a p-adic representation of G_{K} then so is the dual representation

T^{∗} = Hom_{O}(T, O(1)).

We will also write

V^{∗} = Hom_{O}(V, Φ(1)) = Hom_{O}(T, Φ(1)) = T^{∗} ⊗_{O} Φ,
W^{∗} = V^{∗}/T^{∗} = Hom_{O}(T, D(1)).

Example 1.1.4. If ρ : G_{K} → O^{×} is a continuous character as in Example 1.1.2, and T = O_{ρ}, then T^{∗} = O_{ρ^{−1}ε_{cyc}}.

Example 1.1.5. Suppose A is an abelian variety defined over K, and p is a prime different from the characteristic of K. We can take O to be Z_{p} and T to be the p-adic Tate module of A defined by

T_{p}(A) = \underleftarrow{lim}_{n} A_{p^{n}}

where A_{p^{n}} denotes the p^{n}-torsion in A(K̄). Then rank_{Z_{p}}T = 2 dim(A). If A and A′ are isogenous, their Tate modules T = T_{p}(A) and T′ = T_{p}(A′) need not be isomorphic (as G_{K}-modules), but the corresponding Q_{p}-vector spaces V and V′ are isomorphic.

If the endomorphism algebra of A over K contains the ring of integers O_{F} of a number field F, and 𝔭 is a prime of F above p, we can also take Φ = F_{𝔭}, the completion of F at 𝔭, and

T = T_{𝔭}(A) = \underleftarrow{lim}_{n} A_{𝔭^{n}}

which has rank 2 dim(A)/[F : Q] over the ring of integers O of Φ. If A is an elliptic curve with complex multiplication by F ⊂ K, this is another important source of one-dimensional representations.

1.2. Galois Cohomology

Suppose K is a field. If B is a commutative topological group with a continuous action of G_{K}, then we have the continuous cohomology groups

H^{i}(K, B) = H^{i}(G_{K}, B).

If further the action of G_{K} factors through the Galois group Gal(K′/K) for some extension K′ of K, we also write H^{i}(K′/K, B) = H^{i}(Gal(K′/K), B). See Appendix B for the basic facts which we will need about continuous cohomology groups.

Example 1.2.1. We have

H^{1}(K, Q_{p}/Z_{p}) = Hom(G_{K}, Q_{p}/Z_{p}),
H^{1}(K, Z_{p}) = Hom(G_{K}, Z_{p}).

By Kummer theory and Proposition B.2.3, respectively,

H^{1}(K, μ_{p^{∞}}) = K^{×} ⊗ (Q_{p}/Z_{p}),
H^{1}(K, Z_{p}(1)) = \underleftarrow{lim}_{n} H^{1}(K, μ_{p^{n}}) = \underleftarrow{lim}_{n} K^{×}/(K^{×})^{p^{n}} = K^{×} ˆ⊗Z_{p},

where ˆ⊗ denotes the (p-adically) completed tensor product.

Suppose T is a p-adic representation of G_{K} with coefficients in O as in §1.1, and M ∈ O is nonzero. Recall that V = T ⊗ Φ and W = V/T. We will frequently make use of the following exact sequences and commutative diagram.

0 ⟶ W_{M} ⟶ W ⟶^{M} W ⟶ 0  (1.1)

\begin{aligned}0 & ⟶ & T & ⟶^{M} & T & ⟶^{M^{−1}} & W_{M} & ⟶ & 0 \\ & & || & & ↓_{M^{−1}} & ↓ \\ 0 & ⟶ & T & ⟶ & V & ⟶ & W & ⟶ & 0. \end{aligned}  (1.2)

Lemma 1.2.2. Suppose M ∈ O is nonzero.

(i) The exact sequence (1.1) induces an exact sequence

0 ⟶ W^{G_{K}} /MW^{G_{K}} ⟶ H^{1}(K, W_{M}) ⟶ H^{1}(K, W)_{M} ⟶ 0.

(ii) The bottom row of (1.2) induces an exact sequence

V^{G_{K}} ⟶ W^{G_{K}} ⟶ H^{1}(K, T)_{tors} ⟶ 0.

(iii) The kernel of the map H^{1}(K, T) → H^{1}(K, W) induced by the com-

position T ↠ T/MT ⟶^{∼} W_{M} ↪ W is

MH^{1}(K, T) + H^{1}(K, T)_{tors}.

Proof. Assertion (i) is clear, and so is (ii) once we show that the kernel of the natural map H^{1}(K, T) → H^{1}(K, V) is H^{1}(K, T)_{tors}. But this is immediate from Proposition B.2.4, which says that the map H^{1}(K, T) → H^{1}(K, V) induces an isomorphism H^{1}(K, V) ≅ H^{1}(K, T) ⊗ Q_{p}.

The diagram (1.2) induces a commutative diagram with exact rows

H^{1}(K, T) ⟶^{M} H^{1}(K, T) ⟶ H^{1}(K, W_{M})
||  ↓^{ϕ_{1}}  ↓
H^{1}(K, T) ⟶^{ϕ_{2}} H^{1}(K, V) ⟶^{ϕ_{3}} H^{1}(K, W)

where ϕ_{1} is induced by M^{−1} : T → V. Since

ker(ϕ_{3}) = ϕ_{2}(H^{1}(K, T)) = ϕ_{1}(MH^{1}(K, T)),

we see that

ker(ϕ_{3} ∘ ϕ_{1}) = MH^{1}(K, T) + ker(ϕ_{1}) = MH^{1}(K, T) + H^{1}(K, T)_{tors},

which proves (iii).

1.3. Local Cohomology Groups

Unramified local cohomology. Suppose for this section that K is a finite extension of Q_{ℓ} for some ℓ, where we allow ℓ to be either a rational prime or ∞ (in which case Q_{ℓ} = R). Let I denote the inertia subgroup of G_{K} (so I = G_{K} if K is archimedean), let K^{ur} = K̄^{I} be the maximal unramified extension of K, and if K is nonarchimedean let Fr ∈ Gal(K^{ur}/K) denote the Frobenius automorphism.

Definition 1.3.1. Suppose B is a G_{K}-module. We say that B is unramified if I acts trivially on B. We define the subgroup of unramified cohomology classes H_{ur}^{1}(K, B) ⊂ H^{1}(K, B) by

H_{ur}^{1}(K, B) = ker(H^{1}(K, B) ⟶ H^{1}(I, B)).

Note that if T is as in §1.1, then

T is unramified ⟺ V is unramified ⟺ W is unramified,

and if K is nonarchimedean of residue characteristic different from p, then this is equivalent to T^{∗}, V^{∗}, and/or W^{∗} being unramified.

Lemma 1.3.2. Suppose B is a G_{K}-module which is either a finitely generated Z_{p}-module, or a finite-dimensional Q_{p}-vector space, or a discrete torsion Z_{p}-module.

(i) If K is nonarchimedean then

H_{ur}^{1}(K, B) ≅ H^{1}(K^{ur}/K, B^{I}) ≅ B^{I}/(Fr − 1)B^{I}.

(ii) If K is nonarchimedean of residue characteristic different from p,

then

H^{1}(K, B)/H_{ur}^{1}(K, B) ≅ H^{1}(I, B)^{Fr=1}.

(iii) If K is archimedean then H_{ur}^{1}(K, B) = 0.

Proof. Assertion (iii) and the first isomorphism of (i) follow from the inflation-restriction exact sequence (Proposition B.2.5(i)). The second isomorphism of (i) (induced by the map on cocycles c ↦ c(Fr)) is Lemma B.2.8.

The hypotheses on B guarantee (see Propositions B.2.5(ii) and B.2.7) that we have an inflation-restriction exact sequence

0 → H^{1}(K^{ur}/K, B^{I}) → H^{1}(K, B) → H^{1}(I, B)^{Fr=1} → H^{2}(K^{ur}/K, B^{I}).

If K is nonarchimedean then Gal(K^{ur}/K) has cohomological dimension one, so H^{2}(K^{ur}/K, B^{I}) = 0. This proves (ii).

Corollary 1.3.3. Suppose K is nonarchimedean of residue characteristic different from p, and V is a Q_{p}[G_{K}]-module which has finite dimension as a Q_{p}-vector space. Then

(i) dim_{Q_{p}}(H_{ur}^{1}(K, V)) = dim_{Q_{p}}(V^{G_{K}}),

(ii) dim_{Q_{p}}(H^{1}(K, V)/H_{ur}^{1}(K, V)) = dim_{Q_{p}}(H^{2}(K, V)).

Proof. Using Lemma 1.3.2(i) we have an exact sequence

0 ⟶ V^{G_{K}} ⟶ V^{I} ⟶^{Fr−1} V^{I} ⟶ H_{ur}^{1}(K, V) ⟶ 0

which proves (i).

Since the residue characteristic of K is different from p, the inertia group I has a unique maximal p-divisible subgroup I′, and I/I′ ≅ Z_{p} (see [Fr] §8 Corollary 3). Thus both I and Gal(K^{ur}/K) have p-cohomological dimension one. It follows that

H^{m}(K^{ur}/K, H^{n}(I, V)) = 0

if m > 1 or n > 1. Therefore the Hochschild-Serre spectral sequence (Propositions B.2.5(ii) and B.2.7) shows that

H^{1}(K^{ur}/K, H^{1}(I, V)) = H^{2}(K, V).

On the other hand, Lemma 1.3.2 shows that

H^{1}(K^{ur}/K, H^{1}(I, V)) ≅ H^{1}(I, V)/(Fr − 1)H^{1}(I, V),
H^{1}(K, V)/H_{ur}^{1}(K, V) ≅ H^{1}(I, V)^{Fr=1}.

Thus there is an exact sequence

0 → H^{1}(K, V)/H_{ur}^{1}(K, V) → H^{1}(I, V) ⟶^{Fr−1} H^{1}(I, V) → H^{2}(K, V) → 0.

Since dim_{Q_{p}}(H^{1}(I, V)) is finite (Proposition B.2.7(iii)), this proves (ii).

Special subgroups. As above, let K be a finite extension of Q_{ℓ} with ℓ ≤ ∞. Let T be a p-adic representation of G_{K}, let V = T ⊗ Φ, and let W = V/T as in §1.1. Following many authors (for example Bloch and Kato [BK] §3, Fontaine and Perrin-Riou [FPR] §I.3.3, or Greenberg [Gr2]) we define special subgroups H_{f}^{1}(K, ·) of certain cohomology groups H^{1}(K, ·). We assume first that ℓ ≠ p and ℓ ≠ ∞, and we discuss the other cases in Remarks 1.3.6 and 1.3.7 below.

Definition 1.3.4. Suppose ℓ ≠ p and ℓ ≠ ∞. Define the finite part of H^{1}(K, V) by

H_{f}^{1}(K, V) = H_{ur}^{1}(K, V).

Define H_{f}^{1}(K, T) ⊂ H^{1}(K, T) and H_{f}^{1}(K, W) ⊂ H^{1}(K, W) to be the inverse image and image, respectively, of H_{f}^{1}(K, V) under the natural maps

H^{1}(K, T) ⟶ H^{1}(K, V) ⟶ H^{1}(K, W).

For every M ∈ O define H_{f}^{1}(K, W_{M}) ⊂ H^{1}(K, W_{M}) to be the inverse image of H_{f}^{1}(K, W) under the natural map H^{1}(K, W_{M}) → H^{1}(K, W).

Finally, for V, T, W, or W_{M} define the singular quotient of H^{1}(K, ·) by

H_{s}^{1}(K, ·) = H^{1}(K, ·)/H_{f}^{1}(K, ·),

so there are exact sequences

0 ⟶ H_{f}^{1}(K, ·) ⟶ H^{1}(K, ·) ⟶ H_{s}^{1}(K, ·) ⟶ 0.

If A is a Z_{p}-module let A_{div} denote its maximal divisible subgroup.

Lemma 1.3.5. Suppose T is as above, ℓ ≠ p, and ℓ ≠ ∞. Then: (i) H_{f}^{1}(K, W) = H_{ur}^{1}(K, W)_{div}. (ii) H_{ur}^{1}(K, T) ⊂ H_{f}^{1}(K, T) with finite index and H_{s}^{1}(K, T) is torsion-

free. (iii) Writing W = W^{I}/(W^{I})_{div}, there are natural isomorphisms

H_{ur}^{1}(K, W)/H_{f}^{1}(K, W) ⟶^{∼} W/(Fr − 1)W

and

H_{f}^{1}(K, T)/H_{ur}^{1}(K, T) ⟶^{∼} W^{Fr=1}.

(iv) If T is unramified then

H_{f}^{1}(K, T) = H_{ur}^{1}(K, T) and H_{f}^{1}(K, W) = H_{ur}^{1}(K, W).

Proof. It is immediate from the definitions that H_{f}^{1}(K, W) is divisible and H_{s}^{1}(K, T) is torsion-free. The commutative diagram with exact rows and columns

0 ⟶ H_{ur}^{1}(K, T) ⟶ H^{1}(K, T) ⟶ H^{1}(I, T)
↓  ↓
0 ⟶ H_{f}^{1}(K, V) ⟶ H^{1}(K, V) ⟶ H^{1}(I, V)
↓  ↓
0 ⟶ H_{ur}^{1}(K, W) ⟶ H^{1}(K, W) ⟶ H^{1}(I, W)

shows that H_{f}^{1}(K, W) ⊂ H_{ur}^{1}(K, W) and H_{ur}^{1}(K, T) ⊂ H_{f}^{1}(K, T). The rest of assertions (i) and (ii) will follow once we prove (iii), since W^{I}/(W^{I})_{div} is finite.

Note that the image of V^{I} in W^{I} is (W^{I})_{div}. Taking I-cohomology and then Gal(K^{ur}/K)-invariants of the exact sequence 0 → T → V → W → 0 gives an exact sequence

0 ⟶ (W^{I}/(W^{I})_{div})^{Fr=1} ⟶ H^{1}(I, T)^{Fr=1} ⟶ H^{1}(I, V)^{Fr=1}.

Therefore using Lemma 1.3.2 we have

H_{f}^{1}(K, T)/H_{ur}^{1}(K, T)

= ker(H^{1}(K, T)/H_{ur}^{1}(K, T) → H^{1}(K, V)/H_{ur}^{1}(K, V))

= ker(H^{1}(I, T)^{Fr=1} → H^{1}(I, V)^{Fr=1})
= (W^{I}/(W^{I})_{div})^{Fr=1},
H_{ur}^{1}(K, W)/H_{f}^{1}(K, W) = coker(H_{ur}^{1}(K, V) → H_{ur}^{1}(K, W))

= coker(V^{I}/(Fr − 1)V^{I} → W^{I}/(Fr − 1)W^{I})

= W^{I}/((W^{I})_{div} + (Fr − 1)W^{I}).

This proves (iii). If T is unramified then W^{I} = W is divisible, so (iv) is immediate from (iii).

Remark 1.3.6. When the residue characteristic ℓ is equal to p, the choice of a subspace H_{f}^{1}(K, V) is much more subtle. Fortunately, for the purpose of working with Euler systems it is not essential to make such a choice. However, to understand fully the arithmetic significance of the Selmer groups we will define in §1.5, and to get the most out of the applications of Euler systems in Chapter 3, it is necessary to choose a subspace H_{f}^{1}(K, V) in the more difficult case ℓ = p.

In this case, Bloch and Kato define H_{f}^{1}(K, V) using the ring B_{cris} defined by Fontaine ([BK] §3). Namely, they define

H_{f}^{1}(K, V) = ker (H^{1}(K, V) ⟶ H^{1}(K, V ⊗ B_{cris})) .

For our purposes we will allow an arbitrary special subspace of H^{1}(K, V), which we will still denote by H_{f}^{1}(K, V). This notation is not as bad as it may seem: in our applications we will always choose a subspace H_{f}^{1}(K, V) which is the same as the one defined by Bloch and Kato, but we need not (and will not) prove they are the same. One could also choose, for example, H_{f}^{1}(K, V) = 0 or H_{f}^{1}(K, V) = H^{1}(K, V).

Once H_{f}^{1}(K, V) is chosen, we define the groups H_{f}^{1}(K, T), H_{f}^{1}(K, W), and H_{f}^{1}(K, W_{M}) in terms of H_{f}^{1}(K, V) exactly as in Definition 1.3.4.

Remark 1.3.7. If K = R or C then H^{1}(K, V) = 0, so H_{f}^{1}(K, V) = 0 and proceeding as above we are led to define

H_{f}^{1}(K, W) = 0,
H_{f}^{1}(K, T) = H^{1}(K, T),
H_{f}^{1}(K, W_{M}) = ker(H^{1}(K, W_{M}) → H^{1}(K, W)) = W^{G_{K}} /MW^{G_{K}} .

Note that all of these groups are zero unless K = R and p = 2.

Lemma 1.3.8. Suppose M ∈ O is nonzero. Then:

(i) The submodule H_{f}^{1}(K, W_{M}) is the image of H_{f}^{1}(K, T) under the map H^{1}(K, T) ⟶ H^{1}(K, W_{M}) induced by T ↠ M^{−1}T/T = W_{M}.

(ii) If T is unramified, ℓ ≠ p, and ℓ ≠ ∞ then

H_{f}^{1}(K, W_{M}) = H_{ur}^{1}(K, W_{M}).

Proof. The diagram (1.2) gives rise to a commutative diagram with exact rows

\begin{matrix}H^{1}(K, T) & ⟶^{M} & H^{1}(K, T) & ⟶ & H^{1}(K, W_{M}) & ⟶ & H^{2}(K, T) \\ || & & ↓_{M^{−1}} & & ↓ & & || \\ H^{1}(K, T) & ⟶ & H^{1}(K, V) & ⟶ & H^{1}(K, W) & ⟶ & H^{2}(K, T). \end{matrix} (1.3)

It is immediate from this diagram and the definitions that the image of H_{f}^{1}(K, T) is contained in H_{f}^{1}(K, W_{M}).

Suppose c_{W_{M}} ∈ H_{f}^{1}(K, W_{M}). Then the image of c_{W_{M}} in H^{1}(K, W) is the image of some c_{V} ∈ H_{f}^{1}(K, V). Thus (1.3) shows that c_{W_{M}} is the image of some c_{T} ∈ H^{1}(K, T), and the image of c_{T} in H^{1}(K, V) differs from c_{V} by an element c′ of H^{1}(K, T). Therefore c_{T} − Mc′ ∈ H_{f}^{1}(K, T) and c_{T} − Mc′ maps to c_{W_{M}}. This shows that H_{f}^{1}(K, W_{M}) is contained in the image of H_{f}^{1}(K, T), and completes the proof of (i).

If T is unramified, ℓ ≠ p, and ℓ ≠ ∞ then

H_{f}^{1}(K, W_{M}) = image(H_{f}^{1}(K, T)) = image(H_{ur}^{1}(K, T)) ⊂ H_{ur}^{1}(K, W_{M})

by (i) and Lemma 1.3.5(iv). Similarly, if ι_{M} : H^{1}(K, W_{M}) → H^{1}(K, W) is the natural map then Lemma 1.3.5(iv) shows that

H_{f}^{1}(K, W_{M}) = ι_{M}^{−1}(H_{f}^{1}(K, W)) = ι_{M}^{−1}(H_{ur}^{1}(K, W)) ⊃ H_{ur}^{1}(K, W_{M})

which proves (ii).

Remark 1.3.9. We can view W_{M} either as a subgroup of W or as a quotient of T. Lemma 1.3.8(i) says that it makes no difference whether we define H_{f}^{1}(K, W_{M}) as the inverse image of H_{f}^{1}(K, W) (as we did) or as the image of H_{f}^{1}(K, T).

Corollary 1.3.10. There are natural horizontal exact sequences and vertical isomorphisms

0 ⟶ H_{f}^{1}(K, W) ⟶ H^{1}(K, W) ⟶ H_{s}^{1}(K, W) ⟶ 0
||  ||  ||
0 ⟶ \underrightarrow{lim}_{M} H_{f}^{1}(K, W_{M}) ⟶ \underrightarrow{lim}_{M} H^{1}(K, W_{M}) ⟶ \underrightarrow{lim}_{M} H_{s}^{1}(K, W_{M}) ⟶ 0
0 ⟶ H_{f}^{1}(K, T) ⟶ H^{1}(K, T) ⟶ H_{s}^{1}(K, T) ⟶ 0
||  ||  ||
0 ⟶ \underleftarrow{lim}_{M} H_{f}^{1}(K, W_{M}) ⟶ \underleftarrow{lim}_{M} H^{1}(K, W_{M}) ⟶ \underleftarrow{lim}_{M} H_{s}^{1}(K, W_{M}) ⟶ 0

Proof. The groups inside the inverse limits are finite (Proposition B.2.7(ii)), so the horizontal exact sequences are clear.

The isomorphism H^{1}(K, W) = \underrightarrow{lim} H^{1}(K, W_{M}) is a basic fact from Galois cohomology, and the isomorphism H_{f}^{1}(K, W) = \underrightarrow{lim} H_{f}^{1}(K, W_{M}) follows immediately from the definition of H_{f}^{1}(K, W_{M}). The isomorphism H_{s}^{1}(K, W) = \underrightarrow{lim} H_{s}^{1}(K, W_{M}) now follows.

The second set of isomorphisms is similar, except that to handle the inverse limits we use Proposition B.2.3 for the center and Lemma 1.3.8(i) for the left.

1.4. Local Duality

Suppose again that K is a finite extension of Q_{ℓ} for some prime ℓ ≤ ∞, and T is a p-adic representation of G_{K}.

Theorem 1.4.1 (Local duality). Suppose that either K is nonarchimedean and 0 ≤ i ≤ 2, or K is archimedean and i = 1. Then the cup product and the local invariant map induce perfect pairings

\begin{aligned}H^{i}(K, V) & × & H^{2−i}(K, V^{∗}) & ⟶ & H^{2}(K, Φ(1)) & ⟶^{∼} & Φ \\ H^{i}(K, W_{M}) & × & H^{2−i}(K, W_{M}^{∗}) & ⟶ & H^{2}(K, O(1)/MO(1)) & ⟶^{∼} & O/MO \\ H^{i}(K, T) & × & H^{2−i}(K, W^{∗}) & ⟶ & H^{2}(K, D(1)) & ⟶^{∼} & D. \end{aligned}

Proof. See for example [Mi] Corollary I.2.3 or [Se2] §II.5.2 (and use Propositions B.2.3 and B.2.4).

Without fear of confusion, we will denote all of the pairings of Theorem 1.4.1 by ⟨ , ⟩_{K}.

Proposition 1.4.2. Suppose that either K is archimedean, or K is nonarchimedean of residue characteristic ℓ ≠ p. Then H_{f}^{1}(K, V) and H_{f}^{1}(K, V^{∗}) are orthogonal complements under the pairing ⟨ , ⟩_{K}.

Proof. If K is archimedean then all the groups are zero, so there is nothing to prove.

Suppose that K is nonarchimedean of residue characteristic ℓ ≠ p. The pairing

⟨ , ⟩_{K} : H_{f}^{1}(K, V) × H_{f}^{1}(K, V^{∗}) ⟶ Φ

factors through H^{2}(K^{ur}/K, Φ(1)), which is 0 since Gal(K^{ur}/K) has cohomological dimension 1. Thus H_{f}^{1}(K, V) and H_{f}^{1}(K, V^{∗}) are orthogonal. Further, Corollary 1.3.3(i), local duality (Theorem 1.4.1), and Corollary 1.3.3(ii), respectively, give the three equalities

dim_{Φ}(H_{f}^{1}(K, V^{∗})) = dim_{Φ}(H^{0}(K, V^{∗})) = dim_{Φ}(H^{2}(K, V)) = dim_{Φ}(H^{1}(K, V)) − dim_{Φ}(H_{f}^{1}(K, V)),

so H_{f}^{1}(K, V) and H_{f}^{1}(K, V^{∗}) are orthogonal complements.

Proposition 1.4.3. Suppose that either (a) K is archimedean,

(b) K is nonarchimedean of residue characteristic ℓ ≠ p, or

(c) K is nonarchimedean of residue characteristic ℓ = p and we choose subspaces H_{f}^{1}(K, V) and H_{f}^{1}(K, V^{∗}) which are orthogonal complements under the pairing ⟨ , ⟩_{K}.

Then under the pairings ⟨ , ⟩_{K}, (i) H_{f}^{1}(K, T) and H_{f}^{1}(K, W^{∗}) are orthogonal complements, (ii) if M in O is nonzero, then H_{f}^{1}(K, W_{M}) and H_{f}^{1}(K, W_{M}^{∗}) are or-

thogonal complements.

Proof. The definition of the local pairings in terms of cup products shows that the diagram

H^{1}(K, V) × H^{1}(K, V^{∗}) ⟶ Φ
ϕ↑  ↓^{ϕ^{∗}}  ↓
H^{1}(K, T) × H^{1}(K, W^{∗}) ⟶ D.

“commutes”, in the sense that if c ∈ H^{1}(K, T) and d ∈ H^{1}(K, V^{∗}), then

⟨ϕ(c), d⟩_{K} = ⟨c, ϕ^{∗}(d)⟩_{K} ∈ D.

By Proposition 1.4.2, H_{f}^{1}(K, V) and H_{f}^{1}(K, V^{∗}) are orthogonal complements of each other in all cases. Thus if we write ·^{⊥} to denote the orthogonal complement, then since H_{f}^{1}(K, W^{∗}) = ϕ^{∗}(H_{f}^{1}(K, V^{∗})) we see that

H_{f}^{1}(K, W^{∗})^{⊥} = ϕ^{−1}(H_{f}^{1}(K, V^{∗})^{⊥}) = ϕ^{−1}(H_{f}^{1}(K, V)) = H_{f}^{1}(K, T).

This proves (i). The proof of (ii) is similar, using (i), Lemma 1.3.8(i), and the diagram

H^{1}(K, T) × H^{1}(K, W^{∗}) ⟶ D
↓  ↑  ↑
H^{1}(K, W_{M}) × H^{1}(K, W_{M}^{∗}) ⟶ O/MO.

Definition 1.4.4. If K is nonarchimedean of residue characteristic different from p, then there is an exact sequence

0 ⟶ I′ ⟶ I ⟶ Z_{p} ⟶ 0

where I′ has trivial pro-p-part (see [Fr] §8 Corollary 3). It follows that if M is a power of p then I has a unique subgroup of index M (the inverse image of MZ_{p}), and by slight abuse of notation we denote this subgroup by I^{M}.

There is a natural action of Gal(K^{ur}/K) on the cyclic group I/I^{M}. The next lemma is essentially Exercice 2, §IV.2 of [Se3].

Lemma 1.4.5. Suppose that K is nonarchimedean of residue characteristic different from p and that M is a power of p. Then there is a canonical isomorphism of Gal(K^{ur}/K)-modules

I/I^{M} ⟶^{∼} μ_{M}.

Proof. We have isomorphisms

Hom(I/I^{M}, μ_{M}) = Hom(I, μ_{M}) ⟶^{∼} (K^{ur})^{×}/((K^{ur})^{×})^{M} ⟶^{∼} Z/MZ,

given by Kummer theory and (on the right) by the valuation map (the unit group of the ring of integers of K^{ur} is p-divisible). The inverse image of 1 under this composition is the desired isomorphism.

More concretely, the isomorphism is given by

σ ↦ (λ^{1/M})^{σ}/(λ^{1/M})

where λ is any uniformizing parameter of K.

Definition 1.4.6. If M ∈ O is nonzero, we let M̄ ∈ Z^{+} denote the smallest power of p which is divisible in O by M.

Lemma 1.4.7. Suppose that K is nonarchimedean of residue characteristic different from p, that T is unramified, that M ∈ O is nonzero, and that μM̄ ⊂ K. Fix a generator ζ of μM̄ and let σ_{ζ} ∈ I/IM̄ be the inverse image of ζ under the isomorphism of Lemma 1.4.5.

(i) Evaluating cocycles on Fr and σ_{ζ} induces isomorphisms

H_{f}^{1}(K, W_{M}) ⟶^{∼} W_{M}/(Fr − 1)W_{M}, H_{s}^{1}(K, W_{M}) ⟶^{∼} W_{M}^{Fr=1},

respectively. (ii) With an appropriate choice of sign on the right, the diagram

H_{f}^{1}(K, W_{M}^{∗})  × H_{s}^{1}(K, W_{M}) ⟶ O/MO
↓  ↓  ↓^{±1⊗ζ}
W_{M}^{∗}/(Fr − 1)W_{M}^{∗} × (W_{M})^{Fr=1} ⟶ O(1)/MO(1)

commutes, where the first two vertical maps are the isomorphisms of (i), the upper pairing is the paring of Theorem 1.4.1, and the lower pairing is the natural one.

Proof. The first isomorphism of (i) is just a restatement of Lemma 1.3.2(i), since by Lemma 1.3.8(ii) we have H_{f}^{1}(K, W_{M}) = H_{ur}^{1}(K, W_{M}). Similarly, Lemma 1.3.2(ii) shows that

H_{s}^{1}(K, W_{M}) = H^{1}(K, W_{M})/H_{ur}^{1}(K, W_{M}) ≅ H^{1}(I, W_{M})^{Fr=1}.

Lemma 1.4.5 shows that I/IM̄ ≅ μM̄, and we have assumed that G_{K} acts trivially on μM̄, so we conclude that

H_{s}^{1}(K, W_{M}) ≅ Hom(I/IM̄, W_{M})^{Fr=1} ≅ Hom(μM̄, W_{M}^{Fr=1}).

Our choice of generator of μM̄ now completes the proof of (i).

Assertion (ii) can be extracted from Chapter I of [Mi], especially Proposition 0.14, Examples 0.8 and 1.6, and Theorem 2.6.

1.5. Global Cohomology Groups

Suppose for this section that K is a number field, T is a p-adic representation of G_{K}, and V and W are defined in terms of T as in §1.1. We assume in addition that T is unramified outside a finite set of primes of K. (As usual, we say that T is unramified at a place v if the inertia group of v acts trivially on T.) We write K_{v} for the completion of K at a place v, and for all primes v dividing p we fix a subspace H_{f}^{1}(K_{v}, V) of H^{1}(K_{v}, V).

For every place v of K there is a canonical restriction map

H^{1}(K, ·) ⟶ H^{1}(K_{v}, ·),

which we will denote either by c ↦ res_{v}(c) or simply c ↦ c_{v}.

If Σ is a finite set of places of K we write K_{Σ} for the maximal extension of K unramified outside Σ.

Recall that

H_{s}^{1}(K_{v}, W) = H^{1}(K_{v}, W)/H_{f}^{1}(K_{v}, W).

Definition 1.5.1. Suppose Σ is a finite set of places of K. We define some Selmer groups corresponding to Σ as follows. First, define

S_{Σ}(K, W) ⊂ S^{Σ}(K, W) ⊂ H^{1}(K, W)

by

S^{Σ}(K, W) = ker(H^{1}(K, W) ⟶ ⨁_{v∉Σ} H_{s}^{1}(K_{v}, W)),
S_{Σ}(K, W) = ker(S^{Σ}(K, W) ⟶ ⨁_{v∈Σ} H^{1}(K_{v}, W)).

(Note that every element of H^{1}(K, W) restricts to zero in all but finitely many H_{s}^{1}(K_{v}, W) because T is ramified at only finitely many primes.) In other words, S^{Σ}(K, W) consists of all classes c ∈ H^{1}(K, W) satisfying the local conditions

• c_{v} ∈ H_{f}^{1}(K_{v}, W) if v ∉ Σ,

• no restriction for v ∈ Σ,

and S_{Σ}(K, W) has the additional restrictions

• c_{v} = 0 if v ∈ Σ.

When Σ = ∅ is the empty set we write

S(K, W) = S^{∅}(K, W) = S_{∅}(K, W).

Similarly, we define S_{Σ}(K, T) ⊂ S^{Σ}(K, T) ⊂ H^{1}(K, T) by
S^{Σ}(K, T) = ker(H^{1}(K, T) ⟶ ∏_{v∉Σ} H_{s}^{1}(K_{v}, T)),
S_{Σ}(K, T) = ker(S^{Σ}(K, T) ⟶ ⨁_{v∈Σ} H^{1}(K_{v}, T)),

and likewise for S_{Σ}(K, W_{M}) ⊂ S^{Σ}(K, W_{M}) ⊂ H^{1}(K, W_{M}) for every nonzero M in O.

Remark 1.5.2. If Σ contains all primes above p, then the Selmer groups S^{Σ} and S_{Σ} are independent of the choice of subspaces H_{f}^{1}(K_{v}, V) for v dividing p.

Lemma 1.5.3. Suppose Σ contains all infinite places, all primes above p, and all primes of K where T is ramified. If A = T, W, or W_{M} with M ∈ O, then

S^{Σ}(K, A) = H^{1}(K_{Σ}/K, A).

Proof. For every place v ∉ Σ, Lemmas 1.3.5(iv) and 1.3.8(ii) show that H_{f}^{1}(K_{v}, A) = H_{ur}^{1}(K_{v}, A). Therefore, writing I_{v} for an inertia group above v, we have

S^{Σ}(K, A) = ker(H^{1}(K, A) ⟶ ∏_{v∉Σ} Hom(I_{v}, A))
= ker(H^{1}(K, A) ⟶ H^{1}(K_{Σ}, A)) = H^{1}(K_{Σ}/K, A).

Lemma 1.5.4. If M ∈ O is nonzero and Σ is a finite set of primes of K, then the natural map ι_{M} : H^{1}(K, W_{M}) → H^{1}(K, W) induces a surjection

S^{Σ}(K, W_{M}) ↠ S^{Σ}(K, W)_{M}.

Proof. Lemma 1.2.2(i) shows that ι_{M}(H^{1}(K, W_{M})) = H^{1}(K, W)_{M}, and from the definition of H_{f}^{1}(K_{v}, W_{M}) it is clear that

ι_{M}^{−1}(S^{Σ}(K, W)_{M}) = S^{Σ}(K, W_{M}).

Remark 1.5.5. Lemma 1.5.4 need not be true if we replace S^{Σ} by S_{Σ}, because it might not be the case that ι_{M}^{−1}(S_{Σ}(K, W)_{M}) ⊂ S_{Σ}(K, W_{M}).

Proposition 1.5.6. If Σ is a finite set of primes of K then (i) S^{Σ}(K, T) = \underleftarrow{lim}_{M} S^{Σ}(K, W_{M}) and S_{Σ}(K, T) = \underleftarrow{lim}_{M} S_{Σ}(K, W_{M}),

(ii) S^{Σ}(K, W) = \underrightarrow{lim}_{M} S^{Σ}(K, W_{M}) and S_{Σ}(K, W) = \underrightarrow{lim}_{M} S_{Σ}(K, W_{M}).

Proof. We have H^{1}(K, W) = \underrightarrow{lim} H^{1}(K, W_{M}), and by Proposition B.2.3, H^{1}(K, T) = \underleftarrow{lim} H^{1}(K, W_{M}). Corollary 1.3.10 shows that all the local conditions behave well under inverse and direct limits, and the proposition follows.

Lemma 1.5.7. Suppose M ∈ O is nonzero and Σ is a finite set of primes of K. Then

(i) S^{Σ}(K, W_{M}) is finite, (ii) S^{Σ}(K, T) is a finitely generated O-module, (iii) the Pontryagin dual of S^{Σ}(K, W) is a finitely generated O-module.

Proof. Without loss of generality we may enlarge Σ if necessary so that Σ contains all infinite places, all primes above p, and all primes where T is ramified. Then by Lemma 1.5.3, if A is W_{M}, T, or W we have S^{Σ}(K, A) = H^{1}(K_{Σ}/K, A). As is well known (see Proposition B.2.7) these groups have the desired properties.

Remark 1.5.8. Suppose F is a finite extension of K, and Σ is a set of places of K. Write Σ_{F} for the set of primes of F dividing places in Σ. We will often write S^{Σ}(F, ·) instead of S^{Σ_{F}} (F, ·) and S_{Σ}(F, ·) instead of S_{Σ_{F}} (F, ·) for the Selmer groups over F attached to W, W_{M}, and T.

1.6. Examples of Selmer Groups

Again for this section K will denote a number field.

1.6.A. Ideal class groups I. Suppose O = Z_{p} and T = Z_{p} with trivial G_{K}-action.

Proposition 1.6.1. If Σ is a set of places of K containing all primes above p, then

S_{Σ}(K, Q_{p}/Z_{p}) = Hom(A_{K,Σ}, Q_{p}/Z_{p}),
S^{Σ}(K, Q_{p}/Z_{p}) = Hom(Gal(K_{Σ}/K), Q_{p}/Z_{p}),

where A_{K,Σ} is the quotient of the ideal class group of K by the subgroup generated by the classes of primes in Σ.

Proof. Suppose v is a place of K not dividing p. Lemma 1.3.5(iv) (for v finite) and Remark 1.3.7 and Lemma 1.3.2(iii) (for v infinite) show that

H_{f}^{1}(K_{v}, Q_{p}/Z_{p}) = H_{ur}^{1}(K_{v}, Q_{p}/Z_{p}) = Hom(Gal(K_{v}^{ur}/K_{v}), Q_{p}/Z_{p}).

Since H^{1}(K, Q_{p}/Z_{p}) = Hom(G_{K}, Q_{p}/Z_{p}), the proposition follows easily by class field theory.

With an appropriate choice of H_{f}^{1}(K_{v}, Q_{p}) for primes v dividing p, Proposition 1.6.2 below will show that Proposition 1.6.1 holds even when Σ is empty, i.e.,

S(K, Q_{p}/Z_{p}) = Hom(A_{K}, Q_{p}/Z_{p})  (1.4)

where A_{K} is the ideal class group of K.

1.6.B. Ideal class groups II. More generally, suppose that

χ : G_{K} ⟶ O^{×}

is a character of finite prime-to-p order, and let T = O_{χ}, a free rank-one O-module on which G_{K} acts via χ. Let L be an abelian extension of K of degree prime to p such that χ factors through Δ = Gal(L/K). Write D_{χ} = D ⊗ O_{χ} and Φ_{χ} = Φ ⊗ O_{χ}.

Suppose v is a place of K (finite or infinite). If w is a place of L above v let D_{w} and I_{w} denote a decomposition group and inertia group of w, respectively, in G_{K}. The restriction map gives isomorphisms (Corollary B.5.3(ii))

H^{1}(K_{v}, V) ≅ (⊕_{w|v}Hom(D_{w}, V))^{Δ} = (⊕_{w|v}Hom(D_{w}, Φ_{χ}))^{Δ}, (1.5)

and if v ∤ p this identifies

H_{f}^{1}(K_{v}, V) = H_{ur}^{1}(K_{v}, V) = (⊕_{w|v}Hom(D_{w}/I_{w}, V))^{Δ}.  (1.6)

If v | p we take (1.6) as the definition of H_{f}^{1}(K_{v}, V) as well; this agrees with the Bloch-Kato definition of H_{f}^{1} in this case.

Let A_{L} denote the ideal class group of L. When L = K the following proposition reduces to (1.4).

Proposition 1.6.2. S(K, W) ≅ Hom(A_{L}, D_{χ})^{Δ}.

Proof. Since [L : K] is prime to p, the restriction map

H^{1}(K, W) ⟶ H^{1}(L, W)^{Δ} = Hom(G_{L}, D_{χ})^{Δ}

is an isomorphism. Exactly as in (1.5) and (1.6), and using Lemma 1.3.5(i), for every place v of K we have

H^{1}(K_{v}, W) ⟶^{∼}  (⊕_{w|v}Hom(D_{w}, W))^{Δ}
∪  ∪
H_{f}^{1}(K_{v}, W) ⟶^{∼} (⊕_{w|v}(Hom(D_{w}/I_{w}, W))^{Δ})_{div}.

Since each D_{w}/I_{w} is torsion-free, ⊕_{w|v}Hom(D_{w}/I_{w}, W) is divisible. Since Δ has order prime to p, we see that

(⊕_{w|v}Hom(D_{w}/I_{w}, W))^{Δ} = (|Δ|^{−1} ∑_{δ∈Δ} δ)(⊕_{w|v}Hom(D_{w}/I_{w}, W))

is divisible and so H_{f}^{1}(K_{v}, W) = (⊕_{w|v}Hom(D_{w}/I_{w}, W))^{Δ}. Therefore, if H_{L} is the Hilbert class field of L, we have

S(K, W) ≅ {ϕ ∈ Hom(G_{L}, D_{χ})^{Δ} : ϕ(I_{w}) = 0 for every w}
= Hom(Gal(H_{L}/L), D_{χ})^{Δ} = Hom(A_{L}, D_{χ})^{Δ}.

1.6.C. Global units and ideal class groups. Let χ, T = O_{χ}, L, A_{L}, and Δ = Gal(L/K) be as in §1.6.B above. Then T^{∗} = O_{χ^{−1}ε_{cyc}}, i.e., T^{∗} is a free rank-one O-module on which G_{K} acts via χ^{−1}ε_{cyc}, where ε_{cyc} denotes the cyclotomic character. In particular G_{L} acts on T^{∗} by the cyclotomic character.

Definition 1.6.3. Suppose B is an abelian group. We define the p-adic completion of B to be the double dual

Bˆ = Hom(Hom(B, Q_{p}/Z_{p}), Q_{p}/Z_{p})

(with continuous homomorphisms, when B comes with a topology). For example, if B is a Z_{p}-module then Bˆ = B; if B is a finitely generated abelian group then Bˆ = B ⊗_{Z} Z_{p}. In general Bˆ is a Z_{p}-module and there is a canonical map from B to Bˆ .

Now suppose further that B is a Z[Δ]-module. Define the χ-component of B by

B^{χ} = {b ∈ Bˆ ⊗_{Z_{p}} O : γb = χ(γ)b for every γ ∈ Δ}.

We fix once and for all an O-generator of O_{χ^{−1}}, and with this choice we get an isomorphism

B^{χ} ≅ (Bˆ ⊗_{Z_{p}} O_{χ^{−1}})^{Δ}.

Since [L : K] is prime to p, taking χ-components is an exact functor and

Bˆ ⊗_{Z_{p}} O = ⊕_{χ}B^{χ}.

Suppose now that v is a prime of K, and let U_{L,v} denote the local units of L ⊗ K_{v} = ∏_{w|v} L_{w}. (That is, U_{L,v} = ∏_{w|v} O_{w}^{×} where O_{w} is the ring of integers of L_{w}.) The restriction map (Corollary B.5.3(ii)) and Kummer theory (Example 1.2.1) give isomorphisms

H^{1}(K_{v}, V^{∗}) ≅ (⊕_{w|v}H^{1}(L_{w}, V^{∗}))^{Δ}

= (⊕_{w|v}H^{1}(L_{w}, Q_{p}(1)) ⊗ Φ_{χ^{−1}})^{Δ} ≅ ((L ⊗ K_{v})^{×})^{χ} ⊗ Φ.

If v ∤ p then with this identification one can check that

\begin{aligned}H^{1}(K_{v}, V^{∗}) & ⟶^{∼} & ((L ⊗ K_{v})^{×})^{χ} ⊗ Φ \\ ∪ & & ∪ \\ H_{f}^{1}(K_{v}, V^{∗}) & ⟶^{∼} & U_{L,v}^{χ} ⊗ Φ. \end{aligned}  (1.7)

If v | p we take the bottom row of (1.7) as the definition of H_{f}^{1}(K_{v}, V^{∗}). This agrees with the Bloch-Kato definition of H_{f}^{1} in this case. Combining (1.5) and (1.6) with the identifications

⊕_{w|v}D_{w} ≅ (L ⊗ K_{v})^{×}, ⊕_{w|v}I_{w} ≅ U_{L,v}

of local class field theory gives a similar diagram

H^{1}(K_{v}, V) ⟶^{∼} Hom(((L ⊗ K_{v})^{×})^{χ}, Φ)
∪  ∪
H_{f}^{1}(K_{v}, V) ⟶^{∼} Hom(((L ⊗ K_{v})^{×})^{χ}/U_{L,v}^{χ}, Φ).

(1.8)

The local pairing ⟨ , ⟩_{v} is the natural one induced by the identifications of (1.7) and (1.8), and so H_{f}^{1}(K_{v}, V^{∗}) and H_{f}^{1}(K_{v}, V) are orthogonal complements.

Let O_{L} denote the ring of integers of L.

Proposition 1.6.4. (i) There is a natural isomorphism

H^{1}(K, W^{∗}) ⟶^{∼} (L^{×} ⊗ Q_{p}/Z_{p})^{χ}.

(ii) There is an exact sequence

0 ⟶ (O_{L}^{×} ⊗ Q_{p}/Z_{p})^{χ} ⟶ S(K, W^{∗}) ⟶ A_{L}^{χ} ⟶ 0.

Proof. Since [L : K] is prime to p, the restriction map

H^{1}(K, W^{∗}) ⟶^{res_{L/K}} H^{1}(L, W^{∗})^{Δ} = (H^{1}(L, μ_{p^{∞}}) ⊗ O_{χ^{−1}})^{Δ}

≅ H^{1}(L, μ_{p^{∞}})^{χ} ≅ (L^{×} ⊗ Q_{p}/Z_{p})^{χ}

is an isomorphism, which gives (i). It follows easily from (1.7) that for every v there is an isomorphism, compatible with (i),

H_{f}^{1}(K_{v}, W^{∗}) ⟶^{∼} U_{L,v}^{χ} ⊗ Q_{p}/Z_{p}.

Therefore if we define X_{L} to be

{y ⊗ p^{−n} ∈ L^{×} ⊗ Q_{p}/Z_{p} : ord_{w}(y) ≡ 0 (mod p^{n}) for every prime w of L},

then

res_{L/K}(S(K, W^{∗})) ≅ X_{L}^{χ}.

Suppose x ∈ X_{L} is represented by y ⊗ p^{−n} with y ∈ L^{×}. Then the principal fractional ideal yO_{L} is of the form 𝔞^{p^{n}} for some fractional ideal 𝔞. This map x ↦ 𝔞 induces a well-defined surjection from X_{L} to the p-part A_{L}^{(p)} of the ideal class group of L. Thus there is an exact sequence

0 ⟶ O_{L}^{×} ⊗ Q_{p}/Z_{p} ⟶ X_{L} ⟶ A_{L}^{(p)} ⟶ 0,

and taking χ-components gives the exact sequence of the proposition.

Let Σ_{p} denote the set of primes of K above p.

Corollary 1.6.5. If Leopoldt’s conjecture is true for L, then S_{Σ_{p}}(K, W^{∗}) is finite.

Proof. Leopoldt’s conjecture for L is the assertion that the p-adic completion of O_{L}^{×} injects into (L ⊗ Q_{p})^{×}. This implies that the map

(O_{L}^{×} ⊗ Q_{p}/Z_{p})^{χ} ⟶ ((L ⊗ Q_{p})^{×} ⊗ Q_{p}/Z_{p})^{χ} ≅ ⊕_{v|p}H^{1}(K_{v}, W^{∗})

has finite kernel, so the corollary follows from Proposition 1.6.4(ii) and the finiteness of the ideal class group.

Corollary 1.6.6. With notation as above, suppose that K = Q. If χ is odd (i.e., χ sends complex conjugation to −1) then S(Q, W^{∗}) ≅ A_{L}^{χ}.

Proof. Since χ is odd, (O_{L}^{×})^{χ} is finite and so (O_{L}^{×} ⊗ Q_{p}/Z_{p})^{χ} = 0. Thus the corollary follows immediately from Proposition 1.6.4(ii).

1.6.D. Abelian varieties. Let A be an abelian variety defined over K and let T = T_{p}(A) be the p-adic Tate module of A as in Example 1.1.5. (See for example [Si] for the basic facts in the case of elliptic curves.) Then

V = V_{p}(A) = T_{p}(A) ⊗ Q_{p}, W = V_{p}(A)/T_{p}(A) = A_{p^{∞}},

where A_{p^{∞}} is the p-power torsion in A(K̄).

For every place v of K there is a natural injective Kummer map

A(K_{v})ˆ ⊗_{Z_{p}} Q_{p} ↪ H^{1}(K_{v}, V_{p}(A))  (1.9)

where A(K_{v})ˆ denotes the p-adic completion of A(K_{v}). If v is a prime of K above p we define H_{f}^{1}(K_{v}, V_{p}(A)) to be the image of this map. This definition agrees with the Bloch-Kato definition of H_{f}^{1}.

Remark 1.6.7. Let A^{∗} denote the dual abelian variety of A. The Weil pairing shows that V_{p}(A)^{∗} = V_{p}(A^{∗}), and if we define H_{f}^{1}(K_{v}, V_{p}(A^{∗})) in the same way as for A, then H_{f}^{1}(K_{v}, V_{p}(A)) and H_{f}^{1}(K_{v}, V_{p}(A^{∗})) are orthogonal complements under the local pairing ⟨ , ⟩_{K_{v}}.

Note that if we fix a polarization of A, then the Weil pairing gives an isomorphism V_{p}(A^{∗}) ≅ V_{p}(A). This isomorphism identifies H_{f}^{1}(K_{v}, V_{p}(A)) and H_{f}^{1}(K_{v}, V_{p}(A^{∗})).

Proposition 1.6.8. The Selmer group S(K, A_{p^{∞}}) is the usual p-power Selmer group attached to the abelian variety A, and there is an exact sequence

0 ⟶ A(K) ⊗ Q_{p}/Z_{p} ⟶ S(K, A_{p^{∞}}) ⟶ � (A_{/K})_{p^{∞}} ⟶ 0

where � (A_{/K})_{p^{∞}} denotes the p-part of the Tate-Shafarevich group of A over K.

Proof. Suppose v ∤ p. If ℓ is the rational prime below v, then A(K_{v}) has a subgroup of finite index which is a pro-ℓ group, so the p-adic completion A(K_{v})ˆ is finite. Also in this case H_{f}^{1}(K_{v}, V_{p}(A)) = 0 by Corollary 1.3.3(i) and Remark 1.3.7. Therefore for all v (including those above p), H_{f}^{1}(K_{v}, V_{p}(A)) is the image of the map (1.9). It follows that for every v the subgroup H_{f}^{1}(K_{v}, A_{p^{∞}}) is the image of A(K_{v})ˆ ⊗ Q_{p}/Z_{p} under the corresponding Kummer map, and so the definition of S(K, A_{p^{∞}}) coincides with the classical definition of the Selmer group of A.

1.7. Global Duality

As in §1.5 we suppose that K is a number field and T is a p-adic representation of G_{K} ramified at only finitely many primes of K. For all primes v dividing p we also fix special subspaces H_{f}^{1}(K_{v}, V) ⊂ H^{1}(K_{v}, V) and H_{f}^{1}(K_{v}, V^{∗}) ⊂ H^{1}(K_{v}, V^{∗}) which are orthogonal complements under the pairing ⟨ , ⟩_{K_{v}} of Theorem 1.4.1. We will also denote this pairing by ⟨ , ⟩_{v}.

Remark 1.7.1. If v is a place dividing p, and the representation V is potentially semistable (see [FPR] §I.2) at v, then the subspaces H_{f}^{1}(K_{v}, V) and H_{f}^{1}(K_{v}, V^{∗}) defined by Bloch and Kato are orthogonal complements (see [FPR] Proposition I.3.3.9(iii) or [BK] Proposition 3.8).

Definition 1.7.2. If Σ_{0} ⊂ Σ are finite sets of places of K we will write

loc_{Σ} : H^{1}(K, W_{M}) ⟶ ⨁_{v∈Σ} H^{1}(K_{v}, W_{M})
loc_{Σ,Σ_{0}}^{s} : S^{Σ}(K, W_{M}) ⟶ ⨁_{v∈Σ−Σ_{0}} H_{s}^{1}(K_{v}, W_{M})
loc_{Σ,Σ_{0}}^{f} : S_{Σ_{0}}(K, W_{M}) ⟶ ⨁_{v∈Σ−Σ_{0}} H_{f}^{1}(K_{v}, W_{M})

for the respective localization maps.

Theorem 1.7.3 (Poitou-Tate duality). Suppose M ∈ O is nonzero, and Σ_{0} ⊂ Σ are finite sets of places of K.

(i) There are exact sequences

0 ⟶ S^{Σ_{0}}(K, W_{M}) ⟶ S^{Σ}(K, W_{M}) ⟶^{loc_{Σ,Σ_{0}}^{s}} ⨁_{v∈Σ−Σ_{0}} H_{s}^{1}(K_{v}, W_{M}),

0 ⟶ S_{Σ}(K, W_{M}^{∗}) ⟶ S_{Σ_{0}}(K, W_{M}^{∗}) ⟶^{loc_{Σ,Σ_{0}}^{f}} ⨁_{v∈Σ−Σ_{0}} H_{f}^{1}(K_{v}, W_{M}^{∗}).

(ii) The images loc_{Σ,Σ_{0}}^{s}(S^{Σ}(K, W_{M})) and loc_{Σ,Σ_{0}}^{f}(S_{Σ_{0}}(K, W_{M}^{∗})) are or-

thogonal complements with respect to the pairing ∑_{v∈Σ−Σ_{0}} ⟨ , ⟩_{v}. (iii) There is an isomorphism

S_{Σ_{0}}(K, W_{M}^{∗})/S_{Σ}(K, W_{M}^{∗}) ⟶^{∼} Hom_{O}(coker(loc_{Σ,Σ_{0}}^{s}), O/MO).

Proof. Assertion (i) is immediate from the definitions of the Selmer groups involved.

For (ii), recall that by Theorem 1.4.1 and Proposition 1.4.3(ii), the pairing ⟨ , ⟩_{v} induces a nondegenerate pairing on H_{s}^{1}(K_{v}, W_{M})×H_{f}^{1}(K_{v}, W_{M}^{∗}). Suppose first that Σ contains all infinite places, all primes above p, and all primes where T is ramified, so that S^{Σ}(K, W_{M}) = H^{1}(K_{Σ}/K, W_{M}) and S^{Σ}(K, W_{M}^{∗}) = H^{1}(K_{Σ}/K, W_{M}^{∗}) by Lemma 1.5.3. Under these conditions, a part of the Poitou-Tate duality exact sequence ([Mi] Theorem I.4.10 or [T1] Theorem 3.1) is

S^{Σ}(K, W_{M}) ⟶^{loc_{Σ}} ⨁_{v∈Σ} H^{1}(K_{v}, W_{M}) ⟶^{loc_{Σ}^{∨}} S^{Σ}(K, W_{M}^{∗})^{∨}  (1.10)

where S^{Σ}(K, W_{M}^{∗})^{∨} = Hom(S^{Σ}(K, W_{M}^{∗}), O/MO) and the maps are induced by localization and the local pairings between H^{1}(K_{v}, W_{M}) and H^{1}(K_{v}, W_{M}^{∗}). Using Proposition 1.4.3(ii), we can combine (1.10) and (i) to produce a new exact sequence

0 ⟶ S^{Σ_{0}}(K, W_{M}) ⟶ S^{Σ}(K, W_{M}) ⟶^{loc_{Σ,Σ_{0}}^{s}} ⨁_{v∈Σ−Σ_{0}} H_{s}^{1}(K_{v}, W_{M})

⟶^{loc_{Σ,Σ_{0}}^{f}^{∨}} S_{Σ_{0}}(K, W_{M}^{∗})^{∨} ⟶ S_{Σ}(K, W_{M}^{∗})^{∨} ⟶ 0. (1.11)

The exactness in the center proves (ii) in this case. (To see the exactness in the center, note that the dual of the tautological exact sequence

0 ⟶ S_{Σ_{0}}(K, W_{M}^{∗}) ⟶ S^{Σ}(K, W_{M}^{∗})

⟶^{loc_{Σ_{0}} ⊕loc_{Σ−Σ_{0}}^{s}} ⨁_{v∈Σ_{0}} H^{1}(K_{v}, W_{M}^{∗}) ⨁_{v∈Σ−Σ_{0}} H_{s}^{1}(K_{v}, W_{M}^{∗})

is

⨁_{v∈Σ_{0}} H^{1}(K_{v}, W_{M}) ⨁_{v∈Σ−Σ_{0}} H_{f}^{1}(K_{v}, W_{M})

⟶^{(loc_{Σ_{0}} ⊕loc_{Σ−Σ_{0}}^{s})^{∨}} S^{Σ}(K, W_{M}^{∗})^{∨} ⟶ S_{Σ_{0}}(K, W_{M}^{∗})^{∨} ⟶ 0.

Splicing this together with (1.10) and

0 ⟶ ⨁_{v∈Σ_{0}} H^{1}(K_{v}, W_{M}) ⨁_{v∈Σ−Σ_{0}} H_{f}^{1}(K_{v}, W_{M})

⟶ ⨁_{v∈Σ} H^{1}(K_{v}, W_{M}) ⟶ ⨁_{v∈Σ−Σ_{0}} H_{s}^{1}(K_{v}, W_{M}) ⟶ 0

gives (1.11).)

Now suppose Σ is arbitrary, and let Σ′ be a finite set of places containing Σ, all infinite places, all primes above p, and all primes where T is ramified. Then we have an exact sequence (1.11) for each of the pairs Σ ⊂ Σ′ and Σ_{0} ⊂ Σ′, so we obtain a diagram

0  0
↓  ↓
S^{Σ′}(K, W_{M})/S^{Σ_{0}}(K, W_{M}) \overrightarrow{⟶} S^{Σ′}(K, W_{M})/S^{Σ}(K, W_{M})
loc_{Σ′,Σ_{0}}^{s} ↓  ^{loc}_{Σ′,Σ}^{s}↓
⨁v∈Σ_{′}−Σ_{0}H_{s}^{1}(K_{v}, W_{M}) \overrightarrow{⟶} _{⨁v∈Σ_{′}−Σ}H_{s}^{1}(K_{v}, W_{M})
(loc_{Σ′,Σ_{0}}^{f})^{∨}↓  ^{(loc_{Σ′,Σ}^{f})^{∨}}↓
(S_{Σ_{0}}(K, W_{M}^{∗})/S_{Σ′}(K, W_{M}^{∗}))^{∨} \overrightarrow{⟶} (S_{Σ}(K, W_{M}^{∗})/S_{Σ′}(K, W_{M}^{∗}))^{∨}
↓  ↓
0  0

with surjective horizontal maps. The snake lemma gives an exact sequence of kernels of the horizontal maps

0 ⟶ S^{Σ}(K, W_{M})/S^{Σ_{0}}(K, W_{M}) ⟶^{loc_{Σ,Σ_{0}}^{s}} ⊕_{v∈Σ−Σ_{0}}H_{s}^{1}(K_{v}, W_{M})

⟶^{(loc_{Σ,Σ}^{f})^{∨}} (S_{Σ_{0}}(K, W_{M}^{∗})/S_{Σ}(K, W_{M}^{∗}))^{∨} ⟶ 0

and the exactness in the center proves (ii) for Σ_{0} ⊂ Σ. Assertion (iii) is just a restatement of (ii).

Remark 1.7.4. Theorem 1.7.3 will be applied with Σ_{0} equal to the empty set or the set of primes dividing p, and with Σ large enough so that S_{Σ}(K, W_{M}^{∗}) = 0. In that situation, it follows from Theorem 1.7.3(iii) that

|S_{Σ_{0}}(K, W_{M}^{∗})| = |coker(loc_{Σ,Σ_{0}}^{s})|.

Thus if one can produce “enough” cohomology classes in S^{Σ}(K, W_{M}), one obtains a good bound on the size of S_{Σ_{0}}(K, W_{M}^{∗}). The purpose of an Euler system is to construct these classes.

Recall that Σ_{p} denotes the set of primes of K above p.

Corollary 1.7.5. There is an isomorphism

S(K, W^{∗})/S_{Σ_{p}}(K, W^{∗}) ⟶^{∼} Hom_{O}(coker(loc_{Σ_{p}}^{s}), D)

where loc_{Σ_{p}}^{s} is the localization map S^{Σ_{p}}(K, T) → ∏_{v|p} H_{s}^{1}(K_{v}, T).

Proof. We apply Theorem 1.7.3(iii) with Σ = Σ_{p} and with Σ_{0} equal to the empty set, and take the direct limit over M to obtain

\underrightarrow{lim}_{M} (S(K, W_{M}^{∗})/S_{Σ_{p}}(K, W_{M}^{∗})) ≅ \underrightarrow{lim}_{M} Hom_{O}(coker(loc_{Σ_{p},M}^{s}), O/MO)

where loc_{Σ_{p},M}^{s} is the localization map S^{Σ_{p}}(K, W_{M}) → ⊕_{v|p}H_{s}^{1}(K_{v}, W_{M}). By Proposition 1.5.6(ii),

\underrightarrow{lim}_{M} (S(K, W_{M}^{∗})/S_{Σ_{p}}(K, W_{M}^{∗})) = S(K, W^{∗})/S_{Σ_{p}}(K, W^{∗}).

By Proposition 1.5.6(i),

\underleftarrow{lim}_{M} S^{Σ_{p}}(K, W_{M}) = S^{Σ_{p}}(K, T),

and by Corollary 1.3.10,

\underleftarrow{lim}_{M} ⊕_{v|p} H_{s}^{1}(K_{v}, W_{M}) = ⊕_{v|p}H_{s}^{1}(K_{v}, T).

Since all the groups S^{Σ_{p}}(K, W_{M}) and H_{s}^{1}(K_{v}, W_{M}) are finite (Proposition B.2.7(ii) and Lemma 1.5.7), it follows (Proposition B.1.1(ii)) that

\underrightarrow{lim}_{M} Hom_{O}(coker(loc_{Σ_{p},M}^{s}), O/MO) ≅ Hom_{O}(\underleftarrow{lim}_{M} coker(loc_{Σ_{p},M}^{s}), D)

and that

\underleftarrow{lim}_{M} coker(loc_{Σ_{p},M}^{s}) = coker(loc_{Σ_{p}}^{s}).

This completes the proof.

CHAPTER 2 Euler Systems: Definition and Main Results

In this chapter we state our main results. The definition of an Euler system is given in §2.1, and the theorems applying Euler systems to study Selmer groups over number fields and over Z_{p}^{d}-extensions of number fields are given in §2.2 and §2.3, respectively. Examples and applications are given in Chapter 3; the reader might benefit from following along in those examples while reading this chapter. The proofs, using tools to be developed in Chapter 4, will be given in Chapters 5 and 7. In Chapter 9 we discuss some variants and extensions of the definition of Euler system given below.

For similar results see the papers of Kato [Ka2] and Perrin-Riou [PR5]. For a first reading, one might want to restrict below to the case K = Q

(so that the group of global units O_{K}^{×} is finite) and O = Z_{p}. This simplifies the notation, while all the main ideas still appear.

2.1. Euler Systems

Fix a number field K, and let O_{K} denote the ring of integers of K. Fix also a rational prime p and a p-adic representation T of G_{K} as in §1.1, with coefficients in the ring of integers O of some finite extension Φ of Q_{p}. We assume in addition, as in §1.5, that T is unramified outside a finite set of primes of K.

Suppose 𝔮 is a prime of K not dividing p, and T is unramified at 𝔮. Let K(𝔮) denote the maximal p-extension of K inside the ray class field of K modulo 𝔮, let Fr_{𝔮} denote a Frobenius of 𝔮 in G_{K}, and define

P (Fr_{𝔮}^{−1}|T^{∗}; x) = det(1 − Fr_{𝔮}^{−1}x|T^{∗}) ∈ O[x]

(the determinant is well-defined because T^{∗} is unramified at 𝔮).

We will write

K ⊂_{f} F

to indicate that the field F is a finite extension of K.

Definition 2.1.1. Suppose K is an (infinite) abelian extension of K and N is an ideal of K divisible by p and by all primes where T is ramified, such that

(i) K contains K(𝔮) for every prime 𝔮 of K not dividing N, (ii) K contains an extension K_{∞} of K such that

• Gal(K_{∞}/K) ≅ Z_{p}^{d} for some d ≥ 1,

• no (finite) prime of K splits completely in K_{∞}/K.

A collection of cohomology classes

c = {c_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K}

is an Euler system for (T, K, N) if, whenever K ⊂_{f} F ⊂_{f} F′ ⊂ K, then

Cor_{F′/F} (c_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}))c_{F}

where Σ(F′/F) is the set of (finite) primes of K, not dividing N, which ramify in F′ but not in F.

We say a collection c = {c_{F} ∈ H^{1}(F, T)} is an Euler system for T if c is an Euler system for (T, K, N) for some choice of N and K as above.

If K_{∞} is a Z_{p}^{d}-extension of K in which no finite prime of K splits completely, we say a collection c = {c_{F} ∈ H^{1}(F, T)} is an Euler system for (T, K_{∞}) if c is an Euler system for (T, K, N) for some choice of N and K containing K_{∞} as above.

Remark 2.1.2. The condition that no finite prime splits completely in K_{∞}/K is satisfied, for example, if K_{∞} contains the cyclotomic Z_{p}-extension of K.

In general, since Z_{p}^{d} has no proper finite subgroups, to say that a prime does not split completely in K_{∞}/K is equivalent to saying that its decomposition group is infinite. See §9.2 for additional remarks about this assumption.

Note that since we require N to be divisible by p, no Euler factors at primes dividing p enter our picture. It follows from our definition that the Euler system cohomology classes are “universal norms” in the K_{∞}/K direction, i.e., if K ⊂_{f} F ⊂_{f} F′ ⊂ F′K_{∞}, then Σ(F′/F) is empty so

Cor_{F′/F} (c_{F′}) = c_{F} .

On the other hand, one might want to include Euler factors for primes where T is ramified. One could easily modify the definition above to take such Euler factors into account. Alternatively, one can choose an ideal N′ prime to p, replace K by the maximal extension K′ of K in K which is unramified at all primes dividing N′, and replace N by N N′. Then the Euler factors at primes dividing N′ become irrelevant, and no information has been lost when we apply the theorems below (since the conclusions are independent of K and N).

Remark 2.1.3. If 𝔪 is a generalized ideal of K (i.e., 𝔪 can be divisible by archimedean places as well as prime ideals), let K[𝔪] denote the ray class field of K modulo 𝔪. Given K and N as in the definition above, an Euler system for (T, K, N) is equivalent to a collection

{c̃_{𝔪} ∈ H^{1}(K[𝔪] ∩ K, T) : every generalized ideal 𝔪}

satisfying

Cor_{K[𝔪𝔮]∩K/K[𝔪]∩K}(c̃_{𝔪𝔮}) = \begin{cases}P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})c̃_{𝔪} & if 𝔮 ∤ 𝔪N, \\ c̃_{𝔪} & if 𝔮 | 𝔪N. \end{cases}

For, given such a collection, if F is a subfield of K, then we can define

c_{F} = Cor_{K[𝔪]∩K/F} (c̃_{𝔪})

where 𝔪 is the conductor of F/K. One checks easily that the collection {c_{F} } is an Euler system. Conversely, given an Euler system {c_{F} } we can define

c̃_{𝔪} = ∏ P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}) c_{K[𝔪]∩K}

where the product is over primes 𝔮 which divide 𝔪 but do not divide N, and which are unramified in (K[𝔪] ∩ K)/K.

Remark 2.1.4. Suppose now that we are given N and K_{∞}/K as in Definition 2.1.1. If 𝔯 = 𝔮_{1} ⋯ 𝔮_{k} is a product of distinct primes not dividing N, then we define K(𝔯) to be the compositum

K(𝔯) = K(𝔮_{1}) ⋯ K(𝔮_{k}).

We will write 1 for the trivial ideal of O_{K}, and K(1) will denote the maximal p-extension of K inside the Hilbert class field of K. If K ⊂_{f} F ⊂ K_{∞} we let F (𝔯) = F K(𝔯). Let K_{min} be the compositum of K_{∞} and all K(𝔮) for primes 𝔮 not dividing N. Thus K_{min} is the smallest extension of K satisfying the conditions of Definition 2.1.1 for N and K_{∞}/K. Every finite extension of K in K_{min} is contained in F (𝔯) for some squarefree ideal 𝔯 prime to N and some K ⊂_{f} F ⊂ K_{∞}. It follows easily that an Euler system for (T, K_{min}, N) is completely determined by the classes c_{F (𝔯)} with K ⊂_{f} F ⊂ K_{∞} and 𝔯 squarefree and prime to N.

Conversely, suppose we are given a collection

{c_{F (𝔯)} ∈ H^{1}(F (𝔯), T) : K ⊂_{f} F ⊂ K_{∞}, 𝔯 squarefree prime to N }

satisfying (a) if K ⊂_{f} F ⊂_{f} F′ ⊂ K_{∞} and 𝔯 is a squarefree ideal of K prime to N,

then

Cor_{F′(𝔯)/F (𝔯)}(c_{F′(𝔯)}) = c_{F (𝔯)},

(b) if K ⊂_{f} F ⊂ K_{∞}, and 𝔮𝔯 is a squarefree ideal prime to N with 𝔮 prime such that K(𝔮) ≠ K(1), then

Cor_{F (𝔯𝔮)/F (𝔯)}(c_{F (𝔯𝔮)}) = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})c_{F (𝔯)}.

(Note that if K(𝔮) = K(1) if and only if F (𝔯𝔮) = F (𝔯).) Then this collection determines an Euler system: if K ⊂_{f} L ⊂ K_{min} then we can set

c_{L} = Cor_{F (𝔯)/L}(c_{F (𝔯)})

where 𝔯 and F are minimal such that L ⊂ F (𝔯). Thus we may view an Euler system for (T, K_{min}, N) as such a collection {c_{F (𝔯)} ∈ H^{1}(F (𝔯), T)}. This will be the most convenient way to think of an Euler system for the proofs in Chapter 4.

Remark 2.1.5. Kolyvagin’s original method ([Ko2] or [Ru3]) required the Euler system to satisfy an additional “congruence” condition. By expanding on an idea from [Ru6], using our assumption that K contains K_{∞} (i.e., that our Euler system extends “in the p-direction”), we will be able to bypass the need for the congruence condition. In fact, the congruence condition follows easily from our techniques in Chapter 4, and although we do not need it, we will state and prove it in §4.8 (Corollary 4.8.1).

On the other hand, if we assume that our Euler system classes satisfy appropriate congruence conditions then we can remove from Definition 2.1.1(ii) the assumption that K contains K_{∞}. See Chapter 9 for a discussion of this and other possible variations on the definition of an Euler system.

2.2. Results over K

We now come to the fundamental application of Euler systems. We will use the “derivative” classes associated to an Euler system (see §4.4) and the duality theorems from Galois cohomology stated in §1.7 to bound the order of a Selmer group (Theorems 2.2.2, 2.2.3, and 2.2.10). Theorems 2.2.2 and 2.2.3 will be proved in Chapter 5.

Let 𝔭 be the maximal ideal of O and let 𝕜 = O/𝔭 be the residue field. Recall that K(1) is the maximal p-extension of K inside the Hilbert class field of K. We will make use of two different sets of hypotheses on the Galois representation T. Hypotheses Hyp(K, T) are stronger than Hyp(K, V), and will allow us to prove a stronger conclusion.

2.2. RESULTS OVER K 37

Hypotheses Hyp(K, T). (i) There is a τ ∈ G_{K} such that • τ acts trivially on μ_{p^{∞}}, on (O_{K}^{×})^{1/p^{∞}}, and on K(1),

• T/(τ − 1)T is free of rank one over O. (ii) T ⊗ 𝕜 is an irreducible 𝕜[G_{K}]-module.

Hypotheses Hyp(K, V). (i) There is a τ ∈ G_{K} such that • τ acts trivially on μ_{p^{∞}}, on (O_{K}^{×})^{1/p^{∞}}, and on K(1),

• dim_{Φ}(V/(τ − 1)V) = 1. (ii) V is an irreducible Φ[G_{K}]-module.

Definition 2.2.1. If c is an Euler system, we define the index of divisibility of c to be

ind_{O}(c) = sup{n : c_{K} ∈ 𝔭^{n}H^{1}(K, T) + H^{1}(K, T)_{tors}} ≤ ∞,

i.e., 𝔭^{ind^{O}(c)} is the largest power of the maximal ideal 𝔭 by which c_{K} can be divided in H^{1}(K, T)/H^{1}(K, T)_{tors}.

Write ℓ_{O}(B) for the length of an O-module B, so that |B| = |𝕜|^{ℓ_{O}(B)}. We allow ℓ_{O}(B) = ∞.

Define Ω = K(1)K(W)K(μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}), where K(W) denotes the smallest extension of K such that G_{K(W)} acts trivially on W.

Let Σ_{p} denote the set of primes of K above p.

Theorem 2.2.2. Suppose that p > 2 and that T satisfies Hyp(K, T). If c is an Euler system for T then

ℓ_{O}(S_{Σ_{p}}(K, W^{∗})) ≤ ind_{O}(c) + 𝔫_{W} + 𝔫_{W}^{∗}

where

𝔫_{W} = ℓ_{O}(H^{1}(Ω/K, W) ∩ S^{Σ_{p}}(K, W)),
𝔫_{W}^{∗} = ℓ_{O}(H^{1}(Ω/K, W^{∗}) ∩ S_{Σ_{p}}(K, W^{∗})).

Theorem 2.2.3. Suppose that V satisfies Hyp(K, V) and T is not the one-dimensional trivial representation. If c is an Euler system for T and c_{K} ∉ H^{1}(K, T)_{tors}, then S_{Σ_{p}}(K, W^{∗}) is finite.

Note that Theorem 2.2.3 holds even if p = 2.

Remark 2.2.4. Hypotheses Hyp(K, T) are satisfied if the image of the Galois representation on T is “sufficiently large”. They often hold in practice; see the discussion of the examples in the next chapter. If rank_{O}(T) = 1, then (i) holds with τ = 1, and (ii) holds trivially.

Remark 2.2.5. Corollary C.2.2 shows that if V is irreducible as a Φ[G_{K}]- module, then both H^{1}(Ω/K, W) and H^{1}(Ω/K, W^{∗}) are finite unless either T = O with trivial action or T = O(1). Frequently the “error terms” 𝔫_{W} and 𝔫_{W}^{∗} in Theorem 2.2.2 are zero; see the examples in Chapter 3.

Remark 2.2.6. Hypothesis Hyp(K, T)(i) will be used to guarantee the existence of sufficiently many primes 𝔮 of K such that H_{f}^{1}(K_{𝔮}, W_{M}) and H_{s}^{1}(K_{𝔮}, W_{M}^{∗}) are free of rank one over O/MO. This in turn will make it possible to use Theorem 1.7, along with the cohomology classes we will construct from the Euler system in Chapter 4, to bound the Selmer group as in Theorem 2.2.2.

Remark 2.2.7. In the exceptional case of Theorem 2.2.3, when T = O, the Selmer group S_{Σ_{p}}(K, W^{∗}) is finite if and only if Leopoldt’s conjecture holds for K. See Corollary 1.6.5.

Remark 2.2.8. There is always a trivial Euler system, namely the one defined by c_{F} = 0 for every F. But in that case ind_{O}(c) = ∞ so Theorems 2.2.2 and 2.2.3 say nothing.

Remark 2.2.9. Theorem 2.2.2 gives a bound for the size of the “strict” Selmer group S_{Σ_{p}}(K, W^{∗}), not the true Selmer group S(K, W^{∗}). Since we have put no local conditions at p on either our representation T or our Euler system c, that restricted Selmer group is all that the Euler system can “see”. Combining the global duality results from §1.7 with Theorems 2.2.2 and 2.2.3 gives Theorem 2.2.10 below concerning S(K, W^{∗}).

Suppose that, as in §1.7, for every prime v of K dividing p we have subspaces H_{f}^{1}(K_{v}, V) ⊂ H^{1}(K_{v}, V) and H_{f}^{1}(K_{v}, V^{∗}) ⊂ H^{1}(K_{v}, V^{∗}) which are orthogonal complements under the pairing ⟨ , ⟩_{K_{v}}. We write

H^{1}(K_{p}, · ) = ⊕_{v|p}H^{1}(K_{v}, · )

and similarly for H_{f}^{1} and H_{s}^{1} = H^{1}/H_{f}^{1}, and let

loc_{Σ_{p}}^{s} : S^{Σ_{p}}(K, T) ⟶ H_{s}^{1}(K_{p}, T)

be the localization map as in Corollary 1.7.5.

By Corollary B.3.5 (see also Proposition 4.6.1) and Lemma 1.3.5(ii), if c is an Euler system then c_{K} ∈ S^{Σ_{p}}(K, T).

Theorem 2.2.10. Suppose c is an Euler system for T and loc_{Σ_{p}}^{s}(c_{K}) ≠ 0.

(i) Suppose that T is not the one-dimensional trivial representation, that V satisfies Hyp(K, V), and that [H_{s}^{1}(K_{p}, T) : Oloc_{Σ_{p}}^{s}(c_{K})] is finite. Then S(K, W^{∗}) is finite.

(ii) Suppose that p > 2 and T satisfies Hyp(K, T). Let 𝔫_{W} and 𝔫_{W}^{∗} be as in Theorem 2.2.2. Then

ℓ_{O}(S(K, W^{∗})) ≤ ℓ_{O}(H_{s}^{1}(K_{p}, T)/Oloc_{Σ_{p}}^{s}(c_{K})) + 𝔫_{W} + 𝔫_{W}^{∗} .

Proof. We will use Theorems 2.2.2 and 2.2.3 to bound S_{Σ_{p}}(K, W^{∗}), and Corollary 1.7.5 to bound [S(K, W^{∗}) : S_{Σ_{p}}(K, W^{∗})].

For every v, the O-module H_{s}^{1}(K_{v}, T) is torsion-free since by definition it injects into the vector space H_{s}^{1}(K_{v}, V). Hence if loc_{Σ_{p}}^{s}(c_{K}) is not zero then c_{K} ∉ H^{1}(K, T)_{tors}. Now Theorem 2.2.3 shows that S_{Σ_{p}}(K, W^{∗}) is finite, and Corollary 1.7.5 shows that

[S(K, W^{∗}) : S_{Σ_{p}}(K, W^{∗})] = [H_{s}^{1}(K_{p}, T) : Oloc_{Σ_{p}}^{s}(S^{Σ_{p}}(K, T))] (2.1)
≤ [H_{s}^{1}(K_{p}, T) : Oloc_{Σ_{p}}^{s}(c_{K})].

This proves (i).

The definition of S^{Σ_{p}}(K, T) gives an injective map

H^{1}(K, T)/S^{Σ_{p}}(K, T) ↪ ⊕_{v∤p}H_{s}^{1}(K_{v}, T),

so H^{1}(K, T)/S^{Σ_{p}}(K, T) is torsion-free. Since c_{K} ∈ S^{Σ_{p}}(K, T), It follows that for every n ≥ 0,

c_{K} ∈ 𝔭^{n}H^{1}(K, T) + H^{1}(K, T)_{tors} ⇒ c_{K} ∈ 𝔭^{n}S^{Σ_{p}}(K, T) + H^{1}(K, T)_{tors}
⇒ loc_{Σ_{p}}^{s}(c_{K}) ∈ 𝔭^{n}loc_{Σ_{p}}^{s}(S^{Σ_{p}}(K, T)).

Therefore if loc_{Σ_{p}}^{s}(c_{K}) ≠ 0 then

ind_{O}(c) ≤ ℓ_{O}(loc_{Σ_{p}}^{s}(S^{Σ_{p}}(K, T))/Oloc_{Σ_{p}}^{s}(c_{K})),

and so Theorem 2.2.2 shows that

ℓ_{O}(S_{Σ_{p}}(K, W^{∗})) ≤ ℓ_{O}(loc_{Σ_{p}}^{s}(S^{Σ_{p}}(K, T))/Oloc_{Σ_{p}}^{s}(c_{K})) + 𝔫_{W} + 𝔫_{W}^{∗} .

Together with equality (2.1) of Corollary 1.7.5, this proves (ii).

Remark 2.2.11. Note that, although a full Euler system is required to prove Theorems 2.2.2, 2.2.3, and 2.2.10, only the class c_{K} appears in the statements of those theorems.

Remark 2.2.12. The choice of subspace H_{f}^{1}(K_{p}, V) intervenes on both sides of the inequality of Theorem 2.2.10(ii).

Remark 2.2.13. One would like a bound for the order of S(K, W^{∗}) which involves a value of an appropriate L-function. However, Theorems 2.2.2 and 2.2.10 are purely algebraic and never “see” special values of L-functions. One hopes that (as in the examples of Chapter 3) these L-values will arise as loc_{Σ_{p}}^{s}(c_{K}) for some Euler system c, and thereby come into the bound for the order of S(K, W^{∗}) via Theorem 2.2.10. See Chapter 8 for a discussion of a general framework in which one expects Euler systems which are related to L-values to exist.

2.3. Results over K_{∞}

Fix for this section an (infinite) abelian extension K_{∞} of K such that Gal(K_{∞}/K) ≅ Z_{p}^{d} for some d ≥ 1, and such that no finite prime of K splits completely in K_{∞}.

Essentially by proving analogues of Theorem 2.2.2 for each finite extension F of K in K_{∞}, we can pass to the limit and prove an Iwasawa-theoretic version of Theorem 2.2.2. See [Lan] Chapter 5 or [Wa] Chapter 13 for basic background on Iwasawa theory, or [Se1] for the more general situation of Z_{p}^{d}-extensions with d > 1.

Theorems 2.3.2, 2.3.3, and 2.3.4 below will be proved in Chapter 7.

Notation. If K ⊂_{f} F ⊂ K_{∞}, we will write Λ_{F} = O[Gal(F/K)]. Let Γ = Gal(K_{∞}/K) and let Λ denote the Iwasawa algebra

Λ = O[[Γ]] = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}Λ_{F} ,

so Λ is (noncanonically) isomorphic to a power series ring over O in d variables.

We say that a Λ-module B is pseudo-null if B is annihilated by an ideal of Λ of height at least two. A pseudo-isomorphism is a Λ-module homomorphism with pseudo-null kernel and cokernel, and two Λ-modules are pseudo-isomorphic if there is a pseudo-isomorphism between them. If B is a finitely generated torsion Λ-module then there is an injective pseudoisomorphism

⨁_{i} Λ/f_{i}Λ ↪ B

with f_{i} ∈ Λ, and we define the characteristic ideal of B

char(B) = ∏_{i} f_{i}Λ.

The characteristic ideal is well-defined, although the individual f_{i} are not. The individual ideals (elementary divisors) f_{i}Λ are uniquely determined if we add the extra requirement that f_{i+1} | f_{i} for every i. If B is a finitely generated Λ-module which is not torsion, we define char(B) = 0. If

0 ⟶ B′ ⟶ B ⟶ B′′ ⟶ 0

is an exact sequence of finitely generated Λ-modules, then

char(B) = char(B′)char(B′′).

We will need the following weak assumption to rule out some very special bad cases. In particular it is satisfied if K = Q.

2.3. RESULTS OVER K_{∞} 41

Hypothesis Hyp(K_{∞}/K). If rank_{Z_{p}}(Γ) = 1 and G_{K_{∞}} acts either trivially or by the cyclotomic character on V, then either K is a totally real field and Leopoldt’s conjecture holds for K (i.e., the p-adic completion of O_{K}^{×} injects into (O_{K} ⊗ Z_{p})^{×}), or K is an imaginary quadratic field.

We will also write Hyp(K_{∞}, T) (resp. Hyp(K_{∞}, V)) for hypotheses Hyp(K, T) (resp. Hyp(K, V)) with G_{K} replaced by G_{K_{∞}}, i.e.,

Hypotheses Hyp(K_{∞}, T). (i) There is a τ ∈ G_{K_{∞}} such that • τ acts trivially on μ_{p^{∞}}, on (O_{K}^{×})^{1/p^{∞}}, and on K(1),

• T/(τ − 1)T is free of rank one over O. (ii) T ⊗ 𝕜 is an irreducible 𝕜[G_{K_{∞}}]-module.

Hypotheses Hyp(K_{∞}, V). (i) There is a τ ∈ G_{K_{∞}} such that • τ acts trivially on μ_{p^{∞}}, on (O_{K}^{×})^{1/p^{∞}}, and on K(1),

• dim_{Φ}(V/(τ − 1)V) = 1. (ii) V is an irreducible Φ[G_{K_{∞}}]-module.

There are simple implications

Hyp(K_{∞}, T) ⇒ Hyp(K_{∞}, V)
⇓  ⇓
Hyp(K, T) ⇒ Hyp(K, V).

Definition 2.3.1. Recall that D = Φ/O. Define Λ-modules

H_{∞}^{1}(K, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T)
S_{Σ_{p}}(K_{∞}, W^{∗}) = \underrightarrow{lim}_{K ⊂_{f} F ⊂K∞}S_{Σ_{p}}(F, W^{∗})
X_{∞} = Hom_{O}(S_{Σ_{p}}(K_{∞}, W^{∗}), D),

limits with respect to corestriction and restriction maps, respectively. If c is an Euler system let c_{K,∞} = {c_{F} }_{K ⊂_{f} F ⊂K_{∞}} denote the corresponding element of H_{∞}^{1}(K, T) and define an ideal of Λ by

ind_{Λ}(c) = {ϕ(c_{K,∞}) : ϕ ∈ Hom_{Λ}(H_{∞}^{1}(K, T), Λ)} ⊂ Λ.

The ideal ind_{Λ}(c) measures the Λ-divisibility of c_{K,∞}, just as ind_{O}(c) of Definition 2.2.1 measures the O-divisibility of c_{K}.

Recall that c is an Euler system for (T, K_{∞}) if it is an Euler system for (T, K, N) with K_{∞} ⊂ K.

Theorem 2.3.2. Suppose c is an Euler system for (T, K_{∞}), and V satisfies Hyp(K_{∞}, V). If c_{K,∞} does not belong to the Λ-torsion submodule of H_{∞}^{1}(K, T), then X_{∞} is a torsion Λ-module.

Theorem 2.3.3. Suppose c is an Euler system for (T, K_{∞}), and T satisfies hypotheses Hyp(K_{∞}, T) and Hyp(K_{∞}/K). Then

char(X_{∞}) divides ind_{Λ}(c).

Theorem 2.3.4. Suppose c is an Euler system for (T, K_{∞}), and V satisfies hypotheses Hyp(K_{∞}, V) and Hyp(K_{∞}/K). Then there is a nonnegative integer t such that

char(X_{∞}) divides p^{t}ind_{Λ}(c).

Remark 2.3.5. The assertion that X_{∞} is a torsion Λ-module is called the weak Leopoldt conjecture for T. See [Gr2] or [PR4] (§1.3 and Appendice B).

Remark 2.3.6. As with Theorem 2.2.2, these three theorems all give bounds for the size of S_{Σ_{p}}(K_{∞}, W^{∗}) rather than the true Selmer group \underrightarrow{lim} S(F, W^{∗}). Combining these results with the global duality results from §1.7 gives Theorem 2.3.8 below concerning the true Selmer group.

Suppose that for every K ⊂_{f} F ⊂ K_{∞} and every prime w dividing p we have subspaces H_{f}^{1}(F_{w}, V) ⊂ H^{1}(F_{w}, V) and H_{f}^{1}(F_{w}, V^{∗}) ⊂ H^{1}(F_{w}, V^{∗}) which are orthogonal complements under the pairing ⟨ , ⟩_{F_{w}}, as in §1.7. We suppose further that if F ⊂ F′ and w′ | w then

Cor_{F′_{w′} /F_{w}}H_{f}^{1}(F′_{w′}, V) ⊂ H_{f}^{1}(F_{w}, V),
Res_{F′_{w′} /F_{w}}H_{f}^{1}(F_{w}, V^{∗}) ⊂ H_{f}^{1}(F′_{w′}, V^{∗}).

(In fact, the local pairing and our assumptions about orthogonality show that these two inclusions are equivalent.) These conditions ensure that, if K ⊂_{f} F ⊂_{f} F′ ⊂ K, the natural restriction and corestriction maps induce maps

S(F, W^{∗}) ⟶ S(F′, W^{∗}), H_{s}^{1}(F′_{p}, T) ⟶ H_{s}^{1}(F_{p}, T)

where we write

H^{1}(F_{p}, · ) = ⊕_{w|p}H^{1}(F_{w}, · ),

and similarly for H_{f}^{1} and H_{s}^{1} = H^{1}/H_{f}^{1}. Define

S(K_{∞}, W^{∗}) = \underrightarrow{lim}_{K ⊂_{f} F ⊂K∞}S(F, W^{∗}),
H_{∞,s}^{1}(K_{p}, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H_{s}^{1}(F_{p}, T).

Proposition 2.3.7. There is an exact sequence

0 ⟶ H_{∞,s}^{1}(K_{p}, T)/loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T)) ⟶ Hom_{O}(S(K_{∞}, W^{∗}), D) ⟶ X_{∞} ⟶ 0 where loc_{Σ_{p}}^{s} : H_{∞}^{1}(K, T) → H_{∞,s}^{1}(K_{p}, T) is the localization map.

Proof. By Corollary B.3.5,

H_{∞}^{1}(K, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}S^{Σ_{p}}(F, T).

Thus the proposition follows from Corollary 1.7.5 by passing to the (direct) limit and applying Hom_{O}( · , D).

Theorem 2.3.8. Suppose that c is an Euler system for (T, K_{∞}), and V satisfies hypotheses Hyp(K_{∞}, V) and Hyp(K_{∞}/K). Suppose further that loc_{Σ_{p}}^{s}(c_{K,∞}) ∉ H_{∞,s}^{1}(K_{p}, T)_{Λ−tors} and H_{∞,s}^{1}(K_{p}, T)/Λloc_{Σ_{p}}^{s}(c_{K,∞}) is a torsion Λ-module. Then Hom_{O}(S(K_{∞}, W^{∗}), D) is a torsion Λ-module and

(i) there is a nonnegative integer t such that

char(Hom_{O}(S(K_{∞}, W^{∗}), D)) divides p^{t}char(H_{∞,s}^{1}(K_{p}, T)/Λloc_{Σ_{p}}^{s}(c_{K,∞})),

(ii) if T satisfies Hyp(K_{∞}, T) then

char(Hom_{O}(S(K_{∞}, W^{∗}), D)) divides char(H_{∞,s}^{1}(K_{p}, T)/Λloc_{Σ_{p}}^{s}(c_{K,∞})).

Proof. Since loc_{Σ_{p}}^{s}(c_{K,∞}) ∉ H_{∞,s}^{1}(K_{p}, T)_{Λ−tors}, we see that c_{K,∞} cannot belong to H_{∞}^{1}(K, T)_{Λ−tors}. Therefore Theorem 2.3.2 and Proposition 2.3.7 show that Hom_{O}(S(K_{∞}, W^{∗}), D) is a torsion Λ-module and that

char(Hom_{O}(S(K_{∞}, W^{∗}), D)) = char(X_{∞})char(H_{∞,s}^{1}(K_{p}, T)/loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T))).

Our assumptions ensure that loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T)) is a rank-one Λ-module, so there is a map ψ : loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T)) → Λ with pseudo-null cokernel. Then

ψ(loc_{Σ_{p}}^{s}(c_{K,∞}))Λ = char(ψ(loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T)))/ψ(loc_{Σ_{p}}^{s}(c_{K,∞}))Λ)
⊃ char(loc_{Σ_{p}}^{s}(H_{∞}^{1}(K, T))/Λloc_{Σ_{p}}^{s}(c_{K,∞})),

and by definition ind_{Λ}(c) divides ψ ∘ loc_{Σ_{p}}^{s}(c_{K,∞}). The theorem follows easily from these divisibilities and the divisibilities of Theorems 2.3.4 and 2.3.3.

2.4. Twisting by Characters of Finite Order

Suppose c is an Euler system for (T, K, N) as defined in Definition 2.1.1. The consequences of the existence of such an Euler system described in §2.2 and §2.3 do not depend on K (except that, in the case of §2.3, the field K must contain K_{∞}). We could always take K to be the “minimal” field K_{min} described in Remark 2.1.4, and ignore the Euler system classes c_{F} for F ⊄ K_{min}, and still obtain the results stated above.

However, there is a way to make use of the additional information contained in an Euler system for a non-minimal K. Namely, in this section we show how to take an Euler system for (T, K, N) and obtain from it an Euler system for twists T ⊗ χ of T by characters χ of finite order of Gal(K/K) (see below). For example, if K is the maximal abelian extension of K, then we will obtain Euler systems for all twists of T by characters of finite order, and the results of this chapter then give (possibly trivial, possibly not) bounds for all the corresponding Selmer groups.

Suppose χ : G_{K} → O^{×} is a character of finite order. As in Example 1.1.2 we will denote by O_{χ} a free rank-one O-module on which G_{K} acts via χ, and we fix a generator ξ_{χ} of O_{χ} We will write T ⊗χ for the representation T ⊗_{O} O_{χ}.

Definition 2.4.1. Suppose c is an Euler system for (T, K, N) and χ is a character of finite order of Gal(K/K) with values in O^{×}. Let L = K^{ker(χ)} be the field cut out by χ. If K ⊂_{f} F ⊂ K, define c_{F}^{χ} ∈ H^{1}(F, T ⊗ χ) to be the image of c_{F L} under the composition

H^{1}(F L, T) ⟶^{⊗ξ_{χ}} H^{1}(F L, T) ⊗ O_{χ} ≅ H^{1}(F L, T ⊗ χ) ⟶^{Cor} H^{1}(F, T ⊗ χ)

(we get the center isomorphism since G_{F L} is in the kernel of χ).

Proposition 2.4.2. Suppose c is an Euler system for (T, K, N) and

χ : Gal(K/K) ⟶ O^{×}

is a character of finite order. If 𝔣 is the conductor of χ then the collection

{c_{F}^{χ} : K ⊂_{f} F ⊂ K}

defined above is an Euler system for (T ⊗ χ, K, 𝔣N).

Proof. If K ⊂_{f} F ⊂_{f} F′ ⊂ K then using Definition 2.1.1 we have

Cor_{F′/F} (c_{F′}^{χ}) = Cor_{F′L/F} (c_{F′L} ⊗ ξ_{χ})

= Cor_{F L/F} ((Cor_{F′L/F L}c_{F′L}) ⊗ ξ_{χ})

= Cor_{F L/F} (( ∏_{𝔮∈Σ(F′L/F L)}P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}) c_{F L}) ⊗ ξ_{χ})

= Cor_{F L/F} ( ∏_{𝔮∈Σ(F′L/F L)}P (Fr_{𝔮}^{−1}|T^{∗}; χ(Fr_{𝔮})Fr_{𝔮}^{−1})(c_{F L} ⊗ ξ_{χ}))

= ∏_{𝔮∈Σ(F′L/F L)}P (Fr_{𝔮}^{−1}|T^{∗}; χ(Fr_{𝔮})Fr_{𝔮}^{−1}) Cor_{F L/F} (c_{F L} ⊗ ξ_{χ})

= ∏_{𝔮∈Σ(F′L/F L)}P (Fr_{𝔮}^{−1}|(T ⊗ χ)^{∗}; Fr_{𝔮}^{−1}) c_{F}^{χ}

where as usual P (Fr_{𝔮}^{−1}|(T ⊗ χ)^{∗}; x) = det(1 − Fr_{𝔮}^{−1}x|(T ⊗ χ)^{∗}), and

Σ(F′L/F L) = {primes 𝔮 : 𝔮 ∤ N, 𝔮 ramifies in F′L but not in F L}

= {primes 𝔮 : 𝔮 ∤ 𝔣N, 𝔮 ramifies in F′ but not in F }.

This proves the proposition.

Lemma 2.4.3. With notation as in Definition 2.4.1, suppose K ⊂_{f} F ⊂ K_{∞} and L⊂_{f} L′ ⊂ K. If every prime which ramifies in L′/K is already ramified in L/K, then the image of c_{F}^{χ} under the composition

H^{1}(F, T ⊗ χ) ⟶^{Res} H^{1}(F L′, T ⊗ χ) ⟶^{⊗ξ_{χ}^{−1}} H^{1}(F L′, T)

is

∑_{δ∈Gal(F L′/F)}χ(δ)δc_{F L′}.

Proof. Since c is an Euler system, and every prime which ramifies in L′/K ramifies in L/K, we have Cor_{F L′/F L}(c_{F L′}) = c_{F L}. Thus the image of c_{F}^{χ} under the composition above is

(Res_{F L′/F} Cor_{F L/F} (c_{F L} ⊗ ξ_{χ})) ⊗ ξ_{χ}^{−1}

= (Res_{F L′/F} Cor_{F L′/F} (c_{F L′} ⊗ ξ_{χ})) ⊗ ξ_{χ}^{−1}

= ( ∑_{δ∈Gal(F L′/F)}δ(c_{F L′} ⊗ ξ_{χ})) ⊗ ξ_{χ}^{−1}

= ∑_{δ∈Gal(F L′/F)}χ(δ)δc_{F L′}.

CHAPTER 3 Examples and Applications

In this chapter we give the basic examples of Euler systems and their applications, using the results of Chapter 2.

3.1. Preliminaries

Suppose χ is a character of G_{K} into O^{×}. As in Example 1.1.2 we will denote by O_{χ} a free rank-one O-module on which G_{K} acts via χ. Recall that D = Φ/O = O ⊗ (Q_{p}/Z_{p}). We will also write

D_{χ} = D ⊗_{O} O_{χ} = O_{χ} ⊗ (Q_{p}/Z_{p}).

For the first three examples (§§3.2, 3.3, and 3.4) we will assume that χ has finite prime-to-p order. As in §1.6.B and §1.6.C we take T = O_{χ}, and we then have W = D_{χ} and T^{∗} = O(1) ⊗ O_{χ^{−1}} = O_{χ^{−1}ε_{cyc}}, where ε_{cyc} is the cyclotomic character.

Let L = K̄^{kerχ} be the abelian extension of K corresponding to χ, and write Δ = Gal(L/K). Thus Δ is a cyclic group of order prime to p. As in Definition 1.6.3, if B is a Z[Δ]-module we write Bˆ for the p-adic completion of B and B^{χ} for the χ-component of Bˆ ⊗_{Z_{p}} O. We also fix a generator of O_{χ^{−1}}, and this choice determines an isomorphism B^{χ} ≅ (Bˆ ⊗_{Z_{p}} O_{χ^{−1}})^{Δ}.

Lemma 3.1.1. (i) If χ ≠ 1 then H^{1}(L(μ_{p^{∞}})/K, W) = 0.

(ii) If χ is not the character giving the action of G_{K} on μ_{p}, then H^{1}(L(μ_{p^{∞}})/K, W^{∗}) = 0.

Proof. Write Ω = L(μ_{p^{∞}}) as in §2.2. Suppose ρ : G_{K} → O^{×} is a character. Recall that 𝔭 is the maximal ideal of O and 𝕜 = O/𝔭 is the residue field. Write 𝕜_{ρ} = 𝕜 ⊗ O_{ρ}.

Since |Δ| is prime to p, the inflation-restriction sequence shows that

H^{1}(Ω/K, 𝕜_{ρ}) = Hom(Gal(Ω/L), 𝕜_{ρ})^{Δ} = Hom(Gal(Ω/L), 𝕜_{ρ}^{Δ})

(note that Δ acts trivially on Gal(Ω/L) because Ω/K is abelian). Further, if π is a generator of 𝔭, it follows from the exact sequence

0 ⟶ 𝕜_{ρ} ⟶ D_{ρ} ⟶^{π} D_{ρ} ⟶ 0

that

H^{1}(Ω/K, 𝕜_{ρ}) = 0 ⇒ H^{1}(Ω/K, D_{ρ})_{𝔭} = 0 ⇒ H^{1}(Ω/K, D_{ρ}) = 0.

If ρ is not congruent to 1 modulo 𝔭, then 𝕜_{ρ}^{Δ} = 0 and so H^{1}(Ω/K, D_{ρ}) = 0. Applying this with ρ = χ proves (i), and with ρ = χ^{−1}ε_{cyc} proves (ii).

3.2. Cyclotomic Units

The Euler system of cyclotomic units is studied in detail in [Ko2] and [Ru3].

An Euler system for Z_{p}(1). Take K = Q. For every extension F of Q, Kummer theory shows as in Example 1.2.1 that

H^{1}(F, Z_{p}(1)) = \underleftarrow{lim}_{n} H^{1}(F, μ_{p^{n}}) = \underleftarrow{lim}_{n} F^{×}/(F^{×})^{p^{n}} = (F^{×})ˆ  (3.1)

where (F^{×})ˆ is the p-adic completion of F^{×}.

Fix a collection {ζ_{m} : m ∈ Z^{+}} such that ζ_{m} is a primitive m-th root of unity and ζ_{mn}^{n} = ζ_{m} for every m and n. (For example, we could fix an embedding of Q̄ into C and choose ζ_{m} = e^{2πi/m}.) For every m ≥ 1 and every prime ℓ we have the relation

N_{Q(μ_{mℓ})/Q(μ_{m})}(ζ_{mℓ} − 1) = \begin{cases}(ζ_{m} − 1) & if ℓ | m \\ (ζ_{m} − 1)^{1−Fr_{ℓ}^{−1}} & if ℓ ∤ m and m > 1 \\ (−1)^{ℓ−1}ℓ & if m = 1 \end{cases} (3.2)

where Fr_{ℓ} is the Frobenius of ℓ in Gal(Q(μ_{m})/Q) (see for example [Lan] Theorem 6.3.1). For every m ≥ 1 we define

c̃_{m∞} = N_{Q(μ_{mp})/Q(μ_{m})}(ζ_{mp} − 1) ∈ Q(μ_{m})^{×} ⊂ H^{1}(Q(μ_{m}), Z_{p}(1))

and c̃_{m} = N_{Q(μ_{m})/Q(μ_{m})^{+}}(c̃_{m∞}) where Q(μ_{m})^{+} is the maximal real subfield of Q(μ_{m}). The distribution relation (3.2) shows that the collection

{c̃_{m∞}, c̃_{m} : m ∈ Z^{+}}

is an Euler system for (Z_{p}(1), Q^{ab}, p) (see Definition 2.1.1 and Remark 2.1.3), since for every prime ℓ ≠ p we have

det(1 − Fr_{ℓ}^{−1}x|Z_{p}(1)^{∗}) = det(1 − Fr_{ℓ}^{−1}x|Z_{p}) = 1 − x.

Remark 3.2.1. If p | m then (3.2) shows that c̃_{m∞} = ζ_{m} − 1. But if p ∤ m, our definition takes into account that our Euler system must satisfy N_{Q(μ_{mp})/Q(μ_{m})}(c̃_{mp}) = c̃_{m}. This causes us to lose some information, and leads to the unwanted hypothesis χ(p) ≠ 1 in Theorem 3.2.3 below. We can remove this hypothesis either by using Theorem 3.2.10 below (see Remark 3.2.5) or by modifying the definition of Euler system as in Example 9.1.1.

The setting. Let K = Q and K_{∞} = Q_{∞}, the cyclotomic (and only) Z_{p}-extension of Q. As in §3.1 we fix a character χ : G_{Q} → O^{×} of finite prime-to-p order, and we assume for the rest of this section that χ is even and nontrivial.

Let f denote the conductor of χ, and recall that L is the field cut out by χ. We will view χ as a Dirichlet character modulo f in the usual way, so that χ(ℓ) = χ(Fr_{ℓ}) if the prime ℓ does not divide f, and χ(ℓ) = 0 if ℓ | f. For every n ≥ 0 let Q_{n} be the unique subfield of Q_{∞} with degree [Q_{n} : Q] = p^{n}, so Q_{n} ⊂ Q(μ_{p^{n+1}}) and [Q(μ_{p^{n+1}}) : Q_{n}] = p − 1. Let L_{n} = LQ_{n}, and let L_{∞} = LQ_{∞}. Since [L : Q] is prime to p, we have L ∩ Q_{n} = Q for every n so we can identify Δ = Gal(L/Q) with Gal(L_{n}/Q_{n}) for every n.

[figure: Q_{∞}; Δ; L_{∞}; Q_{n}; Δ; p^{n}Z_{p}; L_{n}; p^{n}Z_{p}; Q; Δ; Z/p^{n}Z; L; Z/p^{n}Z]

Let T = O_{χ} as in §3.1, so that T^{∗} = Z_{p}(1) ⊗ χ^{−1}. The restriction map gives an isomorphism (using (3.1))

H^{1}(Q_{n}, T^{∗}) ≅ H^{1}(L_{n}, T^{∗})^{Δ} ≅ ((L_{n}^{×})ˆ ⊗ O_{χ^{−1}})^{Δ} ≅ (L_{n}^{×})^{χ} ⊂ (L_{n}^{×})ˆ ⊗ O. (3.3)

The Euler system c̃ for Z_{p}(1) constructed above gives rise (by Proposition 2.4.2) to an Euler system c = c̃^{χ^{−1}} for (T^{∗}, Q^{ab}, pf). By Lemma 2.4.3, the image of c_{Q} in L^{×} ˆ⊗O under (3.3) is

∏_{δ∈Gal(Q(μ_{f})^{+}/Q)} (δc̃_{f})^{χ^{−1}(δ)} = ∏_{δ∈Gal(Q(μ_{fp})/Q)} (ζ_{fp}^{δ} − 1)^{χ^{−1}(δ)}.  (3.4)

The Selmer group. We have W = D_{χ}. Let Q_{n,p} denote the completion of Q_{n} at the unique prime above p, and as in the example of §1.6.B take H_{f}^{1}(Q_{n,p}, V) = H_{ur}^{1}(Q_{n,p}, V).

For every n let A_{n} be the ideal class group of L_{n}. We will also write A_{L} = A_{0}, the ideal class group of L. By Proposition 1.6.2 we have isomorphisms

S(Q, W) ≅ Hom(A_{L}, D_{χ})^{Δ}, S(Q_{∞}, W) ≅ Hom(\underleftarrow{lim} A_{n}, D_{χ})^{Δ}. (3.5)

The ideal class group of L.

Definition 3.2.2. If n ≥ 0 we let E_{n} denote the group of global units of L_{n}. We define the group of χ-cyclotomic units C_{n,χ} to be the subgroup of E_{n}^{χ} generated over O[Gal(L_{n}/Q)] by

ξ_{n,χ} = \begin{cases}∏_{δ∈Gal(Q(μ_{f})/Q)} (ζ_{f}^{δ} − 1)^{χ^{−1}(δ)} & if n = 0, \\ ∏_{δ∈Gal(Q_{n}(μ_{fp^{n+1}} )/Q_{n})} (ζ_{fp^{n+1}}^{δ} − 1)^{χ^{−1}(δ)} & if n > 0. \end{cases}

We will also write E_{L} = E_{0}, C_{L,χ} = C_{0,χ} and ξ_{L,χ} = ξ_{0,χ}.

The following theorem and its Corollary (3.2.4 below) were first proved by Mazur and Wiles [MW]; the proof given here is due to Kolyvagin [Ko2]. See the additional remarks following the proof.

Theorem 3.2.3. Suppose that χ is an even character of order prime to p. If p > 2 and χ(p) ≠ 1, then

|A_{L}^{χ}| divides [E_{L}^{χ} : C_{L,χ}].

Proof. We will apply Theorem 2.2.2 with the Euler system c constructed from cyclotomic units above. Since rank_{O}T^{∗} = 1, we see that Hyp(Q, T^{∗}) is satisfied with τ = 1. Further, in this case Ω = L(μ_{p^{∞}}), and since χ is nontrivial and even, Lemma 3.1.1 shows that the error terms 𝔫_{W^{∗}} and 𝔫_{W^{∗}}^{∗} in Theorem 2.2.2 are both zero.

By (3.3) we have maps

E_{L}^{χ} ↪ (L^{×})^{χ} ⟶^{∼} H^{1}(Q, T^{∗}).

Identifying ξ_{L,χ} with its image in H^{1}(Q, T^{∗}), it follows from (3.2) and (3.4) that

c_{Q} = ξ_{L,χ}^{1−χ^{−1}(p)}  (3.6)

where χ(p) = 0 if p | f. Since χ(p) ≠ 1 and χ has order prime to p, we have 1 − χ^{−1}(p) ∈ O^{×} so c_{Q} generates C_{L,χ}.

Recall that ind_{O}(c) is the index of divisibility defined in Definition 2.2.1. Since L^{×}/E_{L} is torsion-free, it follows that ind_{O}(c) is the largest power of p by which a generator of C_{L,χ} can be divided in E_{L}^{χ}. Since we have assumed that p > 2, that χ is even, and that χ ≠ 1, the Dirichlet unit theorem (see for example [T5] §I.4) shows that E_{L}^{χ} is free of rank one over O, and we conclude that

ind_{O}(c) = ℓ_{O}(E_{L}^{χ}/C_{L,χ}).

Putting all of this together, Theorem 2.2.2 in this case yields

|S_{Σ_{p}}(Q, W)| divides [E_{L}^{χ} : C_{L,χ}].

Let I denote an inertia group above p and Fr_{p} ∈ G_{Q} a Frobenius element. By Lemma 1.3.2(i),

H_{ur}^{1}(Q_{p}, V) = V^{I}/(Fr_{p} − 1)V^{I} = V^{I}/(χ(p) − 1)V^{I} = 0

since χ(p) ≠ 1. Therefore H_{f}^{1}(Q_{p}, W) = 0 and

S_{Σ_{p}}(Q, W) = S(Q, W) = Hom_{O}(A_{L}^{χ}, D),

the final equality coming from (3.5). This completes the proof.

A well-known argument using the analytic class number formula takes Theorem 3.2.3 and gives the following strengthening.

Corollary 3.2.4 (Mazur & Wiles [MW] Theorem 1.10.1). Suppose χ is an even character of order prime to p. If p > 2 and χ(p) ≠ 1, then

|A_{L}^{χ}| = [E_{L}^{χ} : C_{L,χ}].

Proof. See for example Theorem 4.2 of [Ru3].

Remark 3.2.5. When p divides the order of χ, Theorem 2.2.2 still applies to give a bound for S(Q, W), but (see Proposition 1.6.2) this Selmer group is no longer exactly the ideal class group.

If χ(p) = 1, then (3.6) shows that c_{Q} = 0, so Theorem 2.2.2 is of no use. However, in this case Greenberg ([Gr1] §5) has shown how to deduce the equality of Corollary 3.2.4 from Theorem 3.2.10 below (Iwasawa’s “main conjecture”) which we will prove using Theorem 2.3.3. See also §9.1.

The inverse limit of the ideal class groups. Recall that Λ is the Iwasawa algebra O[[Gal(Q_{∞}/Q)]]. For every n, let L_{n,p} = L_{n} ⊗ Q_{p} and denote by U_{n} the local units of L_{n,p}. Define

A_{∞} = \underleftarrow{lim}_{n} (A_{n})ˆ , E_{∞} = \underleftarrow{lim}_{n} (E_{n})ˆ , C_{∞,χ} = \underleftarrow{lim}_{n} (C_{n,χ})ˆ ,

U_{∞} = \underleftarrow{lim}_{n} (U_{n})ˆ , Y_{∞} = \underleftarrow{lim}_{n} (L_{n,p})ˆ ,

inverse limits with respect to norm maps, where (· )ˆ denotes p-adic completion (Definition 1.6.3). Let E′_{n} be the group of p-units of L_{n} (elements which are units at all primes not dividing p) and E′_{∞} = \underleftarrow{lim} (E′_{n})ˆ . Recall that H_{∞}^{1}(Q, T^{∗}) = \underleftarrow{lim} H^{1}(Q_{n}, T^{∗}), and set H_{∞}^{1}(Q_{p}, T^{∗}) = \underleftarrow{lim} H^{1}(Q_{n,p}, T^{∗}) and H_{∞,s}^{1}(Q_{p}, T^{∗}) = \underleftarrow{lim} H^{1}(Q_{n,p}, T^{∗})/H_{f}^{1}(Q_{n,p}, T^{∗}), where H_{f}^{1}(Q_{n,p}, V^{∗}) is defined as in the example of §1.6.C.

Proposition 3.2.6. (i) With the natural horizontal inclusions and surjections, there are vertical isomorphisms making the following diagram commute.

\xymatrix{Λ\{c_{Q_{n}}\} \ar@{^{(}->}[r] \ar[d]_{≅} & H_{∞}^{1}(Q, T^{∗}) \ar@{^{(}->}[r] \ar[d]_{≅} & H_{∞}^{1}(Q_{p}, T^{∗}) \ar@{->>}[r] \ar[d]_{≅} & H_{∞,s}^{1}(Q_{p}, T^{∗}) \ar[d]_{≅} \\ C_{∞,χ} \ar@{^{(}->}[r] & (E′_{∞})^{χ} \ar@{^{(}->}[r] & Y_{∞}^{χ} \ar@{->>}[r] & Y_{∞}^{χ}/U_{∞}^{χ}.}

(ii) There is a Λ-module isomorphism

Y_{∞}^{χ}/U_{∞}^{χ} ≅ \begin{cases}0 & if χ(p) ≠ 1, \\ O & if χ(p) = 1. \end{cases}

(iii) There is a Λ-module injection (E′_{∞})^{χ}/E_{∞}^{χ} ↪ O.

Proof. Just as for (3.4), Lemma 2.4.3 shows that the image of c_{Q_{n}} in (L_{n}^{×})^{χ} under (3.3) is ξ_{n,χ}, so the left-hand vertical isomorphism is clear. As in the example of §1.6.C, the restriction isomorphism (3.3) identifies

S^{{p}}(Q_{n}, T^{∗}) ≅ (E′_{n})^{χ},

and by Corollary B.3.5

\underleftarrow{lim}_{n} H^{1}(Q_{n}, T^{∗}) = \underleftarrow{lim}_{n} S^{\{p\}}(Q_{n}, T^{∗}),

so we obtain the second vertical isomorphism. With H_{f}^{1} as defined in the example of §1.6.C, we see as in (1.7) that there are restriction isomorphisms (the top row is the local analogue of (3.3))

H^{1}(Q_{n,p}, T^{∗}) ⟶^{∼} (L_{n,p}^{×})^{χ}
∪  ∪
H_{f}^{1}(Q_{n,p}, T^{∗}) ⟶^{∼} U_{n}^{χ}

and the rest of (i) follows. (Note that once we have the vertical isomorphisms, the injectivity of the upper center horizontal map follows from that of the lower center horizontal map; the latter injectivity follows from Leopoldt’s conjecture, which is known in this setting.)

Let Δ_{p} denote the decomposition group of p in Δ. For every m > n there is a commutative diagram in which all maps are isomorphisms

L_{m,p}^{×}/U_{m} ⟶_{∼}^{⊕_{w|p}ord_{w}} ⨁_{w|p} Zw ⟶^{∼} Z[Δ/Δ_{p}]
N_{Lm/Ln} ↓  ^{w↦w|_{Ln}} ↓  ∥
L_{n,p}^{×}/U_{n} ⟶_{∼}^{⊕_{v|p}ord_{v}} ⨁_{v|p} Zv ⟶^{∼} Z[Δ/Δ_{p}],

and so Y_{∞}^{χ}/U_{∞}^{χ} ≅ Z_{p}[Δ/Δ_{p}]^{χ}. Clearly Z_{p}[Δ/Δ_{p}]^{χ} = 0 if χ is nontrivial on Δ_{p}, i.e., if χ(p) ≠ 1, and otherwise Z_{p}[Δ/Δ_{p}]^{χ} ≅ O_{χ}. This proves (ii), and (iii) follows from (ii) since E_{∞}^{χ} is the kernel of the natural map (E′_{∞})^{χ} → Y_{∞}^{χ}/U_{∞}^{χ}.

Theorem 3.2.7. If χ is even and nontrivial then

char(A_{∞}^{χ}) divides char(E_{∞}^{χ}/C_{∞,χ}).

Proof. Hypotheses Hyp(Q_{∞}, T^{∗}) are satisfied with τ = 1, so we can apply Theorem 2.3.3 and Proposition 2.3.7 to conclude that

char(Hom_{O}(S(Q_{∞}, W), D)) divides ind_{Λ}(c) char(H_{∞,s}^{1}(Q_{p}, T^{∗}))

with ind_{Λ}(c) as defined in Definition 2.3.1.

By [Iw3] Theorem 25, the Λ-module Y_{∞}^{χ} is torsion-free, finitely generated, and rank-one. Since (E′_{∞})^{χ} is a nonzero Λ-submodule of Y_{∞}^{χ}, it follows that (E′_{∞})^{χ} is also torsion-free, finitely generated, and rank-one. Combined with the diagram of Proposition 3.2.6(i), it follows easily that ind_{Λ}(c) = char((E′_{∞})^{χ}/C_{∞,χ}), and so using Proposition 3.2.6(iii) we see that ind_{Λ}(c) divides J char(E_{∞}^{χ}/C_{∞,χ}), where J = char(O) is the augmentation ideal of Λ. By (3.5) we have Hom_{O}(S(Q_{∞}, W), D) ≅ A_{∞}^{χ}, and Proposition 3.2.6 shows that char(H_{∞,s}^{1}(Q_{p}, T^{∗})) divides J. Therefore

char(A_{∞}^{χ}) divides J^{2}char(E_{∞}^{χ}/C_{∞,χ}).

Thus to prove the theorem it suffices to show that char(A_{∞}^{χ}) is not divisible by J.

We only sketch the proof. A standard elementary Iwasawa theory argument (see for example [Iw3] §3.1) shows that A_{∞}^{χ}/J A_{∞}^{χ} is a finitely generated Z_{p}-module, that

J | char(A_{∞}^{χ}) ⟺ A_{∞}^{χ}/J A_{∞}^{χ} is infinite,

and that A_{∞}^{χ}/J A_{∞}^{χ} = Gal(M_{∞}/L_{∞}) where M_{∞} is an extension of L_{∞} which is abelian over L. Since χ is even, we know that L is a real abelian field and that Leopoldt’s conjecture holds for L. Therefore class field theory shows that L has no Z_{p}^{2}-extensions, so Gal(M_{∞}/L) has Z_{p}-rank one and [M_{∞} : L_{∞}] must be finite. This completes the proof.

Corollary 3.2.8. Suppose that χ is even and nontrivial, and that p > 2. Then

char(A_{∞}^{χ}) = char(E_{∞}^{χ}/C_{∞,χ}).

Proof. As was the case for Corollary 3.2.4, this follows from Theorem 3.2.7 and the analytic class number formula. See for example [MW] §1.6, or [Ru3] p. 414.

The p-adic L-function. Let ω : G_{Q} → (Z_{p}^{×})_{tors} denote the Teichmüller character giving the action of G_{Q} on μ_{p} (if p is odd) or on μ_{4} (if p = 2). Thus ω^{−1}ε_{cyc} is a character of Gal(Q_{∞}/Q). Fix an embedding O ↪ \overline{Q_{p}} ↪ C so that we can identify complex and p-adic characters of finite order of G_{Q}. With this identification, a character ρ of Gal(Q_{∞}/Q) of finite order extends naturally to an O-algebra homomorphism ρ : Λ → \overline{Q_{p}}.

Let L(s, ρ) denote the Dirichlet L-function attached to a character ρ.

Theorem 3.2.9. Suppose that χ is even and nontrivial, and that p > 2.

(i) There is an element L_{χ} ∈ Λ (the p-adic L-function attached to χ) such that for every k ≥ 1 and every character ρ of finite order of Gal(Q_{∞}/Q),

(ω^{−1}ε_{cyc})^{k}ρ(L_{χ}) = (1 − ω^{−k}ρχ(p)p^{k−1})L(1 − k, ω^{−k}ρχ).

(ii) char(U_{∞}^{χ}/C_{∞,χ}) = L_{χ}Λ.

Proof. See for example [Iw2] §6 or [Wa] Theorem 7.10 for (i), and [Iw1],

[Wa] Theorem 13.56, [Lan] Theorem 7.5.2, or (for the general case) [Gi] Théorème 1 for (ii). (See also §D.2 where we carry out the main computation needed to prove (ii).)

Theorem 3.2.10. Suppose that χ is even and nontrivial, and that p > 2. Let Z_{∞,χ} = Gal(M_{∞}/L_{∞})^{χ}, where M_{∞} is the maximal abelian p-extension of L_{∞} unramified outside primes above p. Then Z_{∞,χ} is a Gal(L/Q)- module and a finitely generated Λ-module, and

char(Z_{∞,χ}) = L_{χ}Λ

where L_{χ} is the p-adic L-function defined in Theorem 3.2.9.

Proof. Class field theory gives an exact sequence (see for example §III.1.7 of [dS])

0 ⟶ E_{∞}^{χ}/C_{∞,χ} ⟶ U_{∞}^{χ}/C_{∞,χ} ⟶ Z_{∞,χ} ⟶ A_{∞}^{χ} ⟶ 0.

Applying Corollary 3.2.8 and Theorem 3.2.9(ii) proves the corollary.

Remark 3.2.11. The case p = 2 was excluded from Corollary 3.2.8 and Theorems 3.2.9 and 3.2.10 because we did not use the best group of cyclotomic units to define our Euler system and to define C_{∞,χ}. By making use of all cyclotomic units one can prove analogues of Corollary 3.2.8 and Theorems 3.2.9 and 3.2.10 for p = 2 as well.

3.3. Elliptic Units

Let K be an imaginary quadratic field, K_{∞} a Z_{p}- or Z_{p}^{2}-extension of K in which no (finite) prime splits completely^{1}, χ : G_{K} → O^{×} a character of finite order, and T = O_{χ} as above. Using elliptic units in abelian extensions of K, exactly as with cyclotomic units in §3.2, we can define an Euler system c_{ell} for Z_{p}(1) over K, from which we get an Euler system for T^{∗}. See [Ru5] §1 and §2 for details.

Keep the notation of §3.2, except that if F is an abelian extension of K we now let C_{F,χ} denote the elliptic units in (F^{×})^{χ}. Then Theorems 2.2.2 and 2.3.3, respectively, prove the following two theorems (compare with [Ru5] Theorems 3.3 and 4.1), exactly as in §3.2.

Theorem 3.3.1. Suppose that p > 2 and that χ(𝔓) ≠ 1 for all primes 𝔓 of K above p. Then

|A_{L}^{χ}| divides [E_{L}^{χ} : C_{L,χ}].

Theorem 3.3.2. If χ(𝔓) ≠ 1 for all primes 𝔓 of K above p, then

char(\underleftarrow{lim} A_{F}^{χ}) divides char(\underleftarrow{lim} (E_{F}^{χ}/C_{F,χ})),

where the inverse limits are over finite extensions F of L in LK_{∞}.

Remarks 3.3.3. As with cyclotomic units, one can use the analytic class number formula to turn the divisibility of Theorem 3.3.1 into an equality.

One can remove the hypothesis that χ(𝔓) ≠ 1 from Theorem 3.3.2 by modifying the definition of an Euler system. See §9.1.

3.4. Stickelberger Elements

The Euler system we present in this section is not the same as the Euler system of Gauss sums introduced by Kolyvagin in [Ko2] (see also [Ru4]), but it has the same applications to ideal class groups. We use Stickelberger’s theorem in the construction of our Euler system, so Gauss sums are implicitly being used.

1In fact, this splitting condition is unnecessary; see §9.2.

Definition 3.4.1. For every integer m ≥ 2, define the Stickelberger element

θ_{m} = ∑_{a∈(Z/mZ)^{×}} (\frac{⟨a⟩}{m} − \frac{1}{2}) γ_{a}^{−1} ∈ Q[Gal(Q(μ_{m})/Q)]

where γ_{a} ∈ Gal(Q(μ_{m})/Q) is the automorphism which sends every m-th root of unity to its a-th power, and ⟨a⟩ ∈ Z is such that 0 ≤ ⟨a⟩ < m and ⟨a⟩ ≡ a (mod m). Also define θ_{1} = 0. It is well-known (and easy to check; see for example [Wa] Lemma 6.9 or [Lan] §2.8) that

if b ∈ Z is prime to 2m, then (b − γ_{b})θ_{m} ∈ Z[Gal(Q(μ_{m})/Q)] (3.7)

and if ℓ is prime, then

θ_{mℓ}|_{Q(μ_{m})} = \begin{cases}(1 − Fr_{ℓ}^{−1})θ_{m} & if ℓ ∤ m, \\ θ_{m} & if ℓ | m. \end{cases}  (3.8)

An Euler system for Z_{p}. Again we take K = Q. For every finite extension F of Q, class field theory shows that

H^{1}(F, Z_{p}) = Hom(G_{F} , Z_{p}) = Hom(A_{F}^{×}/F^{×}, Z_{p})
= Hom(A_{F}^{×}/(F^{×}B_{F}), Z_{p})  (3.9)

where A_{F}^{×} denotes the group of ideles of F and

B_{F} = ∏_{w|∞} F_{w}^{×} × ∏_{w|p}{1} × ∏_{w∤p∞} O_{F,w}^{×} ⊂ A_{F}^{×},

since every (continuous) homomorphism from A_{F}^{×} into Z_{p} must vanish on B_{F}. Further, the map which sends an idele to the corresponding ideal class induces an exact sequence

0 ⟶ U_{F} /Ē_{F} ⟶ A_{F}^{×}/(F^{×}B_{F}) ⟶ A_{F} ⟶ 0  (3.10)

where U_{F} denotes the local units of F ⊗ Q_{p} (i.e., U_{F} = ⊕_{v|p}O_{F,v}^{×}) and Ē_{F} denotes the closure of the global units of F in U_{F}, and A_{F} is the ideal class group of F. We will write Z_{p}[μ_{m}]^{×} = U_{Q(μ_{m})}.

Definition 3.4.2. Fix an integer b prime to 2p (a precise choice will be made later), and for every m ∈ Z^{+} prime to b we use the Stickelberger elements θ_{m} to define

θ̄_{m}^{(b)} = \begin{cases}(b − γ_{b})θ_{m} & if p | m \\ (b − γ_{b})(1 − Fr_{p}^{−1})θ_{m} & if p ∤ m \end{cases} ∈ Z[Gal(Q(μ_{m})/Q)]

(the two separate cases are to ensure, using (3.8), that θ̄_{mp}^{(b)}|_{Q(μ_{m})} = θ̄_{m}^{(b)} for every m). Stickelberger’s theorem (see for example [Wa] Theorem 6.10 or [Lan] Theorem 1.2.3) shows that θ̄_{m}^{(b)}A_{Q(μ_{m})} = 0. Thus, using (3.10) we can view multiplication by θ̄_{m}^{(b)} as a map

A_{Q(μ_{m})}^{×}/(Q(μ_{m})^{×}B_{Q(μ_{m})}) ⟶ Z_{p}[μ_{m}]^{×}/Ē_{Q(μ_{m})},

and we define ϕ_{m} = ϕ_{m}^{(b)} ∈ Hom(A_{Q(μ_{m})}^{×}/(Q(μ_{m})^{×}B_{Q(μ_{m})}), Z_{p}) to be the composition

A_{Q(μ_{m})}^{×}/(Q(μ_{m})^{×}B_{Q(μ_{m})}) ⟶^{θ̄_{m}^{(b)}} Z_{p}[μ_{m}]^{×}/Ē_{Q(μ_{m})}

⟶^{1−c} Z_{p}[μ_{m}]^{×}/(Z_{p}[μ_{m}]^{×})_{tors} ⟶^{λ_{m}} Z_{p}

where c denotes complex conjugation in Gal(Q(μ_{m})/Q), so (1−c)Ē_{Q(μ_{m})} is finite, and λ_{m} is the map defined in Appendix D, Definition D.1.2. Finally, we define c̃′_{m} ∈ H^{1}(Q(μ_{m}), Z_{p}) to be the element corresponding to ϕ_{m} under (3.9).

Proposition 3.4.3. Suppose m is prime to b and ℓ is a prime not dividing b. Then

Cor_{Q(μ_{mℓ})/Q(μ_{m})}(c̃′_{mℓ}) = \begin{cases}(1 − Fr_{ℓ}^{−1})c̃′_{m} & if ℓ ∤ mp, \\ c̃′_{m} & if ℓ | mp. \end{cases}

Proof. It follows from a standard result of class field theory (see for example [T2] §11(13)) that, with the identification (3.9), the map Cor_{Q(μ_{mℓ})/Q(μ_{m})} is induced by the inclusion A_{Q(μ_{m})}^{×} ↪ A_{Q(μ_{mℓ})}^{×}.

Suppose first that ℓ ∤ mp. By Lemma D.1.4 the restriction of λ_{mℓ} to Z_{p}[μ_{m}]^{×} is λ_{m} ∘ (−Fr_{ℓ}), and by (3.8) we have θ̄_{mℓ}^{(b)}|_{Q(μ_{m})} = (1 − Fr_{ℓ}^{−1})θ̄_{m}^{(b)}. Therefore

ϕ_{mℓ}|_{A_{Q(μ_{m})}^{×}} = ϕ_{m} ∘ (−Fr_{ℓ})(1 − Fr_{ℓ}^{−1}) = ϕ_{m} ∘ (1 − Fr_{ℓ}) = (1 − Fr_{ℓ}^{−1})ϕ_{m}

and hence Cor_{Q(μ_{mℓ})/Q(μ_{m})}(c̃′_{mℓ}) = (1−Fr_{ℓ}^{−1})c̃′_{m}. Similarly (but more simply), if ℓ divides mp then Lemma D.1.4 and (3.8) show that ϕ_{mℓ}|_{A_{Q(μ_{m})}^{×}} = ϕ_{m} and then Cor_{Q(μ_{mℓ})/Q(μ_{m})}(c̃′_{mℓ}) = c̃′_{m}.

Remark 3.4.4. Technically we should write c̃′_{m∞} instead of c̃′_{m}, since the ray class field of Q modulo m is the real subfield Q(μ_{m})^{+}. But

Cor_{Q(μ_{m})/Q(μ_{m})^{+}}(c̃′_{m}) = 0 ∈ H^{1}(Q(μ_{m})^{+}, Z_{p})

(because we annihilated all even components in our definition), so we will never need to deal with those classes and there should be no confusion.

For every prime ℓ ≠ p we have

det(1 − Fr_{ℓ}^{−1}x|Z_{p}(1)) = 1 − ℓ^{−1}x.

But Proposition 3.4.3 shows that the collection {c̃′_{m} ∈ H^{1}(Q(μ_{m}), Z_{p})} satisfies a distribution relation with Frobenius polynomials 1 − Fr_{ℓ}^{−1}, not 1 − ℓ^{−1}Fr_{ℓ}^{−1}, so this collection is not an Euler system for the trivial representation Z_{p}. However, since

1 − ℓ^{−1}x ≡ 1 − x (mod (ℓ − 1)Z_{p}[x])

we can modify the classes c̃′_{m} (see Lemma 9.6.1 and Example 9.6.2) to produce a new collection

{c̃_{m} ∈ H^{1}(Q(μ_{m}), Z_{p}) : m ∈ Z^{+} and (m, b) = 1}

which is an Euler system for (Z_{p}, Q^{ab,b}, bp), where Q^{ab,b} denotes the maximal abelian extension of Q unramified outside b. Further (Lemma 9.6.1(ii)), we have c̃_{p^{n}} = c̃′_{p^{n}} for every n.

Note that this Euler system depends on the choice of b.

The setting. As in §3.2 let K = Q, let T = O_{χ} for a character χ of G_{K} of finite prime-to-p order, and keep the rest of the notation of the beginning of §3.2 as well.

For the rest of this section we assume that χ is odd, and we let b be a nonzero integer prime to 2p and to the conductor f of χ. (A precise choice of b will be made later.) Note that these hypotheses imply that p > 2, because there are no odd characters of odd order.

Let Δ = Gal(Q(μ_{f})/Q). Since χ is nontrivial and of order prime to p, we have H^{i}(Δ, O_{χ}) = 0 for every i ≥ 0. Therefore the restriction map gives an isomorphism (compare with (3.9))

H^{1}(Q_{n}, T) = H^{1}(Q_{n}(μ_{f}), O_{χ})^{Δ}

≅ Hom(A_{Q_{n}(μ_{f})}^{×}/Q_{n}(μ_{f})^{×}, O_{χ})^{Δ} ⊂ Hom(A_{Q_{n}(μ_{f})}^{×}, O), (3.11)

the inclusion using our fixed generator of O_{χ}. The Euler system c̃ for Z_{p} constructed above gives rise (by Proposition 2.4.2) to an Euler system c = c̃^{χ} for (T, Q^{ab,b}, bfp). By Lemmas 2.4.3 and 9.6.1(iii), the image under (3.11) of c_{Q} in Hom(A_{Q(μ_{f})}^{×}, O) is

∑_{δ∈Δ} χ(δ)δc̃_{f} = ∑_{δ∈Δ} χ(δ)δc̃′_{f} = ∑_{δ∈Δ} χ(δ)ϕ_{f}^{δ}.  (3.12)

The Selmer group. We have W^{∗} = D_{χ^{−1}ε_{cyc}}. As in §3.2, let L be the fixed field of the kernel of χ, let L_{n} = LQ_{n}, let Q_{n,p} be the completion of Q_{n} above p, let A_{n} be the ideal class group of L_{n}, and let A_{L} = A_{0}, the ideal class group of L.

We take H_{f}^{1}(Q_{n,p}, V) and H_{f}^{1}(Q_{n,p}, V^{∗}) to be as defined in the examples of §1.6.B and §1.6.C, respectively.

Proposition 3.4.5. (i) S(Q, W^{∗}) ≅ A_{L}^{χ},

(ii) S(Q_{∞}, W^{∗}) ≅ \underrightarrow{lim}_{n} A_{n}^{χ}.

Proof. Let E_{n} denote the group of global units of L_{n}. Since χ is odd, E_{n}^{χ} is finite, so (E_{n} ⊗Q_{p}/Z_{p})^{χ} = 0. Now the proposition follows from Proposition 1.6.4(ii).

The minus part of the ideal class group of L. The following theorem (or more precisely, its Corollary 3.4.7) was first proved by Mazur and Wiles in [MW]. A proof using Euler systems, but somewhat different from the one here, was given by Kolyvagin in [Ko2], see also [Ru4].

Define the generalized Bernouilli number

B_{1,χ^{−1}} = \frac{1}{f} ∑_{a=1}^{f} χ^{−1}(a)a = χ(θ_{f}).

Recall that ω : G_{Q} → (Z_{p}^{×})_{tors} is the Teichmüller character giving the action of G_{Q} on μ_{p} (recall also that p ≠ 2, since we have assumed that χ is an odd character of order prime to p).

Theorem 3.4.6. Suppose that χ is an odd character of order prime to p, that χ(p) ≠ 1, and that χ^{−1}ω(p) ≠ 1. Then

|A_{L}^{χ}| ≤ |O/B_{1,χ^{−1}}O|.

Proof. Since χ ≠ ω, we can choose b prime to 2pf so that b−χ(b) ∈ O^{×}. Let c be the Euler system for T constructed above from Stickelberger elements, with this choice of b.

Since T has rank one over O, Hyp(Q, T) is satisfied with τ = 1, so we can apply Theorem 2.2.10 with this Euler system.

As in the proof of Theorem 3.2.3, since χ is odd and different from ω, Lemma 3.1.1 shows that 𝔫_{W} = 𝔫_{W}^{∗} = 0 in Theorem 2.2.2.

Using the definition of H_{f}^{1} in §1.6.B and local class field theory, we have identifications (the top row is the local analogue of (3.11))

\xymatrix{H^{1}(Q_{p}, T) \ar[r]^{∼} \ar@{->>}[d] & Hom(⊕_{w|p}G_{Q(μ_{f})_{w}}, O_{χ})^{Δ} \ar[r]^{∼} \ar@{->>}[d] & Hom(Q_{p}(μ_{f})^{×}, O_{χ})^{Δ} \ar@{->>}[d] \\ H_{s}^{1}(Q_{p}, T) \ar[r]^{∼} & Hom(⊕_{w|p}I_{w}, O_{χ})^{Δ} \ar[r]^{∼} & Hom(Z_{p}[μ_{f}]^{×}, O_{χ})^{Δ}}

where Q_{p}(μ_{f}) = Q(μ_{f}) ⊗ Q_{p} and I_{w} is the inertia group in G_{L_{w}}. Thus

H_{s}^{1}(Q_{p}, T) ≅ Hom(Z_{p}[μ_{f}]^{×}, O_{χ})^{Δ} ≅ Hom(Z_{p}[μ_{f}]^{×}, O)^{χ^{−1}}. (3.13)

With this identification, using (3.12) and Definition 3.4.2 of θ̄_{f}^{(b)}, and writing c for complex conjugation,

loc_{{p},T}^{s} (c_{Q}) = ∑_{δ∈Δ} χ(δ)(λ_{f} ∘ (1 − c)θ̄_{f}^{(b)})^{δ}

= ∑_{δ∈Δ} χ(δ)(λ_{f} ∘ δ^{−1}(1 − c)θ̄_{f}^{(b)})

= λ_{f} ∘ ∑_{δ∈Δ}(χ(δ)δ^{−1})(1 − c)θ̄_{f}^{(b)}

= 2(b − χ(b))(1 − χ^{−1}(p))B_{1,χ^{−1}} ∑_{δ∈Δ} χ(δ)λ_{f}^{δ}.

Since χ^{−1}ω(p) ≠ 1, Lemma D.1.5 shows that ∑_{δ∈Δ} χ(δ)λ_{f}^{δ} generates the (free rank-one) O-module Hom(Z_{p}[μ_{f}]^{×}, O)^{χ^{−1}}. We chose b so that b − χ(b) ∈ O^{×}, and we assumed that χ(p) ≠ 1 and χ has order prime to p, so 1 − χ(p) ∈ O^{×}. Thus (3.13) shows that

Oloc_{{p},T}^{s} (c_{Q}) = B_{1,χ^{−1}}H_{s}^{1}(Q_{p}, T).

Now Theorem 2.2.10 yields

|S(Q, W^{∗})| ≤ [H_{s}^{1}(Q_{p}, T) : B_{1,χ^{−1}}H_{s}^{1}(Q_{p}, T)] = |O/B_{1,χ^{−1}}O|.

Corollary 3.4.7 (Mazur & Wiles [MW] Theorem 1.10.2). With hypotheses as in Theorem 3.4.6,

|A_{L}^{χ}| = |O/B_{1,χ^{−1}}O|.

Proof. As in Corollary 3.2.4, this follows from Theorem 3.4.6 and the analytic class number formula. See for example [Ru4] Theorem 4.3.

Remarks 3.4.8. If χ = ω, then A_{L}^{χ} = 0 and B_{1,χ^{−1}}O = p^{−1}O.

If χ(p) = 1, or χ^{−1}ω(p) = 1 but χ ≠ ω, the equality of Corollary 3.4.7 can be deduced from Theorem 3.4.13 below (Iwasawa’s “main conjecture”). See [MW], §1.10 Theorem 2. See also §9.1.

The p-adic L-function. There is a natural map

χ_{Λ} : O[[Gal(Q_{∞}(μ_{f})/Q)]] = O[Δ][[Gal(Q_{∞}/Q)]] ⟶^{χ} Λ

given by χ on Δ and the identity on Gal(Q_{∞}/Q). Let

⟨ε⟩ = ω^{−1}ε_{cyc} : Gal(Q_{∞}/Q) ⟶ 1 + pZ_{p},

let Tw_{⟨ε⟩} : Λ → Λ be the twisting map induced by

γ ↦ ⟨ε⟩(γ)γ

for γ ∈ Gal(Q_{∞}/Q), and let η ↦ η^{•} denote the involution of Λ induced by γ ↦ γ^{−1} for γ ∈ Gal(Q_{∞}/Q).

Write θ_{fp^{∞}} = {θ_{fp^{n+1}}}_{n}. If b is prime to 2fp then by (3.7) and (3.8),

(b − γ_{b})θ_{fp^{∞}} ∈ Z_{p}[[Gal(Q(μ_{fp^{∞}})/Q)]],

and so by restriction we have χ_{Λ}((b − γ_{b})θ_{fp^{∞}}) ∈ Λ. If χ ≠ ω then we can fix b so that b − χ(b) ∈ O^{×}, and then χ_{Λ}(b − γ_{b}) ∈ Λ^{×}. We will write

χ_{Λ}(θ_{fp^{∞}}) = χ_{Λ}(b − γ_{b})^{−1}χ_{Λ}((b − γ_{b})θ_{fp^{∞}}) ∈ Λ

which is independent of b.

Theorem 3.4.9. If χ is odd and χ ≠ ω, then

χ_{Λ}(θ_{fp^{∞}})^{•} = Tw_{⟨ε⟩}(L_{χ^{−1}ω})

where L_{χ^{−1}ω} is the p-adic L-function defined in Theorem 3.2.9 for the even character χ^{−1}ω.

Proof. This was proved by Iwasawa; see [Iw2] §6 or [Wa] Theorem 7.10. If ρ is a character of finite order of Gal(Q_{∞}/Q), it follows from the definitions that

ρ(χ_{Λ}(θ_{fp^{∞}})^{•}) = ρ^{−1}(χ_{Λ}(θ_{fp^{∞}})) = (1 − χ^{−1}ρ(p))B_{1,χ^{−1}ρ} = (1 − χ^{−1}ρ(p))L(0, χ^{−1}ρ) = ⟨ε⟩ρ(L_{χ^{−1}ω}) = ρ(Tw_{⟨ε⟩}(L_{χ^{−1}ω})).

Since this is true for every ρ, the equality of the theorem holds.

Direct limit of the ideal class groups. The main result of this section, Theorem 3.4.13 below, is equivalent to Theorem 3.2.10 by standard methods of Iwasawa theory (see for example [Ru3] §8), so we will only sketch the proof.

Let U denote the direct limit (not the inverse limit) of the local units of Q_{n}(μ_{f}) ⊗ Q_{p}. Recall that Δ = Gal(Q(μ_{f})/Q) ≅ Gal(Q_{n}(μ_{f})/Q_{n}).

Lemma 3.4.10. There is an isomorphism of Λ-modules

Hom(U, O_{χ})^{Δ} ≅ \begin{cases}Λ & if χ(p) ≠ 1, \\ Λ ⊕ O & if χ(p) = 1. \end{cases}

Sketch of proof. Let Y_{∞} denote the inverse limit of the p-adic completions of the multiplicative groups (Q(μ_{fp^{n}}) ⊗ Q_{p})^{×}. There is a natural Kummer pairing

U × Y_{∞} ⟶ Z_{p}(1)

which leads to a Λ-module isomorphism

(Y_{∞} ⊗ O_{χω^{−1}})^{Gal(Q(μ_{fp})/Q)} ≅ Hom(U, O_{χ})^{Δ} ⊗ O_{⟨ε⟩}.

The lemma then follows from a result of Iwasawa ([Iw3] Theorem 25; see also [Gi] Proposition 1).

Corollary 3.4.11. Suppose that χ is odd and χ ≠ ω. Then we can choose b so that, if c is the Euler system defined above, then

char(H_{∞,s}^{1}(Q_{p}, T)/Λloc_{{p}}^{s}({c_{Q_{n}}}_{n}))

= \begin{cases}Tw_{⟨ε⟩}(L_{χ^{−1}ω}) & if χ^{−1}ω(p) ≠ 1, χ(p) ≠ 1, \\ Tw_{⟨ε⟩}(J L_{χ^{−1}ω}) & if χ^{−1}ω(p) = 1, \\ J Tw_{⟨ε⟩}(L_{χ^{−1}ω}) & if χ(p) = 1, \end{cases}

where J is the augmentation ideal of Λ.

Sketch of proof. For every n, exactly as in (3.13) we have

H_{s}^{1}(Q_{n,p}, T) = Hom(U_{n}, O_{χ})^{Δ}

and so H_{∞,s}^{1}(Q_{p}, T) = Hom(U, O_{χ})^{Δ}. Let

λ_{fp^{∞},χ} = lim_{n→∞} ∑_{δ∈Δ} χ(δ)λ_{fp^{n}}^{δ} ∈ Hom(U, O_{χ})^{Δ}.

One computes, using Lemma 3.4.10, that there are Λ-module isomorphisms

Hom(U, O_{χ})^{Δ}/Λλ_{fp^{∞},χ} ≅ \begin{cases}0 & if χ^{−1}ω(p) ≠ 1, χ(p) ≠ 1, \\ O_{⟨ε⟩} & if χ^{−1}ω(p) = 1, \\ O & if χ(p) = 1. \end{cases}

(The first case follows from Lemma D.1.5; the others require more work.) Also, by the definition of c_{Q_{n}} and Lemma 2.4.3 we have

loc_{{p}}^{s}(c_{Q_{n}}) = ∑_{δ∈Δ} χ(δ)λ_{fp^{n+1}} ∘ δ^{−1}(1 − c)(b − γ_{b})θ_{fp^{n+1}}.

Thus

Λloc_{{p}}^{s}({c_{Q_{n}}}_{n}) = Λλ_{fp^{∞},χ} ∘ 2(χ_{Λ}(b − γ_{b})χ_{Λ}(θ_{fp^{∞}}))
= χ_{Λ}(b − γ_{b})^{•}χ_{Λ}(θ_{fp^{∞}})^{•}Λλ_{fp^{∞},χ}
= χ_{Λ}(θ_{fp^{∞}})^{•}Λλ_{fp^{∞},χ}

since b was chosen so that χ_{Λ}(b−γ_{b}) ∈ Λ^{×}. The corollary now follows from Theorem 3.4.9.

Theorem 3.4.12. If χ is an odd character of order prime to p and χ ≠ ω, then

char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) divides Tw_{⟨ε⟩}(L_{χ^{−1}ω}).

Sketch of proof. Let c be the Euler system constructed above, with b chosen to satisfy Corollary 3.4.11. Since T has rank one over O, we see that Hyp(Q, T) is satisfied with τ = 1. Thus we can apply Theorem 2.3.8(ii), and we conclude (using Proposition 3.4.5(ii) to identify the Selmer group with the direct limit of the ideal class groups) that

char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) divides char(H_{∞,s}^{1}(Q_{p}, T)/Λloc_{\{p\}}^{s}(\{c_{Q_{n}}\}_{n})).

If χ(p) ≠ 1 and χ^{−1}ω(p) ≠ 1, the theorem now follows immediately from Corollary 3.4.11.

The two exceptional cases remain. First suppose that χ^{−1}ω(p) = 1. In this case we conclude from Corollary 3.4.11 that char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) divides Tw_{⟨ε⟩}(J L_{χ^{−1}ω}), so to complete the proof it will suffice to show that Tw_{⟨ε⟩}(J) cannot divide char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)).

Briefly, if Tw_{⟨ε⟩}(J) divides char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) then class field theory and Kummer theory show (see for example [Lan] Chapter 6 or [Wa] §13.5) that there is a divisible subgroup of Q(μ_{fp})^{×} ⊗(Q_{p}/Z_{p}) which generates an unramified extension of Q(μ_{fp^{∞}}). But this would contradict Leopoldt’s conjecture, which holds for Q(μ_{fp}).

Now suppose χ(p) = 1. In this case, if χ_{0} denotes the trivial character then the definition (Theorem 3.2.9) of L_{χ^{−1}ω} shows that

χ_{0}(Tw_{⟨ε⟩}(L_{χ^{−1}ω})) = ω^{−1}ε_{cyc}(L_{χ^{−1}ω}) = (1 − χ(p))L(0, χ) = 0.

In other words, J divides Tw_{⟨ε⟩}(L_{χ^{−1}ω}) so we cannot hope to show in this case that char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) is not divisible by J. Instead, one must “improve” the Euler system c to remove this extra zero. We omit the details.

Theorem 3.4.13 (Mazur & Wiles [MW]). If χ is an odd character of order prime to p and χ ≠ ω then

char(Hom_{O}(\underrightarrow{lim} A_{n}^{χ}, D)) = Tw_{⟨ε⟩}(L_{χ^{−1}ω}).

Proof. This follows from Theorem 3.4.12 by the usual analytic class number argument. See [MW] §1.6, where this equality is deduced from divisibilities opposite to those of Theorem 3.4.12.

3.5. Elliptic Curves

The “Heegner point Euler system” for elliptic curves, used by Kolyvagin in [Ko2], does not fit precisely into the framework we have established. We will discuss in §9.4 how to adapt Definition 2.1.1 to include the system of Heegner points. However, Kato ([Ka3], [Scho]) has constructed an Euler system for the Tate module of a modular^{2} elliptic curve, using Beilinson elements in the K-theory of modular curves. In this section we describe applications of Kato’s Euler system.

2See the footnote on page 3.

The setting. Suppose E is an elliptic curve defined over Q, and take K = Q, K_{∞} = Q_{∞}, O = Z_{p}, and T = T_{p}(E), the p-adic Tate module of E, as in Example 1.1.5. Then V = V_{p}(E) = T_{p}(E) ⊗ Q_{p} and W = E_{p^{∞}}. The Weil pairing gives isomorphisms V ≅ V^{∗}, T ≅ T^{∗}, and W ≅ W^{∗}. As in the previous sections, Q_{n} is the subfield of Q_{∞} with [Q_{n} : Q] = p^{n}, and Q_{n,p} is the completion of Q_{n} at the unique prime above p.

The p-adic cohomology groups. As in §1.6, for every n we let

H_{f}^{1}(Q_{n,p}, V) = image(E(Q_{n,p}) ⊗ Q_{p} ↪ H^{1}(Q_{n,p}, V)).

Since V = V^{∗}, this also fixes a choice of H_{f}^{1}(Q_{n,p}, V^{∗}) ⊂ H^{1}(Q_{n,p}, V^{∗}), and then H_{f}^{1}(Q_{n,p}, V) and H_{f}^{1}(Q_{n,p}, V^{∗}) are orthogonal complements under the local pairing of §1.4.

For every n let Tan(E_{/Q_{n,p}}) denote the tangent space of E_{/Q_{n,p}} at the origin and consider the Lie group exponential map

exp_{E} : Tan(E_{/Q_{n,p}}) ⟶^{∼} E(Q_{n,p}) ⊗ Q_{p}.

Fix a minimal Weierstrass model of E and let ω_{E} denote the corresponding holomorphic differential. Then the cotangent space Cotan(E_{/Q_{n,p}}) is Q_{n,p}ω_{E}, and we let ω_{E}^{∗} be the corresponding dual basis of Tan(E). Let E_{1}(Q_{n,p}) be the kernel of reduction in E(Q_{p}), let 𝔭_{n} be the maximal ideal of Q_{n,p}, and let Ê be the formal group of E. We have a commutative diagram in which all the maps are isomorphisms

\xymatrix{Tan(E_{/Q_{n,p}}) \ar[rr]^{^{exp^{E}}} &  & E(Q_{n,p}) ⊗ Q_{p} \\ Q_{n,p} \ar[u]_{·ω_{E}^{∗}} & Ê(𝔭_{n}) ⊗ Q_{p} \ar[l]_{λ_{E}} \ar[r]^{∼} & E_{1}(Q_{n,p}) ⊗ Q_{p} \ar[u]}

where λ_{E} is the formal group logarithm, and the bottom right isomorphism is defined in [T3] Theorem 4.2. Using these identifications we will also view λ_{E} as a homomorphism from E(Q_{n,p}) to Q_{n,p}.

Since V ≅ V^{∗}, the local Tate pairing gives the second isomorphism in

Hom(E(Q_{n,p}), Q_{p}) ≅ Hom(H_{f}^{1}(Q_{n,p}, V), Q_{p}) ≅ H_{s}^{1}(Q_{n,p}, V).

Thus there is a dual exponential map (see [Ka1] §II.1.2)

exp_{E}^{∗} : H_{s}^{1}(Q_{n,p}, V) ⟶^{∼} Cotan(E_{/Q_{n,p}}) = Q_{n,p}ω_{E}.

Write exp_{ω_{E}}^{∗} : H_{s}^{1}(Q_{n,p}, V) ⟶^{∼} Q_{n,p} for the composition ω_{E}^{∗} ∘ exp_{E}^{∗}. Since H_{s}^{1}(Q_{n,p}, T) injects into H_{s}^{1}(Q_{n,p}, V), we see that exp_{ω_{E}}^{∗} is injective on H_{s}^{1}(Q_{n,p}, T). The local pairing allows us to identify

H_{s}^{1}(Q_{n,p}, V) ⟶^{∼} Hom(E(Q_{n,p}), Q_{p})
↑  ↑
H_{s}^{1}(Q_{n,p}, T) ⟶^{∼} Hom(E(Q_{n,p}), Z_{p}).

(3.14)

Explicitly (see [Ka1] Theorem II.1.4.1(iv)), z ∈ H_{s}^{1}(Q_{n,p}, V) is identified with the map

x ↦ Tr_{Q_{n,p}/Q_{p}}λ_{E}(x) exp_{ω_{E}}^{∗}(z).  (3.15)

Proposition 3.5.1. With notation as above, if p > 2 then

exp_{ω_{E}}^{∗}(H_{s}^{1}(Q_{p}, T)) = [E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}]p^{−1}Z_{p}.

Proof. The diagram (3.14) shows that an element of H_{s}^{1}(Q_{p}, V) belongs to H_{s}^{1}(Q_{p}, T) if and only if its image in Hom(E(Q_{p}), Q_{p}) takes E(Q_{p}) into Z_{p}. Thus by (3.15), we have

exp_{ω_{E}}^{∗}(H_{s}^{1}(Q_{p}, T)) = p^{a}Z_{p}

where λ_{E}(E(Q_{p})) = p^{−a}Z_{p}. If p > 2 then λ_{E}(E_{1}(Q_{p})) = pZ_{p} and, since rank_{Z_{p}}E(Q_{p}) = 1, we see that

[λ_{E}(E(Q_{p})) : λ_{E}(E_{1}(Q_{p})] = [E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}].

The L-functions.

Definition 3.5.2. Let

L(E, s) = ∑_{n=1}^{∞} a_{n}n^{−s} = ∏_{q} ℓ_{q}(q^{−s})^{−1}

denote the Hasse-Weil L-function of E, where ℓ_{q}(q^{−s}) is the usual Euler factor at q. If m ∈ Z^{+} we will also write

L_{m}(E, s) = ∑_{(n,m)=1} a_{n}n^{−s} = ∏_{q∤m} ℓ_{q}(q^{−s})^{−1} = (∏_{q|m} ℓ_{q}(q^{−s}))L(E, s)

for the L-function with the Euler factors dividing m removed. If χ is a character of G_{Q} of conductor f_{χ}, let

L_{m}(E, χ, s) = ∑_{(n,f_{χ}m)=1} χ(n)a_{n}n^{−s} = ∏_{q∤f_{χ}m} ℓ_{q}(q^{−s}χ(q))^{−1}.

When m = 1 we write simply L(E, χ, s), and then we have

L_{m}(E, χ, s) = (∏_{q|m} ℓ_{q}(q^{−s}χ(q)))L(E, χ, s).  (3.16)

If E is modular then these functions all have analytic continuations to C.

The Euler system. Kato has constructed an Euler system in this setting. Let N denote the conductor of E, and let Ω_{E} be the fundamental real period of E (which corresponds to our choice of differential ω_{E}).

Theorem 3.5.3 (Kato [Ka3]; see also [Scho]). Suppose that E is modular. Then there is a positive integer r_{E}, independent of p, and an Euler system c for T_{p}(E) such that for every n ≥ 0 and every character χ of Gal(Q_{n}/Q),

∑_{γ∈Gal(Q_{n}/Q)} χ(γ) exp_{ω_{E}}^{∗}(loc_{{p}}^{s}(c_{Q_{n}}^{γ})) = r_{E}L_{Np}(E, χ, 1)/Ω_{E}.

In particular we have

exp_{ω_{E}}^{∗}(loc_{{p}}^{s}(c_{Q})) = r_{E}L_{Np}(E, 1)/Ω_{E}.

See [Scho], especially §5, for the construction of the Euler system and the proof of the identities in the case where E has good reduction at p. (See also [Ru9] Corollary 7.2 to get from [Scho] Theorem 5.2.6 to the statement above.)

Consequences of Kato’s Euler system. Following Kato, we will apply the results of Chapter 2 to bound the Selmer group of E. Let � (E) be the Tate-Shafarevich group of E.

Theorem 3.5.4 (Kato [Ka3]). Suppose E is modular and E does not have complex multiplication.

(i) If L(E, 1) ≠ 0 then E(Q) and � (E) are finite.

(ii) Suppose L is a finite abelian extension of Q and χ is a character of Gal(L/Q). If L(E, χ, 1) ≠ 0 then E(L)^{χ} and � (E_{/L})^{χ} are finite.

Remarks 3.5.5. We will prove a more precise version of Theorem 3.5.4(i) in Theorem 3.5.11 below. Kato actually constructs an Euler system for (T_{p}(E), Q^{ab,DD′}, NpDD′) for appropriate auxiliary integers D, D′, where Q^{ab,DD′} is the maximal abelian extension of Q unramified outside DD′. Thus (for some choice of D and D′, depending on χ) Proposition 2.4.2 gives an Euler system for T_{p}(E) ⊗ χ for every character χ of G_{Q} of finite order, with properties analogous to those of Theorem 3.5.3. These twisted Euler systems are needed to prove Theorem 3.5.4(ii). For simplicity we will not treat this more general setting here, so we will only prove Theorem 3.5.4(i) below. But the method for (ii) is the same.

Theorem 3.5.4(i) was first proved by Kolyvagin in [Ko2], using a system of Heegner points, along with work of Gross and Zagier [GZ], Bump, Friedberg, and Hoffstein [BFH], and Murty and Murty [MM]. The Euler system proof given here, due to Kato, is self-contained in the sense that it replaces [GZ], [BFH], and [MM] by the calculation of Theorem 3.5.3.

Corollary 3.5.6. Suppose E is modular and E does not have complex multiplication. Then E(Q_{∞}) is finitely generated.

Proof. A theorem of Rohrlich [Ro] shows that L(E, χ, 1) ≠ 0 for almost all characters χ of finite order of Gal(Q_{∞}/Q). A result of Serre ([Se4] Théorème 3) shows that E(Q_{∞})_{tors} is finite, and the corollary follows without difficulty from Theorem 3.5.4(ii). (See for example [RW], pp. 242–243.)

Remark 3.5.7. When E has complex multiplication, the representation T_{p}(E) does not satisfy hypothesis Hyp(Q, V)(i) (see Remark 3.5.10 below), so we cannot apply the results of §2.2 and §2.3 with Kato’s Euler system. However, Theorem 3.5.4 and Corollary 3.5.6 are known in that case, as Theorem 3.5.4 for CM curves can be proved using the elliptic unit Euler system of §3.3. See [CW], [Ru5] §11, and [RW]. See also the final example of §6.5.

Verification of the hypotheses. Fix a Z_{p}-basis of T and let

ρ_{E,p} : G_{Q} ⟶ Aut(T) ⟶^{∼} GL_{2}(Z_{p})

be the p-adic representation of G_{Q} attached to E with respect to this basis.

Proposition 3.5.8. (i) Suppose that E does not have complex multiplication. Then H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) is finite and T_{p}(E) satisfies hypotheses Hyp(Q_{∞}, V).

(ii) Suppose that ρ_{E,p} is surjective. Then H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) = 0 and T_{p}(E) satisfies hypotheses Hyp(Q_{∞}, T).

Proof. The Weil pairing shows that

G_{Q(μ_{p^{∞}})} = ρ_{E,p}^{−1}(SL_{2}(Z_{p})).

If E does not have complex multiplication then Serre’s theorem ([Se4] Théorème 3) says that the image of ρ_{E,p} is open in GL_{2}(Z_{p}). It follows that V_{p}(E) is an irreducible G_{Q_{∞}}-representation, and if ρ_{E,p} is surjective then E_{p} is an irreducible F_{p}[G_{Q_{∞}}]-representation.

It also follows that we can find τ ∈ G_{Q(μ_{p^{∞}})} such that

ρ_{E,p}(τ) = (\begin{matrix}1 & x \\ 0 & 1 \end{matrix})

with x ≠ 0, and such a τ satisfies hypothesis Hyp(Q_{∞}, V)(i). If ρ_{E,p} is surjective we can take x = 1, and then τ satisfies hypothesis Hyp(Q_{∞}, T)(i).

We have

H^{1}(Q(E_{p^{∞}})/Q, E_{p^{∞}}) = H^{1}(ρ_{E,p}(G_{Q}), (Q_{p}/Z_{p})^{2})

which is zero if ρ_{E,p}(G_{Q}) = GL_{2}(Z_{p}), and finite if ρ_{E,p}(G_{Q}) is open in GL_{2}(Z_{p}) (see also Corollary C.2.2). This completes the proof of the proposition.

Remark 3.5.9. Serre’s theorem (see [Se4] Corollaire 1 of Théorème 3) also shows that if E does not have complex multiplication then ρ_{E,p} is surjective for all but finitely many p.

Remark 3.5.10. The conditions on τ in hypothesis Hyp(Q, V)(i) force ρ_{E,p}(τ) to be nontrivial and unipotent. Thus if E has complex multiplication then there is no τ satisfying Hyp(Q, V)(i).

Bounding S(Q, E_{p^{∞}}). Recall that N is the conductor of E.

Theorem 3.5.11. Suppose E is modular, E does not have complex multiplication, and L(E, 1) ≠ 0.

(i) E(Q) and � (E)_{p^{∞}} are finite.

(ii) Suppose in addition that E has good reduction at p, that p does not divide 2r_{E}|Ẽ(F_{p})| (where Ẽ is the reduction of E modulo p and r_{E} is as in Theorem 3.5.3), and that ρ_{E,p} is surjective. Then

|� (E)_{p^{∞}}| divides \frac{L_{N}(E, 1)}{Ω_{E}}.

Proof. Recall that ℓ_{q}(q^{−s}) is the Euler factor of L(E, s) at q, and that by Proposition 1.6.8, S(Q, E_{p^{∞}}) is the usual p-power Selmer group of E.

Since L(E, 1) ≠ 0, and ℓ_{q}(q^{−1}) is easily seen to be nonzero for every q, Theorem 3.5.3 shows that loc_{{p}}^{s}(c_{Q}) ≠ 0. By Proposition 3.5.8(i) and (3.14) we can apply Theorem 2.2.10(i) to conclude that S(Q, E_{p^{∞}}) is finite. This proves (i), and it follows (see for example Proposition 1.6.8) that S(Q, E_{p^{∞}}) = � (E)_{p^{∞}}.

If E has good reduction at p then pℓ_{p}(p^{−1}) = |Ẽ(F_{p})| and

[E(Q_{p}) : E_{1}(Q_{p}) + E(Q_{p})_{tors}] divides |Ẽ(F_{p})|.

Therefore if p ∤ 2r_{E}|Ẽ(F_{p})| then

exp_{ω_{E}}^{∗}(H_{s}^{1}(Q_{p}, T_{p}(E))) =  p^{−1}Z_{p}
∪  ∪
exp_{ω_{E}}^{∗}(Z_{p}loc_{{p}}^{s}(c_{Q})) = p^{−1}(L_{N}(E, 1)/Ω_{E})Z_{p}

by Proposition 3.5.1 and Theorem 3.5.3. By Proposition 3.5.8(ii), if further ρ_{E,p} is surjective then we can apply Theorem 2.2.10(ii) (with 𝔫_{W} = 0 and 𝔫_{W}^{∗} = 0) and (ii) follows.

Remarks 3.5.12. In Corollary 3.5.19 below, we will prove using Iwasawa theory that Theorem 3.5.11(ii) holds for almost all p, even when p divides |Ẽ(F_{p})|. This is needed to prove Theorem 3.5.4(i), since |Ẽ(F_{p})| could be divisible by p for infinitely many p. However, since |Ẽ(F_{p})| < 2p for all primes p > 5, we see that if E(Q)_{tors} ≠ 0 then |Ẽ(F_{p})| is prime to p for almost all p. Thus for curves E with nontrivial rational torsion points, Theorem 3.5.4(i) follows directly from Theorem 3.5.11.

The Euler system techniques we are using give an upper bound for the order of the Selmer group, but no lower bound. In this case there is no analogue of the analytic class number formula that enabled us to go from the Euler system divisibility to equality in Corollaries 3.2.4 and 3.4.7.

The p-adic L-function and the Coleman map. Suppose for this section that E has either good ordinary reduction or multiplicative reduction at p. Let α ∈ Z_{p}^{×} and β = p/α ∈ pZ_{p} be the eigenvalues of Frobenius over F_{p} if E has good ordinary reduction at p, and let (α, β) = (1, p) (resp. (−1, −p)) if E has split (resp. nonsplit) multiplicative reduction.

Write G_{n} = Gal(Q_{n}/Q) = Gal(Q_{n,p}/Q_{p}), and fix a generator {ζ_{p^{n}}}_{n} of \underleftarrow{lim} μ_{p^{n}}. If χ is a character of Gal(Q_{∞}/Q) of conductor p^{n} define the Gauss sum

τ(χ) = ∑_{γ∈Gal(Q(μ_{p^{n}})/Q)}χ(γ)ζ_{p^{n}}^{γ}.

Fix also an embedding of \overline{Q_{p}} into C so that we can identify complex and p-adic characters of G_{Q}.

The following theorem is proved in [MSD] in the case of good ordinary reduction. See [MTT] for the (even more) general statement.

Theorem 3.5.13. Suppose E is modular and E has either good ordinary reduction or multiplicative reduction at p. Let α be as above. Then there are a nonzero integer c_{E} independent of p, and a p-adic L-function L_{E} ∈ c_{E}^{−1}Λ, such that for every character χ of Gal(Q_{∞}/Q) of finite order,

χ(L_{E}) = \begin{cases}(1 − α^{−1})^{2}L(E, 1)/Ω_{E} & if χ = 1 and E has good reduction at p \\ (1 − α^{−1})L(E, 1)/Ω_{E} & if χ = 1 and E is multiplicative at p \\ α^{−n}τ(χ)L(E, χ^{−1}, 1)/Ω_{E} & if χ has conductor p^{n} > 1. \end{cases}

If m ∈ Z^{+}, define

L_{E,m} = ( ∏_{q|m,q≠p} ℓ_{q}(q^{−1}Fr_{q}^{−1}))L_{E} ∈ c_{E}^{−1}Λ.

Using (3.16) and Theorem 3.5.13 one obtains expressions for χ(L_{E,m}) in terms of L_{m}(E, χ^{−1}, 1) similar to those in Theorem 3.5.13.

Proposition 3.5.14. Suppose that E has either good ordinary reduction or multiplicative reduction at p. Then there is a Λ-module map

Col_{∞} : H_{∞,s}^{1}(Q_{p}, T) ↪ Λ

such that for every z = {z_{n}} ∈ H_{∞,s}^{1}(Q_{p}, T) and every nontrivial character χ of G_{n}, we have

χ(Col_{∞}(z)) = α^{−k}τ(χ) ∑_{γ∈G_{n}} χ^{−1}(γ) exp_{ω_{E}}^{∗}(z_{n}^{γ})

where p^{k} is the conductor of χ. If χ_{0} is the trivial character then

χ_{0}(Col_{∞}(z)) = (1 − α^{−1})(1 − β^{−1})^{−1} exp_{ω_{E}}^{∗}(z_{0}).

Further, if E has split multiplicative reduction at p then the image of Col_{∞} is contained in the augmentation ideal of Λ.

Proof. The proof is based on work of Coleman [Co]. See the appendix of [Ru9] for an explicit construction of Col_{∞} in this case, and see §8.1 for a discussion of a generalization due to Perrin-Riou [PR2].

Using the Coleman map Col_{∞} described above, we can relate Kato’s Euler system to the p-adic L-function.

Corollary 3.5.15. With hypotheses as in Theorem 3.5.13, with r_{E} as in Theorem 3.5.3, and with other notation as above, we have

Col_{∞}(loc_{{p}}^{s}({c_{Q_{n}}})) = r_{E}L_{E,N}.

Proof. Fix a character χ of Gal(Q_{∞}/Q) of finite order. Theorem 3.5.3 and Proposition 3.5.14 allow us to compute χ(Col_{∞}(loc_{{p}}^{s}({c_{Q_{n}}}))), the definition (Theorem 3.5.13) of L_{E} and (3.16) allow us to compute χ(r_{E}L_{E,N}), and these values turn out to be equal. For example, if χ is nontrivial then both are equal to

r_{E}α^{−n}τ(χ)L_{Np}(E, χ^{−1}, 1)/Ω_{E}.

When χ = 1, we need to use the fact that ℓ_{p}(p^{−1}) is (1 − α^{−1})(1 − β^{−1}) (resp. (1 − β^{−1})) if E has good (resp. multiplicative) reduction at p. Since χ(Col_{∞}(loc_{{p}}^{s}({c_{Q_{n}}}))) = χ(r_{E}L_{E,N}) for all χ, the corollary follows.

Bounding S(Q_{∞}, E_{p^{∞}}). Recall that N is the conductor of E, and let

Z_{∞} = Hom(S(Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}).

Theorem 3.5.16. Suppose E is modular, E does not have complex multiplication, and E has either good ordinary reduction or nonsplit multiplicative reduction at p. Then Z_{∞} is a finitely generated torsion Λ-module and there is an integer t such that

char(Z_{∞}) divides p^{t}L_{E,N}Λ.

If ρ_{E,p} is surjective and p ∤ r_{E} ∏_{q|N,q≠p} ℓ_{q}(q^{−1}), then char(Z_{∞}) divides L_{E}Λ.

If E has split multiplicative reduction at p, then the same results hold with char(Z_{∞}) replaced by J char(Z_{∞}), where J is the augmentation ideal of Λ.

Proof. Rohrlich [Ro] proved that L_{E} ≠ 0. Thus the theorem follows by combining Propositions 3.5.8 and 3.5.14, Corollary 3.5.15, and Theorem 2.3.8.

Remark 3.5.17. For a discussion of the “extra zero” (the extra factor of J in Theorem 3.5.16) in the case of split multiplicative reduction, see [MTT].

Corollary 3.5.18. Let E be as in Theorem 3.5.16. If p is a prime where E has good ordinary reduction and

p ∤ ∏_{q|N} |E(Q_{q})_{tors}|,

then Z_{∞} has no nonzero finite submodules.

Proof. This corollary is due to Greenberg [Gr2], [Gr3]; we sketch a proof here. Let Σ be the set of places of Q dividing Np∞, and let Q_{Σ} be the maximal extension of Q unramified outside Σ. By Lemma 1.5.3 there is an exact sequence

0 → S(Q_{∞}, E_{p^{∞}}) → H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}) → ⊕_{q∈Σ} ⊕_{v|q} H_{s}^{1}(Q_{∞,v}, E_{p^{∞}}). (3.17)

It follows from local duality (Theorem 1.4.1 and Proposition 1.4.3) that for every place v of Q_{∞}, we have

Hom(H_{s}^{1}(Q_{∞,v}, E_{p^{∞}}), Q_{p}/Z_{p}) ≅ \underleftarrow{lim}_{n} E(Q_{n,v})ˆ

where as usual ( )ˆ denotes p-adic completion.

If v | q for some q ≠ p, and E(Q_{q}) has no p-torsion, then it is not hard to show that E(Q_{∞,v}) has no p-torsion and so E(Q_{n,v})ˆ = 0 for every n. Thus for p as in the statement of the corollary, the Pontryagin dual of (3.17) is

\underleftarrow{lim}_{n} E(Q_{n,p})ˆ ⟶ Hom(H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}) ⟶ Z_{∞} ⟶ 0.

Since Q_{∞}/Q is totally ramified at p, we have

\underleftarrow{lim}_{n} E(Q_{n,p})ˆ = \underleftarrow{lim}_{n} E_{1}(Q_{n,p}) = \underleftarrow{lim}_{n} Ê(𝔭_{n})

and this is free of rank one over Λ (see for example [PR1] Théorème 3.1 or [Schn] Lemma 6, §A.1). It now follows, using the fact that Z_{∞} is a torsion Λ-module (Theorem 3.5.16) and using Propositions 3, 4, and 5 of [Gr2] that Hom(H^{1}(Q_{Σ}/Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}) has no nonzero finite submodules. By the Lemma on p. 123 of [Gr2] the same is true of Z_{∞}.

Corollary 3.5.19. Suppose that E is modular, that E does not have complex multiplication, that E has good reduction at p, that p does not divide 2r_{E} ∏_{q|N} ℓ_{q}(q^{−1})|E(Q_{q})_{tors}| (where r_{E} is as in Theorem 3.5.3), and that ρ_{E,p} is surjective. Then

|� (E)_{p^{∞}}| divides \frac{L(E, 1)}{Ω_{E}}.

Proof. First, if E has good supersingular reduction at p then |Ẽ(F_{p})| is prime to p, so the corollary follows from Theorem 3.5.11(ii).

Thus we may assume that E has good ordinary reduction at p. In this case the corollary is a well-known consequence of Theorem 3.5.16 and Corollary 3.5.18; see for example [PR1] §6 or [Schn] §2 for details. The idea is that if Z_{∞} has no nonzero finite submodules and char(Z_{∞}) divides L_{E}Λ, then

|S(Q_{∞}, E_{p^{∞}})^{Gal(Q_{∞}/Q)}| divides χ_{0}(L_{E,N}),

where χ_{0} denotes the trivial character, and

χ_{0}(L_{E,N}) = (1 − α^{−1})^{2} ∏_{q|N} ℓ_{q}(q^{−1})(L(E, 1)/Ω_{E}).

On the other hand, one can show that the restriction map

S(Q, E_{p^{∞}}) ⟶ S(Q_{∞}, E_{p^{∞}})^{Gal(Q_{∞}/Q)}

is injective with cokernel of order divisible by (1 − α^{−1})^{2}, and the corollary follows.

Remark 3.5.20. The Birch and Swinnerton-Dyer conjecture predicts that the divisibility of Corollary 3.5.19 holds for almost all, but not all, primes p.

Proof of Theorem 3.5.4(i). Suppose E is modular, E does not have complex multiplication, and L(E, 1) ≠ 0. By Theorem 3.5.11, E(Q) is finite and � (E)_{p^{∞}} is finite for every p. By Corollary 3.5.19 (and using Serre’s theorem, see Remark 3.5.9), � (E)_{p^{∞}} = 0 for almost all p. This proves Theorem 3.5.4(i).

Remark 3.5.21. We can also now prove part of Theorem 3.5.4(ii), in the special case where L is contained in Q_{∞} and E has either good ordinary or multiplicative reduction at p. For in that case, Theorem 3.5.16 shows that χ(char(Hom(S(Q_{∞}, E_{p^{∞}}), Q_{p}/Z_{p}))) divides a nonzero multiple of L(E, χ, 1)/Ω_{E}. If L(E, χ, 1) ≠ 0 it follows that S(Q_{∞}, E_{p^{∞}})^{χ} is finite. The kernel of the restriction map S(L, E_{p^{∞}}) → S(Q_{∞}, E_{p^{∞}}) is contained in the finite group H^{1}(Q_{∞}/L, E_{p^{∞}}^{G_{Q∞}}), and so we conclude that both E(L)^{χ} and � (E_{/L})_{p^{∞}}^{χ} are finite.

3.6. Symmetric Square of an Elliptic Curve

Let E be an elliptic curve over Q and let T_{p}(E) be the p-adic Tate module of E. Let T be the symmetric square of T_{p}(E), i.e., the threedimensional Z_{p}-representation of G_{Q} defined by

T = (T_{p}(E) ⊗ T_{p}(E))/{t ⊗ t′ − t′ ⊗ t : t, t′ ∈ T_{p}(E)}.

Suppose τ has eigenvalues α and α^{−1} on T_{p}(E) with α^{2} ≢ 1 (mod p). Then τ ∈ G_{Q(μ_{p^{∞}})} (as in Proposition 3.5.8), and τ has eigenvalues α^{2}, 1, and α^{−2} on T, so τ satisfies hypothesis Hyp(Q, T)(i). If the p-adic representation attached to E is surjective and p > 3 then we can always find such a τ, and further T/pT is an irreducible G_{Q}-module and

H^{1}(Ω/Q, W) = H^{1}(Ω/Q, W^{∗}) = 0.

Thus if we had an Euler system for T, then we could apply Theorem 2.2.10 to study the Selmer group S(Q, W^{∗}). See [Fl] for important progress in this direction.

CHAPTER 4 Derived Cohomology Classes

The proofs of the main theorems stated in Chapter 2 consist of two steps. First we use an Euler system to construct auxiliary cohomology classes which Kolyvagin calls “derivative” classes, and second we use these derived classes along with the duality theorems of §1.7 to bound Selmer groups.

In this chapter we carry out the first of these steps. In §4.2 and §4.3 we define and study the “universal Euler system” associated to T and K_{∞}/K. In §4.4 we construct the Kolyvagin derivative classes, and in §4.5 we state the local properties of these derivative classes, which will be crucial in all the applications. The remainder of this chapter is devoted to the proofs of these properties.

4.1. Setup

Keep the notation of §2.1. We have a fixed number field K, a p-adic representation T of G_{K} with coefficients in the ring of integers O of some finite extension Φ of Q_{p}, and we assume that T is unramified outside a finite set of primes of K.

The letter 𝔮 will always denote a prime of K. For every prime 𝔮 of K not dividing p, we let K(𝔮) denote the maximal p-extension of K inside the ray class field of K modulo 𝔮. Similarly, let K(1) denote the maximal pextension of K inside the Hilbert class field of K. Class field theory shows that K(𝔮)/K(1) is unramified outside 𝔮, totally ramified above 𝔮, and cyclic with Galois group canonically isomorphic to the maximal p-quotient of (O_{K}/𝔮)^{×}/(O_{K}^{×} (mod 𝔮)). Let Γ_{𝔮} = Gal(K(𝔮)/K(1)).

Fix an ideal N of K divisible by p and by all primes where T is ramified, as in Definition 2.1.1. Define

R = R(N) = {squarefree products of primes 𝔮 of K such that 𝔮 ∤ N }.

If 𝔯 ∈ R, say 𝔯 = 𝔮_{1} ⋯ 𝔮_{k}, then we define K(𝔯) to be the compositum

K(𝔯) = K(𝔮_{1}) ⋯ K(𝔮_{k}).

Note that K(𝔯) is contained in, but not in general equal to, the maximal p-extension of K inside the ray class field of K modulo 𝔯. We define

Γ_{𝔯} = Gal(K(𝔯)/K(1)).

Ramification considerations show that the fields K(𝔮) are linearly disjoint over K(1), so there is a natural isomorphism

Γ_{𝔯} ≅ ∏_{primes 𝔮|𝔯}Γ_{𝔮}

where Γ_{𝔮} is identified with the inertia group of 𝔮 in Γ_{𝔯}. If 𝔰 | 𝔯 this allows us to view Γ_{𝔰} as a subgroup of Γ_{𝔯}, as well as a quotient.

Fix a Z_{p}^{d}-extension K_{∞}/K in which no finite prime splits completely, as in Definition 2.1.1. If K ⊂ F ⊂ K_{∞}, let F (𝔯) = F K(𝔯). As in Chapter 2, we will write K ⊂_{f} F to indicate that F is a finite extension of K, and if K ⊂_{f} F ⊂ K_{∞} we let

Γ_{F (𝔯)} = Gal(F (𝔯)/K(1)).

Again, we will often identify Γ_{𝔯} with the subgroup of Γ_{F (𝔯)} generated by the inertia groups of primes dividing 𝔯, and Γ_{F (1)} with the subgroup generated by the inertia groups of primes dividing p, and then (since K_{∞}/K is unramified outside p)

Γ_{F (𝔯)} ≅ Γ_{F (1)} × Γ_{𝔯}.

As above, if 𝔰 | 𝔯 we can also identify Γ_{F (𝔰)} with a subgroup of Γ_{F (𝔯)}. Figure 1 illustrates these fields and Galois groups. For 𝔯 ∈ R define

N_{𝔯} = ∑_{σ∈Γ_{𝔯}} σ ∈ Z[Γ_{𝔯}] ⊂ Z[Gal(K(𝔯)/K)].

If 𝔰 | 𝔯 and K ⊂_{f} F ⊂ K_{∞} we can view N_{𝔰} ∈ Z[Γ_{𝔯}] ⊂ Z[Gal(F (𝔯)/K)] as above, and then N_{𝔯} = N_{𝔰}N_{𝔯/𝔰}.

As in Chapter 2, let Fr_{𝔮} denote a Frobenius of 𝔮 in G_{K}, and

P (Fr_{𝔮}^{−1}|T^{∗}; x) = det(1 − Fr_{𝔮}^{−1}x|T^{∗}) ∈ O[x].

Definition 4.1.1. Suppose K ⊂_{f} F ⊂ K_{∞} and M ∈ O is nonzero. We define R_{F,M} ⊂ R to be the set of all 𝔯 ∈ R such that for every prime 𝔮 dividing 𝔯,

• M | [K(𝔮) : K(1)],

• M | P (Fr_{𝔮}^{−1}|T^{∗}; 1),

• 𝔮 splits completely in F (1)/K.

As in Definition 1.4.6, if M ∈ O is nonzero we let M̄ ∈ Z^{+} denote the smallest power of p which is divisible in O by M.

[figure: K; K(1); F (1); K(𝔯); K(𝔮); F (𝔯); K(𝔯𝔮); F (𝔯𝔮); .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; K_{∞}(1); Γ_{F (1)}; Γ_{F (1)}; Γ_{F (1)}; Γ_{𝔮}; Γ_{𝔮}; Γ_{𝔮}; Γ_{𝔯}; Γ_{𝔯}; Γ_{𝔯}; Γ_{F (𝔯)}]

Figure 1

Lemma 4.1.2. Suppose 𝔮 ∈ R is a prime of K and M ∈ O is nonzero. (i) M | [K(𝔮) : K(1)] ⟺ 𝔮 splits completely in K(μM̄, (O_{K}^{×})^{1/M̄}). (ii) P (Fr_{𝔮}^{−1}|T^{∗}; N(𝔮)Fr_{𝔮}^{−1}) annihilates T. (iii) If M | [K(𝔮) : K(1)] then

P (Fr_{𝔮}^{−1}|T^{∗}; x) ≡ det(1 − Fr_{𝔮}x|W_{M}) (mod M).

(iv) If M | [K(𝔮) : K(1)] then P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}) annihilates W_{M}.

Proof. Class field theory identifies Gal(K(𝔮)/K) with the maximal p-quo-tient of (O_{K}/𝔮)^{×}/(O_{K}^{×} (mod 𝔮)). Thus if 𝔮 ∤ p, then [K(𝔮) : K(1)] divides |(O_{K}/𝔮)^{×}| = (N(𝔮) − 1) and

Fr_{𝔮} fixes μM̄ ⟺ M̄ divides |(O_{K}/𝔮)^{×}| ⟺ M divides |(O_{K}/𝔮)^{×}|.

If Fr_{𝔮} fixes μM̄ we have further

Fr_{𝔮} fixes (O_{K}^{×})^{1/M̄} ⟺ (O_{K}^{×} (mod 𝔮)) ⊂ ((O_{K}/𝔮)^{×})M̄.

This proves (i). One checks easily that

P (Fr_{𝔮}^{−1}|T^{∗}; x) = det(1 − Fr_{𝔮}^{−1}x|T^{∗}) = det(1 − N(𝔮)^{−1}Fr_{𝔮}x|T).

This and the Cayley-Hamilton theorem prove (ii), (iii), and (iv).

The following lemma, together with the Tchebotarev density theorem, will give a large supply of primes in R_{F,M}. By F (W_{M}) we mean the smallest extension of F whose absolute Galois group acts trivially on W_{M} (or equivalently, the fixed field of the kernel of the action of G_{F} on W_{M}).

Lemma 4.1.3. Suppose K ⊂_{f} F ⊂ K_{∞} and M ∈ O is nonzero. Suppose further that τ ∈ G_{K} acts trivially on K_{∞}(1)K(μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}) and that T^{τ=1} ≠ 0. If 𝔮 is a prime of K not dividing N such that the Frobenius Fr_{𝔮} of 𝔮 is a conjugate of τ on F (1)K(W_{M}, μM̄, (O_{K}^{×})^{1/M̄}), then 𝔮 ∈ R_{F,M}.

Proof. Note that 𝔮 is unramified in F (1)K(W_{M}, μM̄, (O_{K}^{×})^{1/M̄})/K. Since Fr_{𝔮} fixes K(μM̄, (O_{K}^{×})^{1/M̄}), it follows from Lemma 4.1.2(i) that M divides [K(𝔮) : K(1)]. Since Fr_{𝔮} fixes F (1), we see that 𝔮 splits completely in F (1)/K. By Lemma 4.1.2(iii) we have

P (Fr_{𝔮}^{−1}|T^{∗}; 1) ≡ det(1 − Fr_{𝔮}|W_{M}) = det(1 − τ|W_{M})

≡ det(1 − τ|T) = 0 (mod M),

the first equality because Fr_{𝔮} is a conjugate of τ on W_{M}, the second because T^{τ=1} ≠ 0. Thus 𝔮 ∈ R_{F,M}.

4.2. The Universal Euler System

We now define the “universal Euler system”. This might also be called a universal distribution for the Euler system distribution relation of Definition 2.1.1. In the special case K = Q and T = Z_{p}(1), it is closely related to the universal ordinary distribution studied by Kubert in [Ku] (see also [Lan] §2.9 or [Wa] §12.3).

Definition 4.2.1. Suppose that 𝔯 ∈ R and K ⊂_{f} F ⊂ K_{∞}. We define an O[Gal(F (𝔯)/K)]-module X_{F (𝔯)} as follows.

First suppose that every prime 𝔮 dividing 𝔯 satisfies K(𝔮) ≠ K(1). For every 𝔰 dividing 𝔯 let x_{F (𝔰)} be an indeterminate. Set X_{F (𝔯)} = Y_{F (𝔯)}/Z_{F (𝔯)} where:

Y_{F (𝔯)} is the free O[Gal(F (𝔯)/K)]-module on generators {x_{F (𝔰)} : 𝔰 | 𝔯}, Z_{F (𝔯)} is the submodule of Y_{F (𝔯)} generated by the relations

σx_{F (𝔰)} = x_{F (𝔰)}  if σ ∈ Gal(F (𝔯)/F (𝔰)) = Γ_{𝔯/𝔰},
N_{𝔮}x_{F (𝔮𝔰)} = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})x_{F (𝔰)} if 𝔮𝔰 | 𝔯.

For general 𝔯, let 𝔯′ be the product of all primes 𝔮 dividing 𝔯 such that K(𝔮) ≠ K(1). Then by definition F (𝔯) = F (𝔯′), and further we have {F (𝔰) : 𝔰 | 𝔯} = {F (𝔰) : 𝔰 | 𝔯′}. We simply let X_{F (𝔯)} be X_{F (𝔯′)} as defined above.

If 𝔰 | 𝔯 and K ⊂_{f} F ⊂_{f} F′ ⊂ K_{∞} then there are natural O[Gal(F′(𝔯)/K)]- module maps

X_{F′(𝔯)} ⟶ X_{F (𝔯)} induced by x_{F′(𝔱)} ↦ x_{F (𝔱)} for 𝔱 | 𝔯,  (4.1)
X_{F (𝔰)} ⟶ X_{F′(𝔯)} induced by x_{F (𝔱)} ↦ N_{F′(𝔯)/F (𝔯)}x_{F′(𝔱)} for 𝔱 | 𝔰. (4.2)

The map (4.1) is clearly surjective, and Proposition 4.3.1(v) below will show that the map (4.2) is injective.

Definition 4.2.2. The universal Euler system (for (T, N , K_{∞}/K)) is

X = X (T, N , K_{∞}/K) = \underrightarrow{lim}_{F,𝔯} X_{F (𝔯)}.

direct limit with respect to the maps (4.2). Using the maps (4.1), (4.2) we also define

X_{∞,𝔯} = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}X_{F (𝔯)} and X_{∞,R} = \underrightarrow{lim}_{𝔯∈R} X_{∞,𝔯}.

For every 𝔯 ∈ R define

H_{∞}^{1}(K(𝔯), T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F (𝔯), T).

Lemma 4.2.3. If c is an Euler system for (T, K, N) with K_{∞} ⊂ K, then sending x_{F (𝔯)} to c_{F (𝔯)} induces G_{K}-equivariant maps

X_{F (𝔯)} ⟶ H^{1}(F (𝔯), T),  X_{∞,𝔯} ⟶ H_{∞}^{1}(K(𝔯), T),
X ⟶ \underrightarrow{lim}_{F,𝔯} H^{1}(F (𝔯), T),  X_{∞,R} ⟶ \underrightarrow{lim}_{𝔯∈R} H_{∞}^{1}(K(𝔯), T),

direct limits with respect to restriction maps.

Proof. This is immediate, since the Euler system classes {c_{F (𝔯)}} satisfy all the relations that the {x_{F (𝔯)}} do (see Definition 2.1.1 and Remark 2.1.4).

Remark 4.2.4. Conversely, although we will not make use of it, it follows from the following lemma that every map

X_{∞,R} ⟶ \underrightarrow{lim}_{𝔯∈R} H_{∞}^{1}(K(𝔯), T)

induces an Euler system for (T, K_{min}, N), where K_{min} is as in Remark 2.1.4.

Lemma 4.2.5. (i) If K ⊂_{f} F ⊂ K_{∞} and 𝔯 ∈ R, then

T^{G_{F (𝔯)}} = T^{G_{F (1)}} and W^{G_{F (𝔯)}} = W^{G_{F (1)}}.

(ii) If 𝔯𝔰 ∈ R then the restriction map induces an isomorphism

H_{∞}^{1}(K(𝔯), T) ≅ H_{∞}^{1}(K(𝔯𝔰), T)^{Γ_{𝔰}}.

Proof. Since Gal(F (𝔯)/F (1)) = Γ_{𝔯} is generated by inertia groups of primes dividing 𝔯, and T is unramified at those primes, Gal(F (𝔯)/F (1)) acts trivially on T^{G_{F (𝔯)}} and W^{G_{F (𝔯)}}. This proves (i).

It is enough to prove (ii) when 𝔰 is prime, and then the general case follows by induction.

Let S be a finite set of places of K containing all places dividing N 𝔯𝔰∞, and let K_{S} be the maximal extension of K unramified outside S. (Recall that N is divisible by p and all primes where T is ramified, so in particular K_{∞}(𝔯𝔰) ⊂ K_{S} and T is a Gal(K_{S}/K)-module). By Propositions B.2.5(ii) and B.2.7(i), we have an inflation-restriction exact sequence

H^{1}(F (𝔯𝔰)/F (𝔯), T^{G_{F (𝔯𝔰)}}) ⟶ H^{1}(K_{S}/F (𝔯), T)

⟶ H^{1}(K_{S}/F (𝔯𝔰), T)^{Γ_{𝔰}} ⟶ H^{2}(F (𝔯𝔰)/F (𝔯), T^{G_{F (𝔯𝔰)}}). (4.3)

By (i) (and using our identification Gal(F (𝔯𝔰)/F (𝔯)) ≅ Γ_{𝔰}),

H^{1}(F (𝔯𝔰)/F (𝔯), T^{G_{F (𝔯𝔰)}}) = H^{1}(F (𝔯𝔰)/F (𝔯), T^{G_{F (1)}})
= Hom(Γ_{𝔰}, T^{G_{F (1)}}) = 0

and similarly (since 𝔰 is prime, so Γ_{𝔰} is cyclic)

H^{2}(F (𝔯𝔰)/F (𝔯), T^{G_{F (𝔯𝔰)}}) = T^{G_{F (1)}}/|Γ_{𝔰}|T^{G_{F (1)}}.

Now pass to the inverse limit over F in (4.3). Using Corollary B.3.6 and our assumption that the decomposition group in Gal(K_{∞}/K) of every finite prime is infinite, we obtain an exact sequence

0 ⟶ H_{∞}^{1}(K(𝔯), T) ⟶ H_{∞}^{1}(K(𝔯𝔰), T)^{Γ_{𝔰}} ⟶ \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}T^{G_{F (1)}}/|Γ_{𝔰}|T^{G_{F (1)}}.

By Lemma B.3.2, this inverse limit is zero, so this proves (ii).

4.3. Properties of the Universal Euler System

Recall that Φ is the field of fractions of O.

Proposition 4.3.1. Suppose 𝔯 ∈ R and K ⊂_{f} F ⊂ K_{∞}. Then (i) X_{F (𝔯)} is a finitely generated free O-module, (ii) X_{F (𝔯)} ⊗ Φ is a free rank-one module over Φ[Gal(F (𝔯)/K)], (iii) X_{F (𝔯)} is a free O[Gal(F (𝔯)/K(𝔯))]-module of rank [K(𝔯) : K], (iv) for every F ⊂_{f} F′ ⊂ K_{∞}, the map (4.1) induces an isomorphism

X_{F′(𝔯)} ⊗_{O[Gal(F′(𝔯)/K)]} O[Gal(F (𝔯)/K)] ⟶^{∼} X_{F (𝔯)},

(v) for every 𝔰 | 𝔯 and F ⊂_{f} F′ ⊂ K_{∞}, the map (4.2) induces an isomor-

phism X_{F (𝔰)} ⟶^{∼} X_{F′(𝔯)}^{Gal(F′(𝔯)/F (𝔰))}.

Proof. Let 𝔯′ be the product of all primes 𝔮 dividing 𝔯 such that Γ_{𝔮} ≠ {1}. Then X_{F (𝔯′)} = X_{F (𝔯)}, F (𝔯′) = F (𝔯), and K(𝔯′) = K(𝔯), so the proposition for 𝔯 is equivalent to the proposition for 𝔯′. Thus without loss of generality we may replace 𝔯 by 𝔯′, i.e., we may simplify the proof by assuming that Γ_{𝔮} ≠ {1} for every 𝔮 dividing 𝔯.

We will prove the proposition by constructing a specific O-basis of X_{F (𝔯)}. Fix a set of representatives A_{1} ⊂ Gal(F (𝔯)/K) of Gal(K(1)/K), and for every prime 𝔮 dividing 𝔯 let A_{𝔮} = Γ_{𝔮} − {1} ⊂ Gal(F (𝔯)/K). For every ideal 𝔰 dividing 𝔯, define a subset A_{F,𝔰} ⊂ Gal(F (𝔯)/K) by

A_{F,𝔰} = Gal(F (𝔯)/K(𝔯)) A_{1} ∏_{primes 𝔮|𝔰}A_{𝔮}

= {g_{F} g_{1} ∏_{𝔮|𝔰} g_{𝔮} : g_{F} ∈ Gal(F (𝔯)/K(𝔯)), g_{1} ∈ A_{1}, 1 ≠ g_{𝔮} ∈ Γ_{𝔮}}

and then define a finite subset B_{F (𝔯)} of X_{F (𝔯)} by

B_{F (𝔯)} = ⋃_{𝔰|𝔯} A_{F,𝔰}x_{F (𝔰)} ⊂ X_{F (𝔯)}.

We will show that B_{F (𝔯)} is an O-basis of X_{F (𝔯)}.

Clearly A_{𝔮} ∪ {N_{𝔮}} generates O[Γ_{𝔮}] over O, so

Gal(F (𝔯)/K(𝔯)) A_{1} ∏_{𝔮|𝔰}(A_{𝔮} ∪ {N_{𝔮}})

generates O[Γ_{F (𝔰)}]. For every 𝔮 dividing 𝔰 we have a relation

N_{𝔮}x_{F (𝔰)} = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})x_{F (𝔰/𝔮)} ∈ O[Gal(F (𝔰/𝔮)/K)]x_{F (𝔰/𝔮)}.

It follows easily, by induction on the number of primes dividing 𝔯, that B_{F (𝔯)} generates X_{F (𝔯)} over O. Further,

|B_{F (𝔯)}| ≤ ∑_{𝔰|𝔯} |A_{F,𝔰}| = [F (1) : K] ∏_{𝔮|𝔯}(|A_{𝔮}| + 1)

= [F (1) : K] ∏_{𝔮|𝔯} |Γ_{𝔮}| = [F (𝔯) : K].

On the other hand, we claim that rank_{O}(X_{F (𝔯)}) ≥ [F (𝔯) : K]. To see this, let Y_{F (𝔯)} and Z_{F (𝔯)} be as in Definition 4.2.1 of X_{F (𝔯)}. One can check directly that the assignment

x_{F (𝔰)} ↦ ∏_{𝔮|(𝔯/𝔰)}N_{𝔮} ∏_{𝔮|𝔰}(|Γ_{𝔮}| + (P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}) − |Γ_{𝔮}|)\frac{N_{𝔮}}{|Γ_{𝔮}|})

induces a well-defined homomorphism from Y_{F (𝔯)} to O[Gal(F (𝔯)/K)] which is zero on Z_{F (𝔯)}. Thus we obtain a map

φ : X_{F (𝔯)} ⊗ Φ ⟶ Φ[Gal(F (𝔯)/K)].

If χ is a character of Gal(F (𝔯)/K) into an algebraic closure of Φ, say χ has conductor exactly 𝔰, then

χ(φ(x_{F (𝔰)})) = ∏_{𝔮|𝔯} |Γ_{𝔮}| ≠ 0.

It follows that φ is surjective, and in particular

rank_{O}(X_{F (𝔯)}) = dim_{Φ}(X_{F (𝔯)} ⊗ Φ) ≥ [F (𝔯) : K] ≥ |B_{F (𝔯)}|.

Since B_{F (𝔯)} generates X_{F (𝔯)} over O, we conclude that equality holds, that B_{F (𝔯)} is an O-basis of X_{F (𝔯)}, that X_{F (𝔯)} is torsion-free, and that φ is an isomorphism. This proves (i) and (ii). Further, since Gal(F (𝔯)/K(𝔯)) permutes the elements of the basis B_{F (𝔯)}, (iii) follows as well.

The map (4.1) defined by x_{F′(𝔰)} ↦ x_{F (𝔰)} induces a surjective map

X_{F′(𝔯)} ⊗_{O[Gal(F′(𝔯)/K)]} O[Gal(F (𝔯)/K)] ↠ X_{F (𝔯)}.

By (iii) applied to F and F′, this map must be injective as well, which proves (iv).

It is immediate from the definitions that the map (4.2) induces a surjection

X_{F (𝔯)} ↠ X_{F′(𝔯)}^{Gal(F′(𝔯)/F (𝔯))},

which then must be injective by (iii). Also we see that B_{F (𝔰)} ⊂ B_{F (𝔯)}, so the map X_{F (𝔰)} → X_{F (𝔯)} is injective and its cokernel is torsion free. By (ii), X_{F (𝔯)}^{Gal(F (𝔯)/F (𝔰))}/X_{F (𝔰)} is finite, so it must be zero. Now (v) follows.

If G is a profinite abelian group, we write O[[G]] = \underleftarrow{lim}_{U⊂G} O[G/U].

Corollary 4.3.2. The O[[Gal(K_{∞}(𝔯)/K(𝔯))]]-module X_{∞,𝔯} is free of rank [K(𝔯) : K], and for every K ⊂_{f} F ⊂ K_{∞} we have

X_{∞,𝔯} ⊗_{O[[Gal(K_{∞}(𝔯)/K)]]} O[Gal(F (𝔯)/K)] ≅ X_{F (𝔯)}.

Proof. This is immediate from Proposition 4.3.1(iii) and (iv).

Lemma 4.3.3. Suppose R is a ring, G is a profinite abelian group, and H is an open subgroup of G. Suppose B is an R[[G]]-module.

(i) Hom_{R[[G]]}(B, R[[G]]) ≅ Hom_{R[[H]]}(B, R[[H]]) as R[[H]]-modules. (ii) If B is free as an R[[H]]-module then Ext_{R[[G]]}^{1}(B, R[[G]]) = 0.

Proof. Write S = R[[H]] and S′ = R[[G]]. Fix a (finite) set C ⊂ G, containing 1, of coset representatives of G/H. Then C is an S-basis of S′, and we let π : S′ → S be the S-module map

∑_{η∈C} a_{η}η ↦ a_{1}.

Define a homomorphism Hom_{S′}(B, S′) → Hom_{S}(B, S) by composition with π. One can check directly that this map is both injective and surjective, which proves (i).

It follows from (i) that Ext_{S′}^{1}(B, S′) = Ext_{S}^{1}(B, S), and if B is free over S this is zero.

Proposition 4.3.4. Suppose that 𝔯 ∈ R, that k ≥ 0, and that M ∈ O is nonzero.

(i) If K ⊂_{f} F ⊂ K_{∞} and G = Gal(F (𝔯)/K), then

Ext_{(O/MO)[G]}^{1}(X_{F (𝔯)}/MX_{F (𝔯)}, (O/MO)[G]^{k}) = 0.

(ii) If G = Gal(K_{∞}(𝔯)/K), then

Ext_{(O/MO)[[G]]}^{1}(X_{∞,𝔯}/MX_{∞,𝔯}, (O/MO)[[G]]^{k}) = 0.

Proof. Apply Lemma 4.3.3(ii) with R = O/MO and with

G = Gal(F (𝔯)/K), H = {1}, B = X_{F (𝔯)}/MX_{F (𝔯)}

to prove (i), and with

G = Gal(K_{∞}(𝔯)/K), H = Gal(K_{∞}(𝔯)/K(𝔯)), B = X_{∞,𝔯}/MX_{∞,𝔯}

to prove (ii). That B is free over R[[H]] is Proposition 4.3.1(i) in the first case, and Corollary 4.3.2 in the second.

Remark 4.3.5. Alternatively, Proposition 4.3.4(i) can be proved by observing that (O/MO)[G] is injective (as a module over itself) when G is finite. However, this is not true for (O/MO)[[Gal(K_{∞}(𝔯)/K)]].

4.4. Kolyvagin’s Derivative Construction

Following Kolyvagin [Ko2], we will associate to an Euler system a collection of “derivative” classes

κ_{[F,𝔯,M]} ∈ H^{1}(F, W_{M})

for every K ⊂_{f} F ⊂ K_{∞}, every nonzero M ∈ O, and every 𝔯 ∈ R_{F,M} (where R_{F,M} is the subset of R given by Definition 4.1.1).

Definition 4.4.1. Fix a generator ξ of \underleftarrow{lim} μ_{p^{n}}, and for every prime 𝔮 of K not dividing p fix a prime 𝔔 of K̄ above 𝔮. We will fix a generator σ_{𝔮} of Γ_{𝔮} as follows.

Let M = |Γ_{𝔮}| = [K(𝔮) : K(1)] and let I_{𝔔} denote the inertia group of 𝔔 in G_{K}. Since M is a power of p and 𝔮 is prime to p, Lemma 1.4.5 shows that I_{𝔔} has a unique cyclic quotient of order M, and this quotient is canonically isomorphic to μ_{M}. Since Γ_{𝔮} itself is a cyclic quotient of I_{𝔔}, this allows us to identify Γ_{𝔮} with μ_{M}. The chosen generator ξ gives us a generator ζ of μ_{M}; we define σ_{𝔮} ∈ Γ_{𝔮} to be the corresponding generator of Γ_{𝔮}. (This definition depends on the choices of 𝔔 and ξ, but we will suppress this dependence from the notation.)

Now define, for every prime 𝔮 not dividing p,

D_{𝔮} = ∑_{i=0}^{|Γ_{𝔮}|−1} iσ_{𝔮}^{i} ∈ Z[Γ_{𝔮}].

If 𝔯 ∈ R and 𝔮 | 𝔯 we view D_{𝔮} ∈ Z[Γ_{𝔯}] and define

D_{𝔯} = ∏_{primes 𝔮|𝔯}D_{𝔮} ∈ Z[Γ_{𝔯}].

If K ⊂_{f} F ⊂ K_{∞}, fix an element N_{F (1)/F} ∈ Z[G_{F}] whose image under restriction to F (1) is the norm element ∑_{γ∈Gal(F (1)/F)} γ ∈ Z[Gal(F (1)/F)], and define

D_{𝔯,F} = N_{F (1)/F} D_{𝔯}.

We have the easy “telescoping” identity

(σ_{𝔮} − 1)D_{𝔮} = |Γ_{𝔮}| − N_{𝔮}.  (4.4)

This is the key step in the following lemma, which in turn is crucial for the construction of the derivative classes.

Lemma 4.4.2. Suppose M ∈ O is nonzero, K ⊂_{f} F ⊂ K_{∞}, and 𝔯 ∈ R_{F,M}. Let x̄_{F (𝔯)} denote the image of x_{F (𝔯)} in X_{F (𝔯)}/MX_{F (𝔯)}. Then

(i) D_{𝔯,F} x̄_{F (𝔯)} ∈ (X_{F (𝔯)}/MX_{F (𝔯)})^{Gal(F (𝔯)/F)}, (ii) D_{𝔯,F} x̄_{F (𝔯)} is independent of the choice of N_{F (1)/F}.

Proof. We will show that

(σ − 1)D_{𝔯}x_{F (𝔯)} ∈ MX_{F (𝔯)} for every σ ∈ Gal(F (𝔯)/F (1)),

and then both assertions of the lemma follow.

The proof is by induction on the number of primes dividing 𝔯. If 𝔯 = 1 there is nothing to prove. In general, suppose 𝔮 is a prime dividing 𝔯, say 𝔯 = 𝔮𝔰. Since 𝔮 ∈ R_{F,M}, we have that Fr_{𝔮} ∈ Gal(F (𝔯)/F (1)) and that M divides both |Γ_{𝔮}| and P (Fr_{𝔮}^{−1}|T^{∗}; 1). Since N_{𝔮}x_{F (𝔯)} = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})x_{F (𝔰)}, using (4.4) and the induction hypothesis we obtain

(σ_{𝔮} − 1)D_{𝔯}x_{F (𝔯)} = (σ_{𝔮} − 1)D_{𝔮}D_{𝔰}x_{F (𝔯)}
= |Γ_{𝔮}|D_{𝔰}x_{F (𝔯)} − P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})D_{𝔰}x_{F (𝔰)}

≡ |Γ_{𝔮}|D_{𝔰}x_{F (𝔯)} − P (Fr_{𝔮}^{−1}|T^{∗}; 1)D_{𝔰}x_{F (𝔰)} (mod (Fr_{𝔮} − 1)D_{𝔰}x_{F (𝔰)})

∈ MX_{F (𝔯)}.

Since Gal(F (𝔯)/F (1)) is generated by the σ_{𝔮}, this proves the lemma.

Remark 4.4.3. The idea of the construction of the derivative class κ_{[F,𝔯,M]} is as follows.

By Lemmas 4.4.2 and 4.2.3, the image of D_{𝔯,F} c_{F (𝔯)} in H^{1}(F (𝔯), W_{M}) is fixed by Gal(F (𝔯)/F). If W^{G_{F (𝔯)}} = 0 then the restriction map

H^{1}(F, W_{M}) ⟶ H^{1}(F (𝔯), W_{M})^{Gal(F (𝔯)/F)}  (4.5)

is an isomorphism, and we define κ_{[F,𝔯,M]} ∈ H^{1}(F, W_{M}) to be the inverse image of D_{𝔯,F} c_{F (𝔯)}.

When W^{G_{F (𝔯)}} ≠ 0, the map (4.5) need not be an isomorphism. The rest of this section will be devoted to showing, using Proposition 4.3.4 and the universal Euler system, that the image of D_{𝔯,F} c_{F (𝔯)} always has a canonical inverse image under (4.5). That inverse image will be our class κ_{[F,𝔯,M]} (see Definition 4.4.10). Our construction will also be quite explicit, so that we can use it to prove the local properties of the derivative classes which we state in §4.5 below.

Fix, for the rest of this section, a nonzero M ∈ O.

Definition 4.4.4. Let 𝕎_{M} = Ind(W_{M}) denote the induced module defined (and called Ind_{{1}}^{G_{K}}(W_{M})) in §B.4:

𝕎_{M} = Maps(G_{K}, W_{M}),

i.e., the O-module of continuous maps (not necessarily homomorphisms) from G_{K} to W_{M}, with G_{K} acting via

(γf)(g) = f(gγ) for all γ, g ∈ G_{K}.

There is a natural G_{K}-module inclusion W_{M} ↪ 𝕎_{M} which sends t to the map g ↦ gt, and we will identify W_{M} with a submodule of 𝕎_{M} using this inclusion.

Proposition 4.4.5. For every 𝔯 ∈ R and every finite extension L of K in K_{∞}(𝔯), there is a canonical map

δ_{L} : (𝕎_{M}/W_{M})^{G_{L}} ⟶ H^{1}(L, W_{M})

such that (i) there is an exact sequence

0 ⟶ W_{M}^{G_{L}} ⟶ 𝕎_{M}^{G_{L}} ⟶ (𝕎_{M}/W_{M})^{G_{L}} ⟶^{δ_{L}} H^{1}(L, W_{M}) ⟶ 0,

(ii) if f ∈ (𝕎_{M}/W_{M})^{G_{L}} and f̂ ∈ 𝕎_{M} lifts f, then δ_{L}(f) is represented by the cocycle

γ ↦ (γ − 1)f̂ ∈ W_{M} for γ ∈ G_{L},

(iii) if K ⊂_{f} L⊂_{f} L′ ⊂ K_{∞}(𝔯) then the following diagram commutes:

\xymatrix{(𝕎_{M}/W_{M})^{G_{L}} \ar[d]^{δ_{L}} \ar@{^{(}->}[r] & (𝕎_{M}/W_{M})^{G_{L′}} \ar[d]^{δ_{L′}} \ar[r]^{N_{L′/L}} & (𝕎_{M}/W_{M})^{G_{L}} \ar[d]^{δ_{L}} \\ H^{1}(L, W_{M}) \ar[r]^{_{Res_{L′}}} & H^{1}(L′, W_{M}) \ar[r]^{_{Cor_{L′/L}}} & H^{1}(L, W_{M}).}

Proof. By Proposition B.4.5, taking G_{L}-cohomology of the exact sequence

0 ⟶ W_{M} ⟶ 𝕎_{M} ⟶ 𝕎_{M}/W_{M} ⟶ 0

gives the exact sequence of (i) and the commutativity of (iii). Assertion (ii) is just the standard calculation of the connecting map in Galois cohomology.

Lemma 4.4.6. Let d = rank_{O}(T), and suppose 𝔯 ∈ R.

(i) If K ⊂_{f} F ⊂ K_{∞} then 𝕎_{M}^{G_{F (𝔯)}} is a free (O/MO)[Gal(F (𝔯)/K)]-mod-ule of rank d.

(ii) Let Λ_{𝔯} = O[[Gal(K_{∞}(𝔯)/K)]]. Then \underleftarrow{lim} 𝕎_{M}^{G_{F (𝔯)}} (inverse limit over K ⊂_{f} F ⊂ K_{∞} with respect to the norm maps) is a free Λ_{𝔯}/MΛ_{𝔯}- module of rank d, and if K ⊂_{f} F′ ⊂ K_{∞} then

\underleftarrow{lim}_{F} 𝕎_{M}^{G_{F (𝔯)}} ⊗_{Λ_{𝔯}} O[Gal(F′(𝔯)/K)] ≅ 𝕎_{M}^{G_{F′(𝔯)}}.

Proof. Let W_{M}^{0} denote the O-module W_{M} with trivial G_{K}-action, and abbreviate H = Gal(F (𝔯)/K). Then there are Galois-equivariant isomorphisms

𝕎_{M}^{G_{F (𝔯)}} = Maps(H, W_{M}) = Maps(H, O/MO) ⊗_{O} W_{M}^{0}.

Since W_{M} is free of rank d over O/MO, and Maps(H, O/MO) is free of rank one over (O/MO)[H], this proves (i). The second assertion follows by taking inverse limits.

If 𝔯 ∈ R we write H_{∞}^{1}(K(𝔯), W_{M}) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F (𝔯), W_{M}).

Proposition 4.4.7. Suppose 𝔯 ∈ R. Then the maps δ_{F (𝔯)} of Proposition 4.4.5 induce an exact sequence

0 ⟶ \underleftarrow{lim}_{F} 𝕎_{M}^{G_{F (𝔯)}} ⟶ \underleftarrow{lim}_{F} (𝕎_{M}/W_{M})^{G_{F (𝔯)}} ⟶^{δ_{𝔯}} H_{∞}^{1}(K(𝔯), W_{M}) ⟶ 0.

Proof. By Lemma 4.4.6(i), 𝕎_{M}^{G_{F (𝔯)}} is finite whenever K ⊂_{f} F ⊂ K_{∞}. Therefore taking inverse limits over F of the exact sequence of Proposition 4.4.5(i) (with respect to norm maps for the first three terms and corestriction for the fourth; see Proposition 4.4.5(iii)) yields a new exact sequence (see Proposition B.1.1(i))

0 ⟶ \underleftarrow{lim}_{F} W_{M}^{G_{F (𝔯)}} ⟶ \underleftarrow{lim}_{F} 𝕎_{M}^{G_{F (𝔯)}}

⟶ \underleftarrow{lim}_{F} (𝕎_{M}/W_{M})^{G_{F (𝔯)}} ⟶^{δ_{𝔯}} H_{∞}^{1}(K(𝔯), W_{M}) ⟶ 0.

By Lemma B.3.2, \underleftarrow{lim} W_{M}^{G_{F (𝔯)}} = 0, and the proposition follows.

Proposition 4.4.8. Suppose c is an Euler system and 𝔯 ∈ R. There is a family of O[G_{K}]-module maps

{d_{F} : X_{F (𝔯)} → (𝕎_{M}/W_{M})^{G_{F (𝔯)}} : K ⊂_{f} F ⊂ K_{∞}}

lifting c, i.e., such that the following diagrams commute

\xymatrix{ & (𝕎_{M}/W_{M})^{G_{F (𝔯)}} \ar[d]^{δ_{F (𝔯)}} & X_{F′(𝔯)} \ar[r]^{d_{F′}} \ar[d]_{N_{F′(𝔯)/F (𝔯)}} & (𝕎_{M}/W_{M})^{G_{F′(𝔯)}} \ar[d]^{N_{F′(𝔯)/F (𝔯)}} \\ X_{F (𝔯)} \ar[r]_{c} \ar[ur]^{d_{F}} & H^{1}(F (𝔯), W_{M}) & X_{F (𝔯)} \ar[r]^{d_{F}} & (𝕎_{M}/W_{M})^{G_{F (𝔯)}}}

where the bottom map on the left sends x_{F (𝔰)} ↦ c_{F (𝔰)} for all 𝔰 dividing 𝔯 as in Lemma 4.2.3, and on the right K ⊂_{f} F ⊂_{f} F′ ⊂ K_{∞}. These conditions determine each d_{F} uniquely up to an element of Hom_{O[G_{K}]}(X_{F (𝔯)}, 𝕎_{M}).

Proof. We first illustrate the proof in a simplified setting. If W_{M}^{G_{F (𝔯)}} = 0, then Proposition 4.4.5(i) becomes a short exact sequence which (abbreviating R = (O/MO)[Gal(F (𝔯)/K)] and X_{F (𝔯)}/M = X_{F (𝔯)}/MX_{F (𝔯)}) induces an exact sequence

0 → Hom_{R}(X_{F (𝔯)}/M, 𝕎_{M}^{G_{F (𝔯)}}) → Hom_{R}(X_{F (𝔯)}/M, (𝕎_{M}/W_{M})^{G_{F (𝔯)}})

⟶^{δ_{F (𝔯)}} Hom_{R}(X_{F (𝔯)}/M, H^{1}(F (𝔯), W_{M})) → Ext_{R}^{1}(X_{F (𝔯)}/M, 𝕎_{M}^{G_{F (𝔯)}}).

By Lemma 4.4.6(i) and Proposition 4.3.4(i), Ext_{R}^{1}(X_{F (𝔯)}/M, 𝕎_{M}^{G_{F (𝔯)}}) = 0, so we can choose a map d_{F} lifting c in this case.

In general, since W_{M}^{G_{F (1)}} may be nonzero, we pass to the limit and use the short exact sequence of Proposition 4.4.7 instead of Proposition 4.4.5(i). Arguing as above, using Lemma 4.4.6(ii) and Propositions 4.4.7 and 4.3.4(ii), and writing Λ_{𝔯} = O[[Gal(K_{∞}(𝔯)/K)]], we obtain an exact sequence

0 ⟶ Hom_{Λ_{𝔯}}(X_{∞,𝔯}/MX_{∞,𝔯}, \underleftarrow{lim}_{F} 𝕎_{M}^{G_{F (𝔯)}})

⟶ Hom_{Λ_{𝔯}}(X_{∞,𝔯}/MX_{∞,𝔯}, \underleftarrow{lim}_{F} (𝕎_{M}/W_{M})^{G_{F (𝔯)}})

⟶^{δ_{𝔯}} Hom_{Λ_{𝔯}}(X_{∞,𝔯}/MX_{∞,𝔯}, H_{∞}^{1}(K(𝔯), W_{M})) ⟶ 0. (4.6)

Therefore there is a map d_{∞} : X_{∞,𝔯} → \underleftarrow{lim}_{F} (𝕎_{M}/W_{M})^{G_{F (𝔯)}} such that

δ_{𝔯} ∘ d_{∞}({x_{F (𝔰)}}_{F}) = {c_{F (𝔰)}}_{F}

for every 𝔰 dividing 𝔯. We define d_{F} to be the composition

X_{F (𝔯)} ⟶^{∼} X_{∞,𝔯} ⊗_{Λ_{𝔯}} O[Gal(F (𝔯)/K)]

⟶^{d_{∞}⊗1} \underleftarrow{lim}_{F′} (𝕎_{M}/W_{M})^{G_{F′(𝔯)}} ⊗_{Λ_{𝔯}} O[Gal(F (𝔯)/K)]

⟶ (𝕎_{M}/W_{M})^{G_{F (𝔯)}}

where the left-hand isomorphism comes from Corollary 4.3.2 and the righthand map is the natural projection. (Explicitly, d_{F} (x_{F (𝔰)}) is the projection of d_{∞}({x_{F′(𝔰)}}) to (𝕎_{M}/W_{M})^{G_{F (𝔯)}}.) It is straightforward to check that these maps have the desired properties. By (4.6), d_{∞} is unique up to an element of Hom_{G_{K}} (X_{∞,𝔯}, \underleftarrow{lim} 𝕎_{M}^{G_{F (𝔯)}}), and it follows that d_{F} is well-defined up to an element of Hom_{O[G_{K}]}(X_{F (𝔯)}, 𝕎_{M}).

Remark 4.4.9. We will only need to use the existence of the maps d_{F} of Proposition 4.4.8 for individual F. The compatibility as F varies (the righthand diagram of the proposition) is needed in order to get the uniqueness portion of the proposition, i.e., to make the map d_{F} well-defined up to an element of Hom_{O[G_{K}]}(X_{F (𝔯)}, 𝕎_{M}).

Definition 4.4.10. Suppose c is an Euler system, M ∈ O is nonzero, K ⊂_{f} F ⊂ K_{∞}, and 𝔯 ∈ R_{F,M}. Fix a map

d = d_{F} : X_{F (𝔯)} ⟶ 𝕎_{M}/W_{M}

in a family lifting c as in Proposition 4.4.8.

Lemma 4.4.2 shows that

d(D_{𝔯,F} x_{F (𝔯)}) ∈ (𝕎_{M}/W_{M})^{G_{F}} ,

where D_{𝔯,F} = N_{F (1)/F} D_{𝔯} is as in Definition 4.4.1, and we define

κ_{[F,𝔯,M]} = δ_{F} (d(D_{𝔯,F} x_{F (𝔯)})) ∈ H^{1}(F, W_{M}).

We can describe this definition with the following diagram

[figure: d(D_{𝔯,F} x_{F (𝔯)}); ∈; (𝕎_{M}/W_{M})^{G_{F (𝔯)}}; δ_{F (𝔯)}; H^{1}(F (𝔯), W_{M}); D_{𝔯,F} c_{F (𝔯)}; ∋; d(D_{𝔯,F} x_{F (𝔯)}); ∈; (𝕎_{M}/W_{M})^{G_{F}}; δ_{F}; H^{1}(F, W_{M}); res; κ_{[F,𝔯,M]}; ∋]

where the commutativity of the inner square is part of Proposition 4.4.5(iii).

Remark 4.4.11. The class κ_{[F,𝔯,M]} is independent of the choice of N_{F (1)/F} used to define D_{𝔯,F}, since by Lemma 4.4.2, D_{𝔯,F} x_{F (𝔯)} ∈ X_{F (𝔯)}/MX_{F (𝔯)} is independent of this choice. The definition of κ_{[F,𝔯,M]} is also independent the choice of d in Proposition 4.4.8. For if d′ is another choice, then d−d′ ∈ Hom_{G_{K}} (X_{F (𝔯)}, 𝕎_{M}), so by Lemma 4.4.2 and Proposition 4.4.5(i),

d(D_{𝔯,F} x_{F (𝔯)}) − d′(D_{𝔯,F} x_{F (𝔯)}) ∈ image((𝕎_{M})^{G_{F}}) = ker(δ_{F}).

Also, note that κ_{[F,𝔯,M]} depends only on the images of the Euler system classes c_{L} in H^{1}(L, W_{M}), not in H^{1}(L, T). However, the extra information in H^{1}(L, T) will be used to prove Theorem 4.5.1 below. See §9.3 for a further discussion in this direction.

The class κ_{[F,𝔯,M]} does depend (because D_{𝔯} does) on the choice of generators σ_{𝔮} of the groups Γ_{𝔮}. Making another set of choices will multiply κ_{[F,𝔯,M]} by a unit in (O/MO)^{×}.

For the next two lemmas, suppose c is an Euler system, M ∈ O is nonzero, K ⊂_{f} F ⊂ K_{∞}, and 𝔯 ∈ R_{F,M} as in Definition 4.4.10.

Lemma 4.4.12. Suppose d : X_{F (𝔯)} → 𝕎_{M}/W_{M} is a lifting of the Euler system c as in Proposition 4.4.8. Let f ∈ 𝕎_{M} be any lifting of d(D_{𝔯,F} x_{F (𝔯)}). Then κ_{[F,𝔯,M]} is represented by the cocycle

γ ↦ (γ − 1)f ∈ W_{M} for γ ∈ G_{F}.

Proof. This is a combination of the definition of κ_{[F,𝔯,M]} above with the explicit description of the connecting map δ_{F} (Proposition 4.4.5(ii)).

Lemma 4.4.13. (i) The class κ_{[F,1,M]} is the image of c_{F} under the

map H^{1}(F, T) → H^{1}(F, W_{M}). (ii) More generally, the restriction of κ_{[F,𝔯,M]} to H^{1}(F (𝔯), W_{M}) is equal

to the image of D_{𝔯,F} c_{F (𝔯)} in H^{1}(F (𝔯), W_{M}).

(iii) If M | M′ and 𝔯 ∈ R_{F,M′} then under the natural maps we have

\xymatrix{H^{1}(F, W_{M′}) \ar[r] & H^{1}(F, W_{M}), & H^{1}(F, W_{M}) \ar[r] & H^{1}(F, W_{M′}). \\ κ_{[F,𝔯,M′]} \ar@{|->}[r] & κ_{[F,𝔯,M]} & κ_{[F,𝔯,M]} \ar@{|->}[r] & (M′/M)κ_{[F,𝔯,M′]}}

Proof. All three assertions follow from Definition 4.4.10. For the first we take 𝔯 = 1, so D_{𝔯,F} = N_{F (1)/F}, and use Proposition 4.4.5(iii) and the Euler system relation Cor_{F (1)/F} (c_{F (1)}) = c_{F}.

4.5. Local Properties of the Derivative Classes

Fix an Euler system c for T. In this section we will state the main results describing the local behavior of the derivative classes κ_{[F,𝔯,M]} of §4.4. We will see (Theorem 4.5.1) that κ_{[F,𝔯,M]} belongs to the Selmer group S^{Σ}(F, W_{M}) where Σ is the set of primes of K dividing p𝔯 (see Remark 1.5.8 for this slight abuse of notation). For our applications it will be crucial to understand (Theorem 4.5.4) the ramification of κ_{[F,𝔯,M]} at primes dividing 𝔯.

The proofs will be given in the remaining sections of this chapter.

Theorem 4.5.1. Suppose that M ∈ O is nonzero, that K ⊂_{f} F ⊂ K_{∞}, and that 𝔯 ∈ R_{F,M}. For every place w of F not dividing p𝔯,

(κ_{[F,𝔯,M]})_{w} ∈ H_{f}^{1}(F_{w}, W_{M}).

In other words,

κ_{[F,𝔯,M]} ∈ S^{Σ_{p𝔯}}(F, W_{M})

where Σ_{p𝔯} is the set of primes of K dividing p𝔯.

Theorem 4.5.1 will be proved in §4.6.

Lemma 4.5.2. Suppose M ∈ O is nonzero and 𝔮 ∈ R_{K,M} is prime. Then there is a unique Q_{𝔮}(x) ∈ (O/MO)[x] such that

P (Fr_{𝔮}^{−1}|T^{∗}; x) ≡ (x − 1)Q_{𝔮}(x) (mod M).

Proof. Take

Q_{𝔮}(x) = \frac{P (Fr_{𝔮}^{−1}|T^{∗}; x) − P (Fr_{𝔮}^{−1}|T^{∗}; 1)}{x − 1}.

Since 𝔮 ∈ R_{K,M}, we know that M divides P (Fr_{𝔮}^{−1}|T^{∗}; 1) so this polynomial has the desired property. The uniqueness comes from the fact that x − 1 is not a zero divisor in (O/MO)[x].

Definition 4.5.3. Suppose M ∈ O is nonzero and 𝔮 ∈ R_{K,M} is prime. The choices of σ_{𝔮} ∈ Γ_{𝔮} (Definition 4.4.1) and Fr_{𝔮} depend on the choice of a prime 𝔔 of K̄ above 𝔮. We use the same choice for both, and we further fix σ̄_{𝔮} in the inertia group of 𝔔 extending σ_{𝔮}.

By Lemma 1.4.7(i) (which applies thanks to Lemma 4.1.2(i)) there are well-defined isomorphisms

α_{𝔮} : H_{s}^{1}(K_{𝔮}, W_{M}) ⟶^{∼} W_{M}^{Fr_{𝔮}=1}
β_{𝔮} : H_{f}^{1}(K_{𝔮}, W_{M}) ⟶^{∼} W_{M}/(Fr_{𝔮} − 1)W_{M}

given on cocycles by

α_{𝔮}(c) = c(σ̄_{𝔮}), β_{𝔮}(c) = c(Fr_{𝔮}).

If 𝔮 ∈ R_{K,M}, then P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}) annihilates W_{M} by Lemma 4.1.2(iv). Thus the polynomial Q_{𝔮} of Lemma 4.5.2 induces a map

Q_{𝔮}(Fr_{𝔮}^{−1}) : W_{M}/(Fr_{𝔮} − 1)W_{M} ⟶ W_{M}^{Fr_{𝔮}=1}.

We define the “finite-singular comparison” map

ϕ_{𝔮}^{fs} : H_{f}^{1}(K_{𝔮}, W_{M}) ⟶ H_{s}^{1}(K_{𝔮}, W_{M})

to be the composition

H_{f}^{1}(K_{𝔮}, W_{M}) ⟶^{β_{𝔮}} W_{M}/(Fr_{𝔮} − 1)W_{M}

⟶^{Q_{𝔮}(Fr_{𝔮}^{−1})} W_{M}^{Fr_{𝔮}=1} ⟶^{α_{𝔮}^{−1}} H_{s}^{1}(K_{𝔮}, W_{M}).

If K ⊂_{f} F ⊂ K_{∞} and 𝔮 ∈ R_{F,M}, then F_{𝔔} = K_{𝔮}, and we can view ϕ_{𝔮}^{fs} as a map from H_{f}^{1}(F_{𝔔}, W_{M}) to H_{s}^{1}(F_{𝔔}, W_{M}). We will still write ϕ_{𝔮}^{fs} in this case, and suppress the dependence on 𝔔.

Theorem 4.5.4. Suppose M ∈ O is nonzero, K ⊂_{f} F ⊂ K_{∞}, 𝔮 is prime, and 𝔯𝔮 ∈ R_{F,M}. If ϕ_{𝔮}^{fs} is the map defined above, and (κ_{[F,𝔯𝔮,M]})_{𝔮}^{s} denotes the image of κ_{[F,𝔯𝔮,M]} in H_{s}^{1}(F_{𝔔}, W_{M}), then

(κ_{[F,𝔯𝔮,M]})_{𝔮}^{s} = ϕ_{𝔮}^{fs}(κ_{[F,𝔯,M]}).

In other words, the singular part of κ_{[F,𝔯𝔮,M]} at 𝔮 is controlled by the (finite) localization of κ_{[F,𝔯,M]} at 𝔮. Theorem 4.5.4 will be proved in §4.7.

Corollary 4.5.5. Suppose M ∈ O is nonzero, 𝔮 is prime, and 𝔯𝔮 ∈ R_{K,M}. Suppose further that W_{M}/(Fr_{𝔮} − 1)W_{M} is free of rank one over O/MO. Then the order of (κ_{[K,𝔯𝔮,M]})_{𝔮}^{s} in H_{s}^{1}(K_{𝔮}, W_{M}) is equal to the order of (κ_{[K,𝔯,M]})_{𝔮} in H_{f}^{1}(K_{𝔮}, W_{M}).

Proof. The maps α_{𝔮} and β_{𝔮} in Definition 4.5.3 are both isomorphisms, and by Lemma 4.1.2(iii) and Corollary A.2.7 (applied with τ = Fr_{𝔮}^{−1} and Q(x) = Q_{𝔮}(x)), so is the map Q_{𝔮}(Fr_{𝔮}^{−1}). Thus ϕ_{𝔮}^{fs} is an isomorphism and the corollary follows from Theorem 4.5.4.

4.6. Local Behavior at Primes Not Dividing p𝔯

For this section fix an Euler system c for T and a nonzero element M ∈ O. If K ⊂_{f} F ⊂ K_{∞} and 𝔯 ∈ R_{F,M}, we need to show that (κ_{[F,𝔯,M]})_{w} ∈ H_{f}^{1}(F_{w}, W_{M}) for every place w of F not dividing p𝔯. When w is archimedean (Lemma 4.6.3), or when w is nonarchimedean and T is unramified at w (Corollary 4.6.2(ii)), this is not difficult. We treat those cases first, and then go on to the general case.

Proposition 4.6.1. Suppose K ⊂_{f} F ⊂ K_{∞} and 𝔯 ∈ R. For every prime Q of F (𝔯) not dividing p, and every γ ∈ G_{K}, we have

(γc_{F (𝔯)})_{Q} ∈ H_{ur}^{1}(F (𝔯)_{Q}, T), (γc̄_{F (𝔯)})_{Q} ∈ H_{f}^{1}(F (𝔯)_{Q}, W_{M})

where c̄_{F (𝔯)} is the image of c_{F (𝔯)} under H^{1}(F (𝔯), T) → H^{1}(F (𝔯), W_{M}).

Proof. Since {γc_{F (𝔯)}}_{F} ∈ H_{∞}^{1}(K(𝔯), T), the first inclusion follows from Corollary B.3.5 and the second from Lemma 1.3.8(i).

Corollary 4.6.2. Suppose K ⊂_{f} F ⊂ K_{∞} and 𝔯 ∈ R_{F,M}. If Q is a prime of F not dividing p𝔯, then

(i) (κ_{[F,𝔯,M]})_{Q} ∈ H_{ur}^{1}(F_{Q}, W_{M}),

(ii) if T is unramified at Q then (κ_{[F,𝔯,M]})_{Q} ∈ H_{f}^{1}(F_{Q}, W_{M}).

Proof. Let D_{𝔯,F} be as in Definition 4.4.10 and write I for an inertia group of Q in G_{F}. Since F (𝔯)/F is unramified at Q we have I ⊂ G_{F (𝔯)}, so by Lemma 4.4.13(ii) the restriction of κ_{[F,𝔯,M]} to I is equal to the image of D_{𝔯,F} c_{F (𝔯)} in H^{1}(I, W_{M}). By Proposition 4.6.1, the latter is zero. This shows that (κ_{[F,𝔯,M]})_{Q} ∈ H_{ur}^{1}(F_{Q}, W_{M}), and if T is unramified at Q then Lemma 1.3.8(ii) shows that H_{f}^{1}(F_{Q}, W_{M}) = H_{ur}^{1}(F_{Q}, W_{M}).

Lemma 4.6.3. Suppose K ⊂_{f} F ⊂ K_{∞} and 𝔯 ∈ R_{F,M}. If w is an infinite place of F, then (κ_{[F,𝔯,M]})_{w} ∈ H_{f}^{1}(F_{w}, W_{M}).

Proof. Let w̃ be a place of F (𝔯) above w. Since F (𝔯)/F ramifies only at primes dividing 𝔯, the place w splits completely in F (𝔯)/F. Thus Lemma 4.4.13(ii) shows that (κ_{[F,𝔯,M]})_{w} is the image of (D_{𝔯,F} c_{F (𝔯)})w̃ under

H^{1}(F (𝔯)w̃, T) = H^{1}(F_{w}, T) ⟶ H^{1}(F_{w}, W_{M}).

By Remark 1.3.7 we have H_{f}^{1}(F_{w}, T) = H^{1}(F_{w}, T), so the lemma follows from Lemma 1.3.8(i).

4.6. LOCAL BEHAVIOR AT PRIMES NOT DIVIDING p𝔯 93

Remark 4.6.4. In the nonarchimedean case, if w is a prime of K not dividing p𝔯, then Corollary 4.6.2(i) shows that (κ_{[F,𝔯,M]})_{w} ∈ H_{ur}^{1}(F_{w}, W_{M}). Unfortunately, for primes w where T is ramified it may not be true that H_{f}^{1}(F_{w}, W_{M}) = H_{ur}^{1}(F_{w}, W_{M}). However, we do get immediately the following corollary, with only a slightly stronger assumption on 𝔯.

Corollary 4.6.5. There is a nonzero m ∈ O, independent of M, such that for every K ⊂_{f} F ⊂ K_{∞}, every 𝔯 ∈ R_{F,Mm}, and every prime Q of F not dividing p𝔯, we have (κ_{[F,𝔯,M]})_{Q} ∈ H_{f}^{1}(F_{Q}, W_{M}).

Proof. Choose m ∈ O such that for every prime 𝔮 of K not dividing p, m annihilates W^{I_{𝔮}}/(W^{I_{𝔮}})_{div}, where I_{𝔮} is an inertia group for 𝔮 in G_{K} and (W^{I_{𝔮}})_{div} is the maximal divisible submodule of W^{I_{𝔮}}. Clearly we can take m to be nonzero, since W^{I_{𝔮}}/(W^{I_{𝔮}})_{div} is always finite and is zero whenever W is unramified at 𝔮.

Suppose K ⊂_{f} F ⊂ K_{∞}. If Q is a prime of F not dividing p, and 𝔮 is the prime of K below Q, then I_{𝔮} is also an inertia group of Q in G_{F}. Therefore Lemma 1.3.5(iii) shows that m annihilates H_{ur}^{1}(F_{Q}, W_{Mm})/H_{f}^{1}(F_{Q}, W_{Mm}), so by Corollary 4.6.2 we have (mκ_{[F,𝔯,Mm]})_{Q} ∈ H_{f}^{1}(F_{Q}, W_{Mm}). Lemma 4.4.13(iii) shows that mκ_{[F,𝔯,Mm]} is the image of κ_{[F,𝔯,M]} in H^{1}(F, W_{Mm}), and the corollary follows.

Corollary 4.6.5 is already strong enough to use in place of Theorem 4.5.1 in proving the theorems of Chapter 2. Thus one could skip the rest of this section if one were so inclined.

To prove Theorem 4.5.1 for primes Q where T may be ramified is much more delicate. We will mimic the construction of κ_{[F,𝔯,M]} locally, and use Proposition 4.6.1 to show that (κ_{[F,𝔯,M]})_{Q} can be constructed inside H^{1}(F_{Q}, T^{I_{Q}}/MT^{I_{Q}}). The theorem will follow directly from this.

Fix for the rest of this section an ideal 𝔯 ∈ R and a prime 𝔮 of K not dividing p𝔯 (but not necessarily in R).

Definition 4.6.6. Fix inertia and decomposition groups I ⊂ D ⊂ G_{K} of 𝔮. If L is a finite extension of K, unramified at 𝔮, let S_{L} denote the set of primes of L above 𝔮 and abbreviate

H^{i}(L_{𝔮}, W_{M}) = ⨁_{Q∈S_{L}} H^{i}(L_{Q}, W_{M}),
H^{i}(L_{𝔮}, T^{I}/MT^{I}) = ⨁_{Q∈S_{L}} H^{i}(L_{Q}, T^{I_{Q}}/MT^{I_{Q}})

where for each Q ∈ S_{L}, we write I_{Q} for the inertia subgroup of G_{L_{Q}}. (Since L/K is unramified at 𝔮, each I_{Q} is conjugate to I.) Write ( · )_{𝔮} or res_{𝔮} : H^{i}(L, W_{M}) → H^{i}(L_{𝔮}, W_{M}) for the sum of the restriction maps. Note that H^{i}(L_{𝔮}, W_{M}) and H^{i}(L_{𝔮}, T^{I}/MT^{I}) are Gal(L/K)-modules: this can be seen directly (every σ ∈ Gal(L/K) induces an isomorphism

H^{i}(L_{Q}, T^{I_{Q}}/MT^{I_{Q}}) ⟶^{∼} H^{i}(L_{σQ}, σ(T^{I_{Q}}/MT^{I_{Q}}))

= H^{i}(L_{σQ}, T^{I_{σQ}}/MT^{I_{σQ}})

for every Q, and summing these maps over Q ∈ S_{L} gives an automorphism of H^{i}(L_{𝔮}, T^{I}/MT^{I}) and similarly for H^{i}(L_{𝔮}, W_{M})), or see Corollary B.5.2.

Write

W_{M}^{f} = T^{I}/MT^{I} ≅ ((W^{I})_{div})_{M} ⊂ (W_{M})^{I} ⊂ W_{M}

and define a G_{K}-submodule 𝕎_{M}^{f} ⊂ 𝕎_{M} by

𝕎_{M}^{f} = Ind(W_{M}^{f}) = Maps(G_{K}, W_{M}^{f}) ⊂ 𝕎_{M}.

As in §B.4, let Ind_{D}(W_{M}) ⊂ 𝕎_{M} denote the G_{K}-submodule of maps satisfying f(hg) = hf(g) for every h ∈ D, and similarly for Ind_{D}(W_{M}^{f}) ⊂ 𝕎_{M}^{f}.

Lemma 4.6.7. If L is a finite extension of K, unramified at 𝔮, then with notation as above we have a natural commutative diagram with exact columns

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ W_{M}^{G_{L}} \ar@{^{(}->}[r] \ar[d] & H^{0}(L_{𝔮}, W_{M}) \ar[d] & H^{0}(L_{𝔮}, W_{M}^{f}) \ar@{^{(}->}[l] \ar[d] \\ 𝕎_{M}^{G_{L}} \ar[r]^{∼} \ar[d] & 𝕎_{M}^{G_{L}} \ar[d] & (𝕎_{M}^{f})^{G_{L}} \ar[l] \ar[d] \\ (𝕎_{M}/W_{M})^{G_{L}} \ar[r] \ar[d]_{δ_{L}} & (𝕎_{M}/Ind_{D}(W_{M}))^{G_{L}} \ar[d]_{δ_{L𝔮}} & (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{L}} \ar[l] \ar[d]_{δ_{L𝔮,W{f \atop M}}} \\ H^{1}(L, W_{M}) \ar[r]^{_{res_{𝔮}}} \ar[d] & H^{1}(L_{𝔮}, W_{M}) \ar[d] & H^{1}(L_{𝔮}, W_{M}^{f}) \ar[l] \ar[d] \\ 0 & 0 & 0.}

Proof. The three columns come from G_{L}-cohomology of the short exact sequences

0 ⟶ W_{M} ⟶ 𝕎_{M} ⟶ 𝕎_{M}/W_{M} ⟶ 0
0 ⟶ Ind_{D}(W_{M}) ⟶ 𝕎_{M} ⟶ 𝕎_{M}/Ind_{D}(W_{M}) ⟶ 0
0 ⟶ Ind_{D}(W_{M}^{f}) ⟶ 𝕎_{M}^{f} ⟶ 𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}) ⟶ 0

respectively (the left-hand column is Proposition 4.4.5(i)), using Corollaries B.4.4 and B.5.2. The horizontal arrows are the natural ones, and the commutativity follows from the functoriality of all the maps involved.

We now need a local analogue of Proposition 4.4.8. If K ⊂_{f} F ⊂ K_{∞}, Q ∈ S_{F (𝔯)}, and 𝔰 | 𝔯, then by Proposition 4.6.1,

(c_{F (𝔰)})_{Q} ∈ H_{ur}^{1}(F (𝔰)_{Q}, T) = H^{1}(F (𝔰)_{Q}^{ur}/F (𝔰)_{Q}, T^{I_{Q}}) ⊂ H^{1}(F (𝔰)_{Q}, T^{I_{Q}}) (4.7)

so (c_{F (𝔰)})_{𝔮} maps naturally to H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}).

Proposition 4.6.8. Suppose c is an Euler system and 𝔯 ∈ R. There are two families of O[G_{K}]-module maps

{d_{F,𝔮} : X_{F (𝔯)} → (𝕎_{M}/Ind_{D}(W_{M}))^{G_{F (𝔯)}} : K ⊂_{f} F ⊂ K_{∞}}
{d_{F,𝔮}^{f} : X_{F (𝔯)} → (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}} : K ⊂_{f} F ⊂ K_{∞}}

lifting c, i.e., such that if K ⊂_{f} F ⊂_{f} F′ ⊂ K_{∞} then

(i) the maps d_{F,𝔮} (resp d_{F,𝔮}^{f}) are compatible with respect to the norm maps

X_{F′(𝔯)} → X_{F (𝔯)}, (𝕎_{M}/Ind_{D}(W_{M}))^{G_{F′(𝔯)}} → (𝕎_{M}/Ind_{D}(W_{M}))^{G_{F (𝔯)}},

(𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F′(𝔯)}} → (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}},

(ii) for every K ⊂_{f} F ⊂ K_{∞} and every 𝔰 dividing 𝔯, the compositions

X_{F (𝔯)} ⟶^{d_{F,𝔮}} (𝕎_{M}/Ind_{D}(W_{M}))^{G_{F (𝔯)}} ⟶^{δ_{F (𝔯)𝔮}} H^{1}(F (𝔯)_{𝔮}, W_{M})

X_{F (𝔯)} ⟶^{d_{F,𝔮}^{f}} (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}} ⟶^{δ_{F (𝔯)𝔮,W{f \atop M}}} H^{1}(F (𝔯)_{𝔮}, W_{M}^{f})

both (using (4.7) for the latter) send x_{F (𝔰)} to (c_{F (𝔰)})_{𝔮}.

These two conditions determine every d_{F,𝔮} (resp. d_{F,𝔮}^{f}) uniquely up to an element of Hom_{O[G_{K}]}(X_{F (𝔯)}, 𝕎_{M}) (resp. Hom_{O[G_{K}]}(X_{F (𝔯)}, 𝕎_{M}^{f})).

Proof. For each K ⊂_{f} F ⊂ K_{∞} we have maps (see Lemma 4.2.3)

x_{F (𝔰)} ↦  c_{F (𝔰)}  ↦  (c_{F (𝔰)})_{𝔮}
X_{F (𝔯)} ⟶ H^{1}(F (𝔯), T) ⟶ H^{1}(F (𝔯)_{𝔮}, T)
↓  ↓
H^{1}(F (𝔯), W_{M}) ⟶ H^{1}(F (𝔯)_{𝔮}, W_{M}).

and by (4.7) the composition X_{F (𝔯)} → H^{1}(F (𝔯)_{𝔮}, W_{M}) factors through a G_{K}-equivariant map

X_{F (𝔯)} ⟶ H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}).  (4.8)

To prove the proposition we need to lift these to maps

X_{F (𝔯)} ⟶ 𝕎_{M}/Ind_{D}(W_{M}), X_{F (𝔯)} ⟶ 𝕎_{M}^{f}/Ind_{D}(W_{M}^{f})

in the center and right-hand columns, respectively, of the diagram of Lemma 4.6.7 with L = F (𝔯). We will do this by mimicking the proof of Proposition 4.4.8. We describe the proof only for the right-hand column; the other proof is exactly the same (and see Remark 4.6.9 below).

Since we have assumed that the decomposition group of 𝔮 in K_{∞}/K is infinite, we can find a Z_{p}-extension K′_{∞} of K in K_{∞} such that K′_{∞} has only finitely many primes above 𝔮. Then for each finite extension L of K we see that \underrightarrow{lim}_{K ⊂_{f} F ⊂K′_{∞}L} H^{0}(F (𝔯)_{𝔮}, W_{M}^{f}) is finite, so by Lemma B.3.2,

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞} H^{0}(F (𝔯)_{𝔮}, W_{M}^{f}) = \underleftarrow{lim}_{K ⊂_{f} L⊂K∞} \underleftarrow{lim}_{KL ⊂_{f} F ⊂K′_{∞}L} H^{0}(F (𝔯)_{𝔮}, W_{M}^{f}) = 0

(inverse limits with respect to the norm maps). Proposition B.2.7(ii) shows that each H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}) is finite, so exactly as in Proposition 4.4.7 the inverse limit over K ⊂_{f} F ⊂ K_{∞} of the right-hand column of the diagram of Lemma 4.6.7 is a short exact sequence

0 ⟶ \underleftarrow{lim}_{F} (𝕎_{M}^{f})^{G_{F (𝔯)}} ⟶ \underleftarrow{lim}_{F} (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}}

⟶ \underleftarrow{lim}_{F} H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}) ⟶ 0.

Taking the inverse limit over F of (4.8) yields a map

X_{∞,𝔯} ⟶ \underleftarrow{lim}_{F} H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}),  (4.9)

and exactly as in Lemma 4.4.6 we see that the (O/MO)[[Gal(K_{∞}(𝔯)/K)]]- module \underleftarrow{lim} (𝕎_{M}^{f})^{G_{F (𝔯)}} is free of finite rank. Just as in Proposition 4.4.8, Proposition 4.3.4 now shows that (4.9) lifts to a map

X_{∞,𝔯} ⟶ \underleftarrow{lim}_{F} (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}}.

Also as in Proposition 4.4.8, Corollary 4.3.2 shows that this in turn induces maps

d_{F,𝔮}^{f} : X_{F (𝔯)} ⟶ (𝕎_{M}^{f}/Ind_{D}(W_{M}^{f}))^{G_{F (𝔯)}}

having the desired properties. The uniqueness is clear from the diagram of Lemma 4.6.7.

Remark 4.6.9. To construct the maps d_{F,𝔮} in Proposition 4.6.8 it suffices to construct either the global maps d_{F} of Proposition 4.4.8 or the “unramified” maps d_{F,𝔮}^{f} of Proposition 4.6.8 and then compose them with the appropriate map to (𝕎_{M}/Ind_{D}(W_{M}))^{G_{F (𝔯)}} in the diagram of Lemma 4.6.7.

In fact, that is how our proof of Theorem 4.5.1 will proceed. We construct the maps d_{F} and d_{F,𝔮}^{f} lifting our Euler system c. This gives us two different constructions of d_{F,𝔮} and we compare them using the uniqueness assertion of Proposition 4.6.8.

Proof of Theorem 4.5.1. Keep the notation from the beginning of this section, so M ∈ O is nonzero and we now suppose that 𝔯 ∈ R_{F,M}. Fix a prime 𝔮 of K, not necessarily in R. Fix a lifting d : X_{F (𝔯)} → 𝕎_{M}/W_{M} (resp. d_{𝔮}^{f} : X_{F (𝔯)} → 𝕎_{M}^{f}/Ind_{D}(W_{M}^{f})) of c as in Proposition 4.4.8 (resp. Proposition 4.6.8). Write d_{𝔮} (resp d′_{𝔮}) for the image of d (resp d_{𝔮}^{f}) in Hom(X_{F (𝔯)}, 𝕎_{M}/Ind_{D}(W_{M})) in the diagram of Lemma 4.6.7. From the uniqueness portion of Proposition 4.6.8 it follows that

d_{𝔮} − d′_{𝔮} ∈ image(Hom(X_{F (𝔯)}, 𝕎_{M}^{G_{F (𝔯)}})).

In particular, Lemma 4.4.2 shows that, in the center column of the diagram of Lemma 4.6.7 with L = F, we have

d_{𝔮}(D_{𝔯,F} x_{F (𝔯)}) − d′_{𝔮}(D_{𝔯,F} x_{F (𝔯)}) ∈ image(𝕎_{M}^{G_{F}}) = ker(δ_{F_{𝔮}}).

By definition κ_{[F,𝔯,M]} = δ_{F} (d(D_{𝔯,F} x_{F (𝔯)})), so the diagram of Lemma 4.6.7 shows that (κ_{[F,𝔯,M]})_{𝔮} is equal to the image of κ_{[F,𝔯,M]}^{(𝔮,f)} in H^{1}(F_{𝔮}, W_{M}), where

κ_{[F,𝔯,M]}^{(𝔮,f)} = δ_{F_{𝔮},W_{M}^{f}} (d_{𝔮}^{f}(D_{𝔯,F} x_{F (𝔯)})) ∈ H^{1}(F_{𝔮}, W_{M}^{f}).

As in Lemma 4.4.13, the restriction of κ_{[F,𝔯,M]}^{(𝔮,f)} to H^{1}(F (𝔯)_{𝔮}, W_{M}^{f}) is the image of (D_{𝔯,F} c_{F (𝔯)})_{𝔮} under the map

H^{1}(F (𝔯)_{𝔮}, T^{I}) ⟶ H^{1}(F (𝔯)_{𝔮}, W_{M}^{f})

(using (4.7) to view (D_{𝔯,F} c_{F (𝔯)})_{𝔮} ∈ H^{1}(F (𝔯)_{𝔮}, T^{I})). Suppose Q is a prime of F above 𝔮, and I_{Q} is an inertia group of Q in G_{F}. Then I_{Q} ⊂ G_{F (𝔯)}, and as in Proposition 4.6.1 the restriction of (D_{𝔯,F} c_{F (𝔯)})_{𝔮} to I_{Q} is zero. We conclude that the restriction of κ_{[F,𝔯,M]}^{(𝔮,f)} to I_{Q} is zero and hence

(κ_{[F,𝔯,M]}^{(𝔮,f)})_{Q} ∈ H^{1}(F_{Q}^{ur}/F_{Q}, T^{I_{Q}}/MT^{I_{Q}}).

Since Gal(F_{Q}^{ur}/F_{Q}) has cohomological dimension one, taking Gal(F_{Q}^{ur}/F_{Q})- cohomology of the short exact sequence

0 ⟶ T^{I_{Q}} ⟶^{M} T^{I_{Q}} ⟶ T^{I_{Q}}/MT^{I_{Q}} ⟶ 0

gives a surjective map

H_{ur}^{1}(F_{Q}, T) = H^{1}(F_{Q}^{ur}/F_{Q}, T^{I_{Q}}) ↠ H^{1}(F_{Q}^{ur}/F_{Q}, T^{I_{Q}}/MT^{I_{Q}}).

Thus we conclude finally that (κ_{[F,𝔯,M]})_{Q} lies in the image of H_{ur}^{1}(F_{Q}, T), so by Lemmas 1.3.5(ii) and 1.3.8(i),

(κ_{[F,𝔯,M]})_{Q} ∈ H_{f}^{1}(F_{Q}, W_{M}).

4.7. Local Behavior at Primes Dividing 𝔯

Fix for this section an Euler system c for T, a nonzero M ∈ O, an ideal 𝔯 ∈ R, a prime 𝔮 ∈ R (which may or may not divide 𝔯), and a finite extension F of K in K_{∞}.

Fix a prime 𝔔 of K̄ above 𝔮 and let I ⊂ D be the inertia and decomposition groups, respectively, of 𝔔 in G_{K}. Since K(𝔮)/K(1) is totally ramified at 𝔮, the natural map I → Γ_{𝔮} is surjective, so we can choose a lift of σ_{𝔮} to I which we will also denote by σ_{𝔮}. With this choice we will view

N_{𝔮} = ∑_{i=0}^{[K(𝔮):K(1)]−1} σ_{𝔮}^{i}, D_{𝔮} = ∑_{i=0}^{[K(𝔮):K(1)]−1} iσ_{𝔮}^{i} ∈ Z[I].

However, writing m = [K(𝔮) : K(1)], we no longer have σ_{𝔮}^{m} = 1 in I, so instead of the identity (4.4) we have

(σ_{𝔮} − 1)D_{𝔮} = mσ_{𝔮}^{m} − N_{𝔮}  (4.10)

in Z[I]. Fix also some choice Fr_{𝔮} ∈ D of Frobenius for 𝔔, and fix a lift of the element N_{F (1)/F} of Definition 4.4.10 to Z[G_{F}], so that we can view D_{𝔯,F} ∈ Z[G_{F}].

Lemma 4.7.1. Suppose d : X_{F (𝔯)} → 𝕎_{M}/W_{M} is a lifting of c as in Proposition 4.4.8, and d̂(x_{F (𝔯)}) ∈ 𝕎_{M} is a lifting of d(x_{F (𝔯)}). Then for every γ ∈ G_{K} and ρ, ρ′ ∈ D,

ρρ′γd̂(x_{F (𝔯)}) = ρ′ργd̂(x_{F (𝔯)}).

4.7. LOCAL BEHAVIOR AT PRIMES DIVIDING 𝔯 99

Proof. Let I_{F (𝔯)} = I ∩ G_{F (𝔯)}. Since T is unramified at 𝔮, Proposition 4.6.1 shows that

res_{I_{F (𝔯)}}(γc_{F (𝔯)}) = 0 in H^{1}(I_{F (𝔯)}, T) = Hom(I_{F (𝔯)}, T).

Thus every cocycle representing γc_{F (𝔯)} vanishes on I_{F (𝔯)}. In particular by Proposition 4.4.5(ii),

(σ − 1)γd̂(x_{F (𝔯)}) = 0 in 𝕎_{M}, for every σ ∈ I_{F (𝔯)}.  (4.11)

Since D/I and Gal(F (𝔯)/K) are abelian, the commutator subgroup of D is contained in both I and G_{F (𝔯)}. In particular if we apply (4.11) with σ = ρ^{−1}ρ′^{−1}ρρ′ ∈ I_{F (𝔯)}, the lemma follows.

Remark 4.7.2. Suppose that ρ and ρ′ belong to G_{K}, but not necessarily to D. Then ρρ′d(x_{F (𝔯)}) = ρ′ρd(x_{F (𝔯)}), because d is G_{K}-equivariant and the action of G_{K} on x_{F (𝔯)} factors through an abelian extension of K. However, the action of G_{K} on d̂(x_{F (𝔯)}) will not in general factor through an abelian extension of K so it is not in general true that ρρ′d̂(x_{F (𝔯)}) = ρ′ρd̂(x_{F (𝔯)}). However, Lemma 4.7.1 shows that this does hold if ρ, ρ′ ∈ D. We will use this repeatedly below.

Note that Lemma 4.7.1 is true whether or not 𝔮 divides 𝔯.

Theorem 4.5.4 which will follow easily from the following lemma.

Lemma 4.7.3. Suppose that 𝔮 ∈ R_{K,M} and that 𝔮 does not divide 𝔯. Fix a lifting d : X_{F (𝔯𝔮)} → 𝕎_{M}/W_{M} of c as in Proposition 4.4.8, and fix liftings d̂(x_{F (𝔯𝔮)}), d̂(x_{F (𝔯)}) ∈ 𝕎_{M} of d(x_{F (𝔯𝔮)}) and d(x_{F (𝔯)}), respectively. Then for every γ ∈ G_{K},

N_{𝔮}γd̂(x_{F (𝔯𝔮)}) = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})γd̂(x_{F (𝔯)}).

Proof. We will abbreviate P_{𝔮}(x) = P (Fr_{𝔮}^{−1}|T^{∗}; x). Note that

N_{𝔮}γd(x_{F (𝔯𝔮)}) = P_{𝔮}(Fr_{𝔮}^{−1})γd(x_{F (𝔯)})

since d is G_{K}-equivariant and N_{𝔮}x_{F (𝔯𝔮)} = P_{𝔮}(Fr_{𝔮}^{−1})x_{F (𝔯)}, so

N_{𝔮}γd̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂(x_{F (𝔯)}) ∈ W_{M}.

First we will show that N_{𝔮}γd̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂(x_{F (𝔯)}) is independent of the choices of d and d̂. Suppose we replace d by some other choice d′. By Proposition 4.4.8, d′ = d + d_{0} with d_{0} ∈ Hom_{G_{K}} (X_{F (𝔯𝔮)}, 𝕎_{M}). Therefore if we choose liftings d̂′(x_{F (𝔯𝔮)}) and d̂′(x_{F (𝔯)}) ∈ 𝕎_{M} of d′(x_{F (𝔯𝔮)}) and d′(x_{F (𝔯)}), respectively, they must satisfy

d̂′(x_{F (𝔯𝔮)}) = d̂(x_{F (𝔯𝔮)}) + d_{0}(x_{F (𝔯𝔮)}) + t,
d̂′(x_{F (𝔯)}) = d̂(x_{F (𝔯)}) + d_{0}(x_{F (𝔯)}) + t′

where t, t′ ∈ W_{M}. Thus, since d_{0} is G_{K}-equivariant,

(N_{𝔮}γd̂′(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂′(x_{F (𝔯)}))
− (N_{𝔮}γd̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂(x_{F (𝔯)}))

= d_{0}(γ(N_{𝔮}x_{F (𝔯𝔮)} − P_{𝔮}(Fr_{𝔮}^{−1})x_{F (𝔯)})) + N_{𝔮}γt − P_{𝔮}(Fr_{𝔮}^{−1})γt′

= N_{𝔮}γt − P_{𝔮}(Fr_{𝔮}^{−1})γt′.

Since σ_{𝔮} fixes W_{M}, and M divides [K(𝔮) : K(1)] (because 𝔮 ∈ R_{K,M}), we see that N_{𝔮} annihilates W_{M}. By Lemma 4.1.2(iv), P_{𝔮}(Fr_{𝔮}^{−1}) annihilates W_{M} as well. Therefore N_{𝔮}γt − P_{𝔮}(Fr_{𝔮}^{−1})γt′ = 0 as desired.

Next we will make useful choices of d̂(x_{F (𝔯)}) and d̂(x_{F (𝔯𝔮)}). Choose k ∈ Z^{+} so that Fr_{𝔮}^{k} is the identity on both F (𝔯𝔮) and W_{M}, and let k_{p} be the largest power of p dividing k. Since the decomposition group of 𝔮 in Gal(K_{∞}/K) is infinite, we can fix a finite extension F′ of F in K_{∞} such that the decomposition group of 𝔮 in F′(𝔯𝔮)/F (𝔯𝔮) has order divisible by k_{p}M. Choose a lifting d : X_{F′(𝔯𝔮)} → 𝕎_{M}/W_{M} of c as in Proposition 4.4.8.

Let H ⊂ Gal(F′(𝔯𝔮)/F (𝔯𝔮)) be the subgroup generated by Fr_{𝔮}^{k}. Fix a set B ⊂ G_{F (𝔯𝔮)} of coset representatives of Gal(F′(𝔯𝔮)/F (𝔯𝔮))/H. Write

N′ = ∑_{i=0}^{|H|−1} Fr_{𝔮}^{ki}, N′′ = ∑_{β∈B} β ∈ Z[G_{F (𝔯𝔮)}].

The product N′N′′ restricts to the norm from F′(𝔯𝔮) to F (𝔯𝔮), so in particular

N′N′′x_{F′(𝔯𝔮)} = x_{F (𝔯𝔮)} and N′N′′x_{F′(𝔯)} = x_{F (𝔯)}  (4.12)

in X_{F′(𝔯𝔮)}.

Choose liftings d̂(x_{F′(𝔯𝔮)}), d̂(x_{F′(𝔯)}) ∈ 𝕎_{M} of d(x_{F′(𝔯𝔮)}), d(x_{F′(𝔯)}) ∈ 𝕎_{M}/W_{M}, respectively, and define

d̂(x_{F (𝔯𝔮)}) = γ^{−1}N′N′′γd̂(x_{F′(𝔯𝔮)}), d̂(x_{F (𝔯)}) = γ^{−1}N′N′′γd̂(x_{F′(𝔯)}).

It follows from (4.12) that these are liftings of d(x_{F (𝔯𝔮)}) and d(x_{F (𝔯)}), respectively, to 𝕎_{M}. We will prove the lemma by showing that with these choices, N_{𝔮}γd̂(x_{F (𝔯)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂(x_{F (𝔯𝔮)}) = 0.

Note that N′, P_{𝔮}(Fr_{𝔮}^{−1}), and N_{𝔮} all belong to O[D] because Fr_{𝔮} and σ_{𝔮} do, so by Lemma 4.7.1 these elements commute in their action on N′′γd̂(x_{F′(𝔯𝔮)}) and N′′γd̂(x_{F′(𝔯)}). Thus

N_{𝔮}γd̂(x_{F (𝔯)}) − P_{𝔮}(Fr_{𝔮}^{−1})γd̂(x_{F (𝔯𝔮)})
= N_{𝔮}N′N′′γd̂(x_{F′(𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})N′N′′γd̂(x_{F′(𝔯)})

= N′(N_{𝔮}N′′γd̂(x_{F′(𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})N′′γd̂(x_{F′(𝔯)})) ∈ N′W_{M}, the last inclusion because N_{𝔮}N′′γd̂(x_{F′(𝔯𝔮)})−P_{𝔮}(Fr_{𝔮}^{−1})N′′γd̂(x_{F′(𝔯)}) ∈ 𝕎_{M} projects to N′′γd(N_{𝔮}x_{F′(𝔯𝔮)} − P_{𝔮}(Fr_{𝔮}^{−1})x_{F′(𝔯)}) = 0 in 𝕎_{M}/W_{M}. Since Fr_{𝔮}^{k} fixes W_{M}, we have

N′W_{M} ⊂ |H| W_{M}.

Now observe that H has index dividing k_{p} in the decomposition group of 𝔮 in F′(𝔯𝔮)/F (𝔯𝔮), so in particular M divides |H|. This completes the proof.

Remark 4.7.4. The proof of Lemma 4.7.3 used in an essential way the Euler system classes c_{F′(𝔯𝔮)} and c_{F′(𝔯)} for F ⊂_{f} F′ ⊂ K_{∞}, and not just c_{F (𝔯𝔮)} and c_{F (𝔯)}.

Proof of Theorem 4.5.4. Keep the notation from the beginning of this section, and suppose now that 𝔯𝔮 ∈ R_{F,M}. Choose Q_{𝔮} ∈ O[x] as in Lemma 4.5.2, so that Q_{𝔮}(x)(x − 1) ≡ P (Fr_{𝔮}^{−1}|T^{∗}; x) (mod M) . To prove the theorem we need to show that, for some (or equivalently, for every) choice of cocycles representing κ_{[F,𝔯,M]} and κ_{[F,𝔯𝔮,M]}, we have

Q_{𝔮}(Fr_{𝔮}^{−1})κ_{[F,𝔯,M]}(Fr_{𝔮}) = κ_{[F,𝔯𝔮,M]}(σ_{𝔮}) ∈ W_{M}.

Fix d : X_{F (𝔯𝔮)} → 𝕎_{M}/W_{M} lifting c as in Proposition 4.4.8, and choose liftings d̂(x_{F (𝔯)}), d̂(x_{F (𝔯𝔮)}) ∈ 𝕎_{M} of d(x_{F (𝔯)}), d(x_{F (𝔯𝔮)}) ∈ 𝕎_{M}/W_{M}, respectively. Lemma 4.4.12 shows that

κ_{[F,𝔯,M]}(Fr_{𝔮}) = (Fr_{𝔮} − 1)D_{𝔯,F} d̂(x_{F (𝔯)}) ∈ W_{M},
κ_{[F,𝔯𝔮,M]}(σ_{𝔮}) = (σ_{𝔮} − 1)D_{𝔮}D_{𝔯,F} d̂(x_{F (𝔯𝔮)}) ∈ W_{M}.

Also

Q_{𝔮}(Fr_{𝔮}^{−1})(Fr_{𝔮}^{−1} − 1)κ_{[F,𝔯,M]}(Fr_{𝔮}) = P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})κ_{[F,𝔯,M]}(Fr_{𝔮}) = 0

by Lemma 4.1.2(iv). Thus, using Lemma 4.7.1 repeatedly to commute elements of O[D], and using (4.10), we see

Q_{𝔮}(Fr_{𝔮}^{−1})κ_{[F,𝔯,M]}(Fr_{𝔮}) − κ_{[F,𝔯𝔮,M]}(σ_{𝔮})
= Q_{𝔮}(Fr_{𝔮}^{−1})Fr_{𝔮}^{−1}κ_{[F,𝔯,M]}(Fr_{𝔮}) − κ_{[F,𝔯𝔮,M]}(σ_{𝔮})

= Q_{𝔮}(Fr_{𝔮}^{−1})Fr_{𝔮}^{−1}(Fr_{𝔮} − 1)D_{𝔯,F} d̂(x_{F (𝔯)}) − (σ_{𝔮} − 1)D_{𝔮}D_{𝔯,F} d̂(x_{F (𝔯𝔮)})

= −P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})D_{𝔯,F} d̂(x_{F (𝔯)}) + N_{𝔮}D_{𝔯,F} d̂(x_{F (𝔯𝔮)})
− [K(𝔮) : K(1)]σ_{𝔮}^{[K(𝔮):K(1)]}D_{𝔯,F} d̂(x_{F (𝔯𝔮)}).

Since 𝔮 ∈ R_{F,M} we have M | [K(𝔮) : K(1)]. Thus by Lemma 4.7.3 we conclude that Q_{𝔮}(Fr_{𝔮}^{−1})κ_{[F,𝔯,M]}(Fr_{𝔮}) − κ_{[F,𝔯𝔮,M]}(σ_{𝔮}) = 0 in W_{M}, as desired.

4.8. The Congruence

Although we will not need it, we can now prove the following corollary (the “congruence condition” for an Euler system) which was promised in Remark 2.1.5. We again write P_{𝔮}(x) = P (Fr_{𝔮}^{−1}|T^{∗}; x).

Corollary 4.8.1. Suppose that c is an Euler system for T and 𝔮 ∈ R is prime. If K ⊂_{f} F ⊂ K_{∞} and 𝔯𝔮 ∈ R, then for every prime Q of F (𝔯𝔮) dividing 𝔮,

(c_{F (𝔯𝔮)})_{Q} = \frac{P_{𝔮}(Fr_{𝔮}^{−1}) − P_{𝔮}(N(𝔮)Fr_{𝔮}^{−1})}{[K(𝔮) : K(1)]} (c_{F (𝔯)})_{Q} ∈ H^{1}(F (𝔯𝔮)_{Q}, T).

Proof. Write

R(x) = \frac{P_{𝔮}(x) − P_{𝔮}(N(𝔮)x)}{[K(𝔮) : K(1)]}.

Since [K(𝔮) : K(1)] divides (N(𝔮) − 1), we see that R(x) ∈ O[x].

Keep the notation and setting from the beginning of the previous section, and let

c = c_{F (𝔯𝔮)} − R(Fr_{𝔮}^{−1})c_{F (𝔯)} ∈ H^{1}(F (𝔯𝔮), T).

For every nonzero M ∈ O let (c)_{Q,M} be the image of c in H^{1}(F (𝔯𝔮)_{Q}, W_{M}). Proposition B.2.3 shows that H^{1}(F (𝔯𝔮)_{Q}, T) = \underleftarrow{lim} H^{1}(F (𝔯𝔮)_{Q}, W_{M}), so to prove the corollary it will suffice to show that (c)_{Q,M} = 0 for every M.

Fix an M divisible by [K(𝔮) : K(1)], and a lifting

d : X_{F (𝔯𝔮)} → 𝕎_{M}/W_{M}

of c as in Proposition 4.4.8. Choose d̂(x_{F (𝔯)}), d̂(x_{F (𝔯𝔮)}) ∈ 𝕎_{M} lifting d(x_{F (𝔯)}), d(x_{F (𝔯𝔮)}) ∈ 𝕎_{M}/W_{M}, respectively. Fix a Frobenius element Fr_{𝔮} of 𝔮 corresponding to a prime of K̄ above Q. Then a Frobenius element for Q in G_{F (𝔯𝔮)} is given by φ = Fr_{𝔮}^{k} for some k. By Proposition 4.6.1, (c)_{Q,M} ∈ H_{ur}^{1}(F (𝔯𝔮)_{Q}, W_{M}), and by Lemma 1.3.2(i) there is an isomorphism

\begin{matrix}H_{ur}^{1}(F (𝔯𝔮)_{Q}, W_{M}) & ⟶^{∼} & W_{M}/(φ − 1)W_{M} \\ (c)_{Q,M} & ↦ & c(φ). \end{matrix}

Proposition 4.4.5(ii) shows that

γ ↦ (γ − 1)(d̂(x_{F (𝔯𝔮)}) − R(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)})) ∈ W_{M}

is a cocycle representing (c)_{Q,M}, so

(c)_{Q,M} = 0 ⟺ (φ − 1)(d̂(x_{F (𝔯𝔮)}) − R(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)})) ∈ (φ − 1)W_{M}.

Note that

(φ − 1)d̂(x_{F (𝔯)}), (φ − 1)d̂(x_{F (𝔯𝔮)}) ∈ W_{M}

and

N_{𝔮}, φ − 1, P_{𝔮}(Fr_{𝔮}^{−1}) ∈ O[D].

Therefore

[K(𝔮) : K(1)](φ − 1)(d̂(x_{F (𝔯𝔮)}) − R(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)}))
= N_{𝔮}(φ − 1)d̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})(φ − 1)d̂(x_{F (𝔯)})
= (φ − 1)(N_{𝔮}d̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)}))

the first equality since σ_{𝔮} fixes W_{M} and P_{𝔮}(N(𝔮)Fr_{𝔮}^{−1}) annihilates W_{M} (Lemma 4.1.2(ii)), and the second by Lemma 4.7.1. Lemma 4.7.3 shows that the image of N_{𝔮}d̂(x_{F (𝔯𝔮)}) − P_{𝔮}(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)}) under the projection W_{M} ↠ W_{[K(𝔮):K(1)]} is zero, and we conclude that

[K(𝔮) : K(1)](φ − 1)(d̂(x_{F (𝔯𝔮)}) − R(Fr_{𝔮}^{−1})d̂(x_{F (𝔯)})) ∈ [K(𝔮) : K(1)](φ − 1)W_{M}.

It follows that (c)_{Q,M/[K(𝔮):K(1)]} = 0, and since this holds for every M, the corollary is proved.

Example 4.8.2. Suppose T = Z_{p}(1). Then for every 𝔯 ∈ R and every prime Q of F (𝔯) not dividing p (see Example 1.2.1)

H^{1}(F (𝔯), T) = (F (𝔯)^{×})ˆ , H^{1}(F (𝔯)_{Q}, T) = (F (𝔯)_{Q}^{×})ˆ ≅ (𝕜_{Q}^{×})ˆ

where ( · )ˆ denotes the p-adic completion and 𝕜_{Q} is the residue field of F (𝔯) modulo Q. In this case P_{𝔮}(x) = det(1 − Fr_{𝔮}^{−1}x|Z_{p}) = 1 − x, so

\frac{P_{𝔮}(Fr_{𝔮}^{−1}) − P_{𝔮}(N(𝔮)Fr_{𝔮}^{−1})}{[K(𝔮) : K(1)]} = \frac{N(𝔮) − 1}{[K(𝔮) : K(1)]}Fr_{𝔮}^{−1}.

Thus viewing c_{F (𝔯)}, c_{F (𝔯𝔮)} ∈ (F (𝔯)^{×})ˆ , Corollary 4.8.1 in this case says

c_{F (𝔯𝔮)} = c_{F (𝔯)}^{\frac{N(𝔮)−1}{[K(𝔮):K(1)]} Fr_{𝔮}^{−1}} in (𝕜_{Q}^{×})ˆ .  (4.13)

For the Euler system of cyclotomic units discussed in §3.2, (4.13) is the “p-part” of the congruence

1 − ζ_{rq} ≡ 1 − ζ_{r}^{Fr_{q}^{−1}} modulo every prime of Q̄ above q

(which in turn follows from the observation that ζ_{q} ≡ 1 modulo every prime above q).

CHAPTER 5 Bounding the Selmer Group

In this chapter we will prove Theorems 2.2.2 (in §5.2) and 2.2.3 (in §5.3). For every power M of p we will choose inductively a finite subset Σ of primes in R_{K,M}. As 𝔯 runs through products of primes in Σ, Theorem 4.5.1 shows that the derivative cohomology classes κ_{[K,𝔯,M]} defined in Chapter 4 belong to S^{Σ∪Σ_{p}}(K, W_{M}), where Σ_{p} is the set of primes of K above p, and Theorem 4.5.4 tells us about the singular parts of these classes at primes in Σ. This information and the duality results of §1.7 will allow us to bound the index [S_{Σ_{p}}(K, W_{M}^{∗}) : S_{Σ∪Σ_{p}}(K, W_{M}^{∗})]. By taking Σ large enough so that S_{Σ∪Σ_{p}}(K, W_{M}^{∗}) = 0, and letting M go to infinity, we will obtain the bound of Theorem 2.2.2.

5.1. Preliminaries

Keep the notation of §2.1 and §2.2. Fix an Euler system c for (T, K, N) for some K and N. If M is a power of p we will write R_{M} = R_{K,M} (as defined in Definition 4.1.1), the set of ideals in R divisible only by primes 𝔮 such that 𝔮 ∤ N, both [K(𝔮) : K(1)] and P (Fr_{𝔮}^{−1}|T^{∗}; 1) are divisible by M, and 𝔮 splits completely in K(1). If 𝔯 ∈ R_{M} then κ_{[𝔯,M]} ∈ H^{1}(K, W_{M}) will denote the derivative class κ_{[K,𝔯,M]} defined in §4.4.

Suppose B is an O-module. Recall that 𝔭 is the maximal ideal of O and ℓ_{O}(B) denotes the length of B. If b ∈ B, define

order(b, B) = inf{n ≥ 0 : 𝔭^{n}b = 0} ≤ ∞,

the exponent of the smallest power of 𝔭 which annihilates b. Recall that (Definition 2.2.1) ind_{O}(c) is the largest integer n such that c_{K} is divisible by 𝔭^{n} in H^{1}(K, T)/H^{1}(K, T)_{tors}. We will suppose that ind_{O}(c) is finite, or else there is nothing to prove.

If M ∈ O is nonzero, we let ι_{M} : H^{1}(K, W_{M}) → H^{1}(K, W) denote the map induced by the inclusion of W_{M} in W. If L is an extension of K and η ∈ H^{1}(K, W_{M}^{∗}), we write (η)_{L} for the restriction of η to L, and similarly with W_{M} in place of W_{M}^{∗}.

Lemma 5.1.1. Suppose M is a power of p and ord_{𝔭}M ≥ ind_{O}(c). Then

order(ι_{M}(κ_{[1,M]}), H^{1}(K, W)) = ord_{𝔭}M − ind_{O}(c).

Proof. Lemma 4.4.13(i) shows that ι_{M}(κ_{[1,M]}) is the image of c_{K} under the composition

H^{1}(K, T) ⟶ H^{1}(K, W_{M}) ⟶ H^{1}(K, W).

The kernel of this composition is MH^{1}(K, T) + H^{1}(K, T)_{tors} by Lemma 1.2.2(iii), so

order(ι_{M}(κ_{[1,M]}), H^{1}(K, W)) = order(c_{K}, H^{1}(K, T)/(MH^{1}(K, T) + H^{1}(K, T)_{tors}).

Since H^{1}(K, T)/H^{1}(K, T)_{tors} is a torsion-free O-module, it follows from the definition (Definition 2.2.1) of ind_{O}(c) that

order(c_{K}, H^{1}(K, T)/(MH^{1}(K, T) + H^{1}(K, T)_{tors})) = ord_{𝔭}M − ind_{O}(c).

This proves the lemma.

5.2. Bounding the Order of the Selmer Group

We divide the proof of Theorem 2.2.2 into two main steps. The first step (Lemma 5.2.3) is to produce inductively a sequence of primes of K with useful properties. The second step (Lemma 5.2.5) is to show that the Kolyvagin derivative classes we construct with these primes generate a subgroup which has large image when we project to the singular part of the local cohomology groups. Once this is accomplished, we have only to plug this information into Theorem 1.7.3, the global duality theorem, and we obtain the desired bound.

Suppose throughout this section that p > 2 and that T satisfies hypotheses Hyp(K, T). Fix a τ ∈ G_{K} as in Hyp(K, T)(i), i.e., τ acts trivially on K(1)K(μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}) and T/(τ − 1)T is free of rank one over O. Then for every power M of p the O/MO-module W_{M}/(τ − 1)W_{M} is free of rank one, and by Corollary A.2.6, so is W_{M}^{τ=1} and hence so is

W_{M}^{∗}/(τ − 1)W_{M}^{∗} ≅ Hom(W_{M}^{τ=1}, μ_{M} ⊗ O).

Lemma 5.2.1. Fix a power M of p. Suppose L is a Galois extension of K such that G_{L} acts trivially on W_{M} and on W_{M}^{∗}. If κ ∈ H^{1}(K, W_{M}) and η ∈ H^{1}(K, W_{M}^{∗}), then there is an element γ ∈ G_{L} satisfying

(i) order(κ(γτ), W_{M}/(τ − 1)W_{M})) ≥ order((κ)_{L}, H^{1}(L, W_{M})), (ii) order(η(γτ), W_{M}^{∗}/(τ − 1)W_{M}^{∗}) ≥ order((η)_{L}, H^{1}(L, W_{M}^{∗})).

Proof. For γ ∈ G_{L}, the image of κ(γτ) in W_{M}/(τ − 1)W_{M} is well-defined independent of the choice of cocycle representing κ, and

κ(γτ) ≡ κ(γ) + κ(τ) (mod (τ − 1)W_{M}),  (5.1)

and similarly for η.

Define subsets of G_{L}

B_{κ} = {γ ∈ G_{L} :

order(κ(γτ), W_{M}/(τ − 1)W_{M}) < order((κ)_{L}, Hom(G_{L}, W_{M}))}

B_{η} = {γ ∈ G_{L} :

order(η(γτ), W_{M}^{∗}/(τ − 1)W_{M}^{∗}) < order((η)_{L}, Hom(G_{L}, W_{M}^{∗}))}.

Every γ ∈ G_{L} − (B_{κ} ∪ B_{η}) satisfies the conclusions of the lemma, so we need only show that B_{κ} ∪ B_{η} is a proper subset of G_{L}.

Define a subgroup J of G_{L} by

J = {γ ∈ G_{L} : order(κ(γ), W_{M}/(τ − 1)W_{M}) < order((κ)_{L}, Hom(G_{L}, W_{M}))}.

By (5.1), if γ, γ′ ∈ B_{κ} then γ^{−1}γ′ ∈ J. Therefore B_{κ} either is empty or is a coset of J.

Write d = order((κ)_{L}, Hom(G_{L}, W_{M})), and consider the image κ(G_{L}) of κ on G_{L}. Since (κ)_{L} ∈ Hom(G_{L}, W_{M})^{Gal(L/K)}, we see that

γ(κ(h)) = κ(γhγ^{−1}) for every h ∈ G_{L} and γ ∈ G_{K},

so κ(G_{L}) is a G_{K}-stable subgroup of W_{𝔭^{d}}, not contained in W_{𝔭^{d−1}}. By hypothesis Hyp(K, T)(ii), W_{𝔭} = T ⊗ 𝕜 is irreducible so 𝔭^{d−1}κ(G_{L}) = W_{𝔭} and therefore Oκ(G_{L}) = W_{𝔭^{d}}. Since W_{M}/(τ − 1)W_{M} ≅ O/MO, we conclude that

Oκ(J) ⊂ W_{𝔭^{d−1}} + (τ − 1)W_{M} ⫋ W_{𝔭^{d}} = Oκ(G_{L})

and so J has index at least p in G_{L}.

In exactly the same way, B_{η} either is empty or is a coset of a subgroup of G_{L} of index at least p. Since p > 2, we cannot have B_{κ} ∪B_{η} = G_{L}. This completes the proof.

Remark 5.2.2. The end of the previous proof is the only place where we use the assumption that p > 2 in Theorem 2.2.2.

Let

Ω = K(1)K(W)K(μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}})

where K(W) denotes the smallest extension of K such that G_{K(W)} acts trivially on W. Note that K(W^{∗}) ⊂ Ω.

Lemma 5.2.3. Fix a power M of p. Suppose C is a finite subset of H^{1}(K, W_{M}^{∗}) and let k = |C|.

Then there is a finite set Σ = {𝔮_{1}, …, 𝔮_{k}} of primes of K such that for 1 ≤ i ≤ k,

(i) 𝔮_{i} ∈ R_{M}, (ii) Fr_{𝔮_{i}} is in the conjugacy class of τ in Gal(K(W_{M})/K), (iii) writing 𝔯_{j} = ∏_{t=1}^{j} 𝔮_{t} (so 𝔯_{0} = 1), we have

order((κ_{[𝔯_{i−1},M]})_{𝔮_{i}}, H_{f}^{1}(K_{𝔮_{i}}, W_{M})) ≥ order((κ_{[𝔯_{i−1},M]})_{Ω}, H^{1}(Ω, W_{M})),

(iv) {η ∈ C : (η)_{𝔮} = 0 for every 𝔮 ∈ Σ} ⊂ H^{1}(Ω/K, W_{M}^{∗}).

Proof. Number the elements of C so that C = {η_{1}, η_{2}, …, η_{k}}. We will choose the 𝔮_{i} inductively to satisfy (i), (ii), (iii), and

(η)_{𝔮_{i}} ∈ H_{f}^{1}(K_{𝔮_{i}}, W_{M}^{∗}) for every η ∈ C,  (5.2)
order((η_{i})_{𝔮_{i}}, H_{f}^{1}(K_{𝔮_{i}}, W_{M}^{∗})) ≥ order((η_{i})_{Ω}, H^{1}(Ω, W_{M}^{∗})).  (5.3)

Suppose 1 ≤ i ≤ k and we have chosen {𝔮_{1}, …, 𝔮_{i−1}} satisfying (i), (ii), (iii), (5.2), and (5.3). We need to find 𝔮_{i} also satisfying these properties. Define N′ to be the (finite) product of N and all primes 𝔮 of K such that {(η)_{𝔮} : η ∈ C} ⊄ H_{f}^{1}(K_{𝔮}, W_{M}^{∗}). (Recall that N is divisible by p and all primes where W_{M} is ramified.)

Let L = K(1)K(W_{M}, μ_{M}, (O_{K}^{×})^{1/M}), so L is a finite extension of K contained in Ω, and G_{L} acts trivially on both W_{M} and W_{M}^{∗}. Apply Lemma 5.2.1 with this L, with κ = κ_{[r_{i−1},M]}, and with η = η_{i}, to produce an element γ ∈ G_{L}. Let L′ denote the (finite) extension of L which is the fixed field of

ker((κ_{[r_{i−1},M]})_{L}) ∩ ker((η_{i})_{L})

where we view (κ_{[r_{i−1},M]})_{L} ∈ Hom(G_{L}, W_{M}) and (η_{i})_{L} ∈ Hom(G_{L}, W_{M}^{∗}). Let 𝔮_{i} be a prime of K not dividing N′𝔯_{i−1}, whose Frobenius in L′/K, for some choice of prime above 𝔮_{i}, is γτ. The Tchebotarev theorem guarantees the existence of infinitely many such primes.

Property (i) holds by Lemma 4.1.3, and (ii) and (5.2) are immediate from the definition. By Lemma 1.4.7(i), evaluating cocycles at Fr_{𝔮_{i}} induces an isomorphism

H_{f}^{1}(K_{𝔮_{i}}, W_{M}) ≅ W_{M}/(Fr_{𝔮_{i}} − 1)W_{M} = W_{M}/(τ − 1)W_{M}

and similarly for W_{M}^{∗}, so (iii) and (5.3) follow from Lemma 5.2.1(i) and (ii).

It remains only to check (iv). Define Σ = {𝔮_{1}, …, 𝔮_{k}}, and suppose that for some i, we have (η_{i})_{𝔮} = 0 for every 𝔮 ∈ Σ. Then in particular (η_{i})_{𝔮_{i}} = 0, so (5.3) shows that

η_{i} ∈ ker(H^{1}(K, W_{M}^{∗}) → H^{1}(Ω, W_{M}^{∗})) = H^{1}(Ω/K, W_{M}^{∗}).

Definition 5.2.4. Suppose Σ is a finite set of primes in R. For every M we have an exact sequence

0 ⟶ S^{Σ_{p}}(K, W_{M}) ⟶ S^{Σ∪Σ_{p}}(K, W_{M}) ⟶^{loc_{Σ,W_{M}}^{s}} ⨁_{𝔮∈Σ} H_{s}^{1}(K_{𝔮}, W_{M})

(5.4)

where we recall that

H_{s}^{1}(K_{𝔮}, W_{M}) = H^{1}(K_{𝔮}, W_{M})/H_{f}^{1}(K_{𝔮}, W_{M})

and loc_{Σ,W_{M}}^{s} is the sum of the localization maps (in Theorem 1.7.3 the map loc_{Σ,W_{M}}^{s} was denoted loc_{Σ∪Σ_{p},Σ_{p}}^{s}). We define loc_{Σ,W}^{s} in exactly the same way with W_{M} replaced by W.

If c ∈ H^{1}(K, W_{M}) and 𝔮 is a prime, we write (c)_{𝔮}^{s} for the projection of the localization (c)_{𝔮} into H_{s}^{1}(K_{𝔮}, W_{M}).

If 𝔞 is an ideal of K let Σ_{𝔞} denote the set of primes dividing 𝔞. Let

𝔫_{W} = ℓ_{O} (H^{1}(Ω/K, W) ∩ S^{Σ_{p}}(K, W))

as in Theorem 2.2.2.

Lemma 5.2.5. Suppose that 𝔪 = 𝔭^{n} is a nonzero ideal of O, that k ∈ Z^{+}, and that M is a power of p satisfying

ord_{𝔭}M ≥ n + (k + 1)𝔫_{W} + ind_{O}(c).

Suppose further that

Σ = {𝔮_{1}, …, 𝔮_{k}} ⊂ R_{M}

is a finite set of primes of K such that for 1 ≤ i ≤ k, (a) Fr_{𝔮_{i}} is in the conjugacy class of τ in Gal(K(W_{M})/K),

(b) writing 𝔯_{j} = ∏_{t=1}^{j} 𝔮_{t} (so 𝔯_{0} = 1), we have

order((κ_{[𝔯_{i−1},M]})_{𝔮_{i}}, H_{f}^{1}(K_{𝔮_{i}}, W_{M})) ≥ order((κ_{[𝔯_{i−1},M]})_{Ω}, H^{1}(Ω, W_{M})).

Then the map loc_{Σ,W_{𝔪}}^{s} of (5.4) satisfies

ℓ_{O} (coker(loc_{Σ,W_{𝔪}}^{s})) ≤ ind_{O}(c) + 𝔫_{W} .

Remark 5.2.6. Since the proof of Lemma 5.2.5 is a rather technical calculation, we first give a proof under the mild additional hypotheses

W^{G_{K}} = 0 and H^{1}(Ω/K, W) = 0.  (5.5)

We will follow this immediately by the general proof; we include the easier special case only because it makes the important ideas clearer.

Proof of Lemma 5.2.5 under the assumption (5.5). Note that by assumption (a) of the lemma, W_{M}/(Fr_{𝔮_{i}} − 1)W_{M} is free of rank one over O/MO for every i. Therefore we can apply Corollary 4.5.5 with 𝔮 = 𝔮_{i} and 𝔯 = 𝔯_{i−1} to relate κ_{[𝔯_{i},M]} and κ_{[𝔯_{i−1},M]}. This will be the key to the proof.

By Lemma 1.2.2(i) and (5.5), all of the maps

H^{1}(K, W_{𝔪}) ⟶^{ι_{𝔪,M}} H^{1}(K, W_{M}) ⟶^{ι_{M}} H^{1}(K, W) ⟶^{( )_{Ω}} H^{1}(Ω, W)

are injective. Therefore for 0 ≤ i ≤ k we can define

𝔡_{i} = order(κ_{[𝔯_{i},M]}, H^{1}(K, W_{M})) = order(ι_{M}(κ_{[𝔯_{i},M]}), H^{1}(K, W))
= order((ι_{M}(κ_{[𝔯_{i},M]}))_{Ω}, H^{1}(Ω, W))
= order((κ_{[𝔯_{i},M]})_{Ω}, H^{1}(Ω, W_{M})).

By Lemma 5.1.1,

𝔡_{0} = ord_{𝔭}M − ind_{O}(c) ≥ n.  (5.6)

For i ≥ 1,

𝔡_{i} ≥ order((κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W_{M})) = order((κ_{[𝔯_{i−1},M]})_{𝔮_{i}}, H_{f}^{1}(K_{𝔮_{i}}, W_{M})) ≥ 𝔡_{i−1}, (5.7)

the equality by Corollary 4.5.5, and the inequality on the right by assumption (b) of the lemma. Combining (5.6) and (5.7) we see that 𝔡_{i} ≥ n for every i.

It follows from Lemma 1.5.4 and the injectivity of ι_{M} that the homomorphism ι_{𝔪,M} : H^{1}(K, W_{𝔪}) → H^{1}(K, W_{M}) sends S^{Σ_{p𝔯_{i}}} (K, W_{𝔪}) onto S^{Σ_{p𝔯_{i}}} (K, W_{M})_{𝔪}. Theorem 4.5.1 shows that κ_{[𝔯_{i},M]} ∈ S^{Σ_{p𝔯_{i}}} (K, W_{M}), so for each i ≥ 1 we can choose κ̄_{i} ∈ S^{Σ_{p𝔯_{i}}} (K, W_{𝔪}) such that Oι_{𝔪,M}(κ̄_{i}) = 𝔭^{𝔡_{i}−n}κ_{[𝔯_{i},M]}.

If 1 ≤ i ≤ k let A^{(i)} denote the O-submodule of H^{1}(K, W_{𝔪}) generated by {κ̄_{1}, …, κ̄_{i}}, and let A^{(0)} = 0. Then

A^{(i)} ⊂ S^{Σ_{p𝔯_{i}}} (K, W_{𝔪}) ⊂ S^{Σ∪Σ_{p}}(K, W_{𝔪})

so for 1 ≤ i ≤ k, writing loc_{Σ}^{s} for loc_{Σ,W_{𝔪}}^{s}, restriction to 𝔮_{i} induces a surjective map

loc_{Σ}^{s}(A^{(i)})/loc_{Σ}^{s}(A^{(i−1)}) ↠ O(κ̄_{i})_{𝔮_{i}}^{s} ⊂ H_{s}^{1}(K_{𝔮_{i}}, W_{𝔪}).

Hence for 1 ≤ i ≤ k, (5.7) shows that

ℓ_{O}(loc_{Σ}^{s}(A^{(i)})/loc_{Σ}^{s}(A^{(i−1)})) ≥ order((κ̄_{i})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W_{𝔪}))

≥ order((κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W_{M})) − (𝔡_{i} − n)

≥ n + 𝔡_{i−1} − 𝔡_{i}.

Using the filtration

loc_{Σ}^{s}(A^{(k)}) ⊃ loc_{Σ}^{s}(A^{(k−1)}) ⊃ ⋯ ⊃ loc_{Σ}^{s}(A^{(1)}) ⊃ loc_{Σ}^{s}(A^{(0)}) = 0

we conclude, using (5.6) and the trivial estimate 𝔡_{k} ≤ ord_{𝔭}M, that

ℓ_{O}(loc_{Σ}^{s}(S^{Σ∪Σ_{p}}(K, W_{𝔪}))) ≥ ℓ_{O}(loc_{Σ}^{s}(A^{(k)}))

≥ ∑_{i=1}^{k}(n + 𝔡_{i−1} − 𝔡_{i}) = kn + 𝔡_{0} − 𝔡_{k} ≥ kn − ind_{O}(c).

For every prime 𝔮 ∈ R_{M}, we have H_{s}^{1}(K_{𝔮}, W_{𝔪}) = W_{𝔪}^{Fr_{𝔮}=1} by Lemma 1.4.7(i), so

ℓ_{O}(⊕_{𝔮∈Σ}H_{s}^{1}(K_{𝔮}, W_{𝔪})) = kℓ_{O}(W_{𝔪}^{τ=1}) = kℓ_{O}(W_{𝔪}/(τ − 1)W_{𝔪}) = kn.

Thus

ℓ_{O}(coker(loc_{Σ}^{s})) ≤ ind_{O}(c)

as desired.

Proof of Lemma 5.2.5 in general. Recall that ι_{M} is the natural map from H^{1}(K, W_{M}) to H^{1}(K, W). For 0 ≤ i ≤ k define

𝔡′_{i} = order(ι_{M}(κ_{[𝔯_{i},M]}), H^{1}(K, W)),
𝔡_{i} = order((κ_{[𝔯_{i},M]})_{Ω}, H^{1}(Ω, W_{M})).

By Lemma 5.1.1,

𝔡′_{0} = ord_{𝔭}M − ind_{O}(c) ≥ n + (k + 1)𝔫_{W} .  (5.8)

Since 𝔭^{𝔡_{i}}(κ_{[𝔯_{i},M]})_{Ω} = 0, we see that

𝔭^{𝔡_{i}}ι_{M}(κ_{[𝔯_{i},M]}) ⊂ H^{1}(Ω/K, W).

By Theorem 4.5.1, 𝔭^{𝔡_{i}}ι_{M}(κ_{[𝔯_{i},M]}) ∈ S^{Σ_{p𝔯_{i}}} (K, W). Lemma 1.3.5(iv) shows that for every 𝔮 ∤ N we have H_{f}^{1}(K_{𝔮}, W) = H_{ur}^{1}(K_{𝔮}, W), and Ω/K is unramified outside primes dividing N, so H^{1}(Ω/K, W) ⊂ S^{Σ_{N}} (K, W). Therefore, since 𝔯_{i} is prime to N we conclude that

𝔭^{𝔡_{i}}ι_{M}(κ_{[𝔯_{i},M]}) ∈ H^{1}(Ω/K, W) ∩ S^{Σ_{p𝔯_{i}}} (K, W)

= H^{1}(Ω/K, W) ∩ S^{Σ_{p}}(K, W). (5.9)

Therefore for every i, we have 𝔭^{𝔡_{i}+𝔫_{W}} ι_{M}(κ_{[𝔯_{i},M]}) = 0, so

𝔫_{W} + 𝔡_{i} ≥ 𝔡′_{i}.  (5.10)

Suppose i ≥ 1. If I_{𝔮_{i}} is an inertia group of 𝔮_{i}, then (using Lemmas 1.3.8(ii), 1.3.5(iv), and 1.3.2(ii)) we have a commutative diagram

H_{s}^{1}(K_{𝔮_{i}}, W_{M}) = H^{1}(K_{𝔮_{i}}, W_{M})/H_{ur}^{1}(K_{𝔮_{i}}, W_{M}) ⊂ Hom(I_{𝔮_{i}}, W_{M})
↓^{ι_{M}}  ↓  ∩
H_{s}^{1}(K_{𝔮_{i}}, W) = H^{1}(K_{𝔮_{i}}, W)/H_{ur}^{1}(K_{𝔮_{i}}, W) ⊂ Hom(I_{𝔮_{i}}, W).

Therefore the map ι_{M} : H_{s}^{1}(K_{𝔮_{i}}, W_{M}) → H_{s}^{1}(K_{𝔮_{i}}, W) is injective. This gives the first equality of

𝔡′_{i} ≥ order(ι_{M}(κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W))
= order((κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W_{M}))
= order((κ_{[𝔯_{i−1},M]})_{𝔮_{i}}, H_{f}^{1}(K_{𝔮_{i}}, W_{M})) ≥ 𝔡_{i−1},  (5.11)

the second equality comes from Corollary 4.5.5 and assumption (a) of the lemma, and the final inequality comes from assumption (b). Combining (5.11) with (5.8) and (5.10), we conclude by induction that

𝔡_{i} ≥ 𝔡′_{0} − (i + 1)𝔫_{W} ≥ n.

For 0 ≤ i ≤ k let A^{(i)} denote the O-submodule of H^{1}(K, W) generated by

{𝔭^{𝔡_{j}−n}ι_{M}(κ_{[𝔯_{j},M]}) : 0 ≤ j ≤ i},

and write A = A^{(k)}. By Theorem 4.5.1,

A^{(i)} ⊂ S^{Σ_{p𝔯_{i}}} (K, W),  (5.12)

so for 1 ≤ i ≤ k restriction to 𝔮_{i} induces a surjective map

loc_{Σ,W}^{s} (A^{(i)})/loc_{Σ,W}^{s} (A^{(i−1)}) ↠ 𝔭^{𝔡_{i}−n}(κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s} ⊂ H_{s}^{1}(K_{𝔮_{i}}, W).

For 1 ≤ i ≤ k, (5.11) shows that

order(𝔭^{𝔡_{i}−n}(κ_{[𝔯_{i},M]})_{𝔮_{i}}^{s}, H_{s}^{1}(K_{𝔮_{i}}, W_{M})) ≥ 𝔡_{i−1} − 𝔡_{i} + n,

so using the filtration

loc_{Σ,W}^{s} (A) = loc_{Σ,W}^{s} (A^{(k)}) ⊃ loc_{Σ,W}^{s} (A^{(k−1)}) ⊃ ⋯

⋯ ⊃ loc_{Σ,W}^{s} (A^{(1)}) ⊃ loc_{Σ,W}^{s} (A^{(0)})

we conclude that

ℓ_{O}(loc_{Σ,W}^{s} (A)) ≥ ∑_{i=1}^{k}(𝔡_{i−1} − 𝔡_{i} + n) = kn + 𝔡_{0} − 𝔡_{k} ≥ kn + 𝔡_{0} − ord_{𝔭}M. (5.13)

Since 𝔪 = 𝔭^{n}, (5.9) shows that

𝔪A ⊂ H^{1}(Ω/K, W) ∩ S^{Σ_{p}}(K, W).  (5.14)

Let A_{𝔪} denote the submodule of A killed by 𝔪. By (5.12) and Lemma 1.5.4,

A_{𝔪} ⊂ S^{Σ∪Σ_{p}}(K, W)_{𝔪} = ι_{𝔪}(S^{Σ∪Σ_{p}}(K, W_{𝔪})).  (5.15)

From the exact diagram

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & ker(loc_{Σ,W}^{s}) ∩ A_{𝔪} \ar[r] \ar[d] & A_{𝔪} \ar[r] \ar[d] & loc_{Σ,W}^{s} (A_{𝔪}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & ker(loc_{Σ,W}^{s}) ∩ A \ar[r] & A \ar[r] \ar[d] & loc_{Σ,W}^{s} (A) \ar[r] & 0 \\  &  & 𝔪A \ar[d] \\  &  & 0}

we see that

ℓ_{O}(loc_{Σ,W}^{s} (A_{𝔪})) = ℓ_{O}(loc_{Σ,W}^{s} (A)) + ℓ_{O}((ker(loc_{Σ,W}^{s}) ∩ A)/(ker(loc_{Σ,W}^{s}) ∩ A_{𝔪})) − ℓ_{O}(𝔪A). (5.16)

By (5.12) with i = 0, we have

A^{(0)} = 𝔭^{𝔡_{0}−n}ι_{M}(κ_{[1,M]}) ⊂ ker(loc_{Σ,W}^{s}).

Since A^{(0)} is a cyclic O-module, we conclude using (5.8) that

ℓ_{O}((ker(loc_{Σ,W}^{s}) ∩ A)/(ker(loc_{Σ,W}^{s}) ∩ A_{𝔪})) ≥ ℓ_{O}(A^{(0)}) − ℓ_{O}(A^{(0)} ∩ A_{𝔪})
≥ (𝔡′_{0} − (𝔡_{0} − n)) − n
= ord_{𝔭}M − ind_{O}(c) − 𝔡_{0}.

Combining this with (5.15), (5.16), (5.13), and (5.14) yields

ℓ_{O}(loc_{Σ,W_{𝔪}}^{s}(S^{Σ∪Σ_{p}}(K, W_{𝔪}))) ≥ ℓ_{O}(loc_{Σ,W}^{s} (A_{𝔪}))
≥ (kn + 𝔡_{0} − ord_{𝔭}M) + (ord_{𝔭}M − ind_{O}(c) − 𝔡_{0}) − 𝔫_{W}
= kn − ind_{O}(c) − 𝔫_{W} .

For every prime 𝔮 ∈ R_{M}, we have H_{s}^{1}(K_{𝔮}, W_{𝔪}) = W_{𝔪}^{Fr_{𝔮}=1} by Lemma 1.4.7(i), so

ℓ_{O}(⊕_{𝔮∈Σ}H_{s}^{1}(K_{𝔮}, W_{𝔪})) = kℓ_{O}(W_{𝔪}^{τ=1}) = kℓ_{O}(W_{𝔪}/(τ − 1)W_{𝔪}) = kn.

Thus

ℓ_{O}(coker(loc_{Σ,W_{𝔪}}^{s}))

= ℓ_{O}(⊕_{𝔮∈Σ}H_{s}^{1}(K_{𝔮}, W_{𝔪})) − ℓ_{O}(loc_{Σ,W_{𝔪}}^{s}(S^{Σ∪Σ_{p}}(K, W_{𝔪})))
≤ ind_{O}(c) + 𝔫_{W}

as desired.

Proof of Theorem 2.2.2. Fix a nonzero ideal 𝔪 = 𝔭^{n} of O. Let C be the image of S_{Σ_{p}}(K, W_{𝔪}^{∗}) (which is finite by Lemma 1.5.7(i)) in H^{1}(K, W_{M}^{∗}) where M is a power of p large enough so that

ord_{𝔭}M > n + (|S_{Σ_{p}}(K, W_{𝔪}^{∗})| + 1)𝔫_{W} + ind_{O}(c)

(if ind_{O}(c) is infinite then there is nothing to prove). Apply Lemma 5.2.3 with this group C, let Σ be a set of primes of K produced by that lemma, and apply Lemma 5.2.5 with this set Σ.

Combining the inequality of Lemma 5.2.5 with Theorem 1.7.3(iii) shows that

ℓ_{O}(S_{Σ_{p}}(K, W_{𝔪}^{∗})/S_{Σ∪Σ_{p}}(K, W_{𝔪}^{∗})) ≤ 𝔫_{W} + ind_{O}(c).

Therefore

ℓ_{O}(ι_{𝔪}(S_{Σ_{p}}(K, W_{𝔪}^{∗}))) ≤ ℓ_{O}(ι_{𝔪}(S_{Σ∪Σ_{p}}(K, W_{𝔪}^{∗}))) + 𝔫_{W} + ind_{O}(c).

Lemma 5.2.3(iv) shows that

ι_{𝔪}(S_{Σ∪Σ_{p}}(K, W_{𝔪}^{∗})) ⊂ H^{1}(Ω/K, W^{∗}) ∩ S_{Σ_{p}}(K, W^{∗}),

and by definition 𝔫_{W}^{∗} = ℓ_{O}(H^{1}(Ω/K, W^{∗}) ∩ S_{Σ_{p}}(K, W^{∗})), so we see that

ℓ_{O}(ι_{𝔪}(S_{Σ_{p}}(K, W_{𝔪}^{∗}))) ≤ ind_{O}(c) + 𝔫_{W} + 𝔫_{W}^{∗} .

Since this holds for every 𝔪, and S_{Σ_{p}}(K, W^{∗}) = \underrightarrow{lim}_{𝔪} ι_{𝔪}(S_{Σ_{p}}(K, W_{𝔪}^{∗})), The-

orem 2.2.2 follows.

5.3. Bounding the Exponent of the Selmer Group

The proof of Theorem 2.2.3 is similar to that of Theorem 2.2.2; it is easier in the sense that one can work with a single prime 𝔮 instead of a finite set of primes, but more difficult in the sense that one must keep track of extra “error terms”.

The idea is as follows. Given η ∈ S_{Σ_{p}}(K, W_{M}^{∗}), we use Lemma 5.3.1 below to choose a prime 𝔮 such that

• H_{f}^{1}(K_{𝔮}, W_{M}^{∗}) and H_{s}^{1}(K_{𝔮}, W_{M}) are “almost” isomorphic to O/MO,

• order((κ_{[𝔮,M]})_{𝔮}^{s}, H_{s}^{1}(K_{𝔮}, W_{M})) is approximately ord_{𝔭}M − ind_{O}(c),

• order((η)_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M}^{∗})) is approximately order(η, H^{1}(K, W_{M}^{∗})).

Since the Kolyvagin derivative class κ_{[𝔮,M]} belongs to S^{Σ_{p𝔮}}(K, W_{M}), the duality Theorem 1.7.3 shows that

order((κ_{[𝔮,M]})_{𝔮}^{s}, H_{s}^{1}(K_{𝔮}, W_{M})) + order((η)_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M}^{∗}))

is “approximately” bounded by ord_{𝔭}M, and so order(η, H^{1}(K, W_{M}^{∗})) is “approximately” bounded by ind_{O}(c). Since η ∈ S_{Σ_{p}}(K, W_{M}^{∗}) is arbitrary, if we can bound all the error terms independently of M, this will prove Theorem 2.2.3. In the remainder of this section we sketch the details of this argument.

Keep the notation of §5.1 and §5.2. Suppose the Euler system c satisfies the hypotheses Hyp(K, V), and fix a τ ∈ G_{K} as in hypothesis Hyp(K, V)(i). We now allow p = 2.

Let a be the smallest positive integer such that 𝔭^{a} annihilates the maximal G_{K}-stable O-submodule of (τ − 1)W and of (τ − 1)W^{∗}. Hypothesis Hyp(K, V)(ii) ensures that a is finite, since any divisible G_{K}-stable subgroup of (τ − 1)W would be the image of a G_{K}-stable subgroup of (τ − 1)V, which must be zero.

We have the following variant of Lemma 5.2.1.

Lemma 5.3.1. Fix a power M of p. Suppose L is a Galois extension of K such that G_{L} acts trivially on W_{M} and on W_{M}^{∗}. If

κ ∈ H^{1}(K, W_{M}), η ∈ H^{1}(K, W_{M}^{∗})

then there is an element γ ∈ G_{L} satisfying (i) order(κ(γτ), W_{M}/(τ − 1)W_{M})) ≥ order((κ)_{L}, H^{1}(L, W_{M})) − a − 1, (ii) order(η(γτ), W_{M}^{∗}/(τ − 1)W_{M}^{∗}) ≥ order((η)_{L}, H^{1}(L, W_{M}^{∗})) − a − 1.

Proof. The proof is similar to that of Lemma 5.2.1, once we note that every G_{K}-submodule of W_{M} which projects to zero in W_{M}/(τ − 1)W_{M} is killed by 𝔭^{a}, and similarly for W_{M}^{∗}. The extra ‘1’ takes care of the case p = 2.

Let Ω = K(1)K(W, μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}) as in §5.2.

Lemma 5.3.2. If T ≠ O and T ≠ O(1) then both H^{1}(Ω/K, W) and H^{1}(Ω/K, W^{∗}) are finite.

Proof. This is Corollary C.2.2 applied with F = K.

Proof of Theorem 2.2.3. If T = O(1) then Proposition 1.6.1 shows that

S_{Σ_{p}}(K, W^{∗}) ⊂ Hom(A_{K}, D),

where A_{K} is the ideal class group of K, so S_{Σ_{p}}(K, W^{∗}) is finite. The theorem assumes that T ≠ O, so by Lemma 5.3.2 we may assume from now on that H^{1}(Ω/K, W) and H^{1}(Ω/K, W^{∗}) are finite. Let

n = max{ℓ_{O}(H^{1}(Ω/K, W)), ℓ_{O}(H^{1}(Ω/K, W^{∗}))}.

Suppose M is a power of p and η ∈ S_{Σ_{p}}(K, W_{M}^{∗}). Apply Lemma 5.3.1 with L = K(1)K(W_{M}, μ_{M}, (O_{K}^{×})^{1/M}) ⊂ Ω, with this η, and with κ = κ_{[1,M]} ∈ H^{1}(K, W_{M}), and let γ ∈ G_{L} be an element satisfying the conclusions of that lemma. Then since H^{1}(Ω/K, W) is the kernel of the restriction map H^{1}(K, W) → H^{1}(Ω, W),

order(κ_{[1,M]}(γτ), W_{M}/(τ − 1)W_{M}))
≥ order(ι_{M}(κ_{[1,M]})_{Ω}, H^{1}(Ω, W)) − a − 1

≥ order(ι_{M}(κ_{[1,M]}), H^{1}(K, W)) − a − 1 − n

= ord_{𝔭}M − ind_{O}(c) − a − 1 − n  (5.17)

by Lemma 5.1.1. Similarly

order(η(γτ), W_{M}^{∗}/(τ − 1)W_{M}^{∗}) ≥ order(η, H^{1}(K, W_{M}^{∗})) − a − 1 − n. (5.18)

Let L′ denote the fixed field of

ker((κ_{[1,M]})_{L}) ∩ ker((η)_{L})

and, using the Tchebotarev theorem, choose a prime 𝔮 of K, not dividing N, whose Frobenius in L′/K, for some choice of prime above 𝔮, is γτ. By Lemma 4.1.3, we have 𝔮 ∈ R_{M}.

As in the proof of Lemma 5.2.3, we conclude from (5.17) and (5.18) that

order((κ_{[1,M]})_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M})) ≥ ord_{𝔭}M − ind_{O}(c) − a − 1 − n

and

order((η)_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M}^{∗})) ≥ order(η, H^{1}(K, W_{M}^{∗})) − a − 1 − n. (5.19)

Let b = ℓ_{O}(W^{τ=1}/(W^{τ=1})_{div}), where (W^{τ=1})_{div} is the maximal divisible submodule of W^{τ=1}. By Theorem 4.5.4 and Corollary A.2.6,

order((κ_{[𝔮,M]})_{𝔮}^{s}, H_{s}^{1}(K_{𝔮}, W_{M})) ≥ order((κ_{[1,M]})_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M})) − 2b
≥ ord_{𝔭}M − ind_{O}(c) − a − 1 − n − 2b.

By Lemma 1.4.7(i),

ℓ_{O}(H_{s}^{1}(K_{𝔮}, W_{M})) = ℓ_{O}((W_{M})^{τ=1}) = ℓ_{O}((W^{τ=1})_{M}) ≤ ord_{𝔭}M + b.

Thus, applying Theorem 1.7.3(iii) with

Σ = Σ_{p𝔮}, Σ_{0} = Σ_{p}, η ∈ S_{Σ_{p}}(K, W_{M}^{∗}),

we conclude that

order((η)_{𝔮}, H_{f}^{1}(K_{𝔮}, W_{M}^{∗})) ≤ ℓ_{O}(coker(loc_{Σ_{p𝔮},Σ_{p}}^{s}))

≤ ℓ_{O}(H_{s}^{1}(K_{𝔮}, W_{M})) − order((κ_{[𝔮,M]})_{𝔮}^{s}, H_{s}^{1}(K_{𝔮}, W_{M}))

≤ ind_{O}(c) + a + 1 + n + 3b

since κ_{[𝔮,M]} ∈ S^{Σ_{p𝔮}}(K, W_{M}). Combining this with (5.19) shows that

order(η, H^{1}(K, W_{M}^{∗})) ≤ 2 + 2a + 3b + 2n + ind_{O}c.

This inequality holds for every M and every η ∈ S_{Σ_{p}}(K, W_{M}^{∗}). Since S_{Σ_{p}}(K, W^{∗}) is the direct limit of the S_{Σ_{p}}(K, W_{M}^{∗}), if we set

𝔪 = 𝔭^{2+2a+3b+2n+ind_{O}c}

then we conclude that 𝔪S_{Σ_{p}}(K, W^{∗}) = 0.

As is well-known, this implies that S_{Σ_{p}}(K, W^{∗}) is finite: Lemma 1.5.4 shows that

S_{Σ_{p}}(K, W^{∗}) = S_{Σ_{p}}(K, W^{∗})_{𝔪} ⊂ S(K, W^{∗})_{𝔪} = ι_{𝔪}(S(K, W_{𝔪}^{∗}))

and the latter is finite by Lemma 1.5.7(i).

CHAPTER 6 Twisting

In this chapter we extend the methods of §2.4 to twist Euler systems by characters of infinite order. This will be used in Chapter 7 when we prove Theorems 2.3.2, 2.3.3, and 2.3.4. If ρ is a character of Gal(K_{∞}/K), then

• Theorem 6.3.5 says that an Euler system c for (T, K_{∞}) gives rise to an Euler system c^{ρ} for (T ⊗ ρ, K_{∞}),

• Theorem 6.4.1 shows that the theorems of §2.3 hold for T and c if and only if they hold for T ⊗ ρ and c^{ρ}, and

• Lemma 6.1.3 allows us to choose a particular ρ which avoids certain complications.

We keep the setting of Chapter 2, so K is a number field, T is a p-adic representation of G_{K} ramified at only finitely many primes, and K_{∞} is an abelian extension of K satisfying Gal(K_{∞}/K) ≅ Z_{p}^{d}. Let Γ = Gal(K_{∞}/K), and recall that Λ is the Iwasawa algebra

Λ = O[[Γ]] = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}O[Gal(F/K)],

a complete local noetherian unique factorization domain. The characteristic ideal char(B) of a finitely generated Λ-module B was defined in §2.3.

6.1. Twisting Representations

Definition 6.1.1. Suppose ρ : G_{K} → O^{×} is a continuous character, possibly of infinite order. As in Example 1.1.2 we will write O_{ρ} for a free rank-one O-module with G_{K} acting via ρ, and if B is a G_{K}-module we will abbreviate

B ⊗ ρ = B ⊗_{O} O_{ρ}.

Then B ⊗ ρ is isomorphic to B as an O-module but not (in general) as a G_{K}-module.

If ρ : Γ → O^{×} define

Tw_{ρ} : Λ ⟶^{∼} Λ

to be the O-linear isomorphism induced by γ ↦ ρ(γ)γ for γ ∈ Γ.

Lemma 6.1.2. Suppose B is a finitely generated torsion Λ-module and ρ : Γ → O^{×} is a character. Then B ⊗ ρ is a finitely generated torsion Λ-module and

(i) Tw_{ρ}(char(B ⊗ ρ)) = char(B), (ii) if f ∈ Λ then f · (B ⊗ ρ) = 0 ⟺ Tw_{ρ}(f) · B = 0.

Proof. If f ∈ Λ and ξ_{ρ} ∈ O_{ρ} then

f · (b ⊗ ξ_{ρ}) = (Tw_{ρ}(f)b) ⊗ ξ_{ρ}.

The lemma follows easily from this, along with (for (i)) the fact that twisting preserves the heights of ideals of Λ.

Lemma 6.1.3. (i) Suppose B is G_{K}-module, free of finite rank over O, and γ_{1}, …, γ_{k} are elements of G_{K} whose projections to Γ are nontrivial. Then the set

{ρ ∈ Hom(Γ, O^{×}) : (B ⊗ ρ)^{γ_{i}^{p^{n}} =1} = 0 for 1 ≤ i ≤ k and every n ≥ 0}

contains an open dense subset of Hom(Γ, O^{×}). (ii) Suppose B is a finitely generated torsion Λ-module. Then the set

{ρ ∈ Hom(Γ, O^{×}) : (B ⊗ ρ) ⊗_{Λ} O[Gal(F/K)] is finite for every K ⊂_{f} F ⊂ K_{∞}}

is dense in Hom(Γ, O^{×}).

Proof. Consider (i) first. Recall that Φ is the field of fractions of O, and let Φ̄ denote an algebraic closure of Φ. For each i define

R_{i} = {eigenvalues of γ_{i} acting on B ⊗ Φ̄},
P_{i} = {x ∈ O^{×} : xR_{i} ∩ μ_{p^{∞}} ≠ ∅},
Z_{i} = {ρ ∈ Hom(Γ, O^{×}) : ρ(γ_{i}) ∉ P_{i}}.

Each R_{i} is finite, and μ_{p^{∞}} ∩ O^{×} is finite, so each P_{i} is finite and thus Z = ∩_{i}Z_{i} is an open dense subset of Hom(Γ, O^{×}). We will show that Z is contained in the set of (i).

Suppose ζ ∈ μ_{p^{∞}}. Then

ζ is an eigenvalue of γ_{i} acting on (B ⊗ ρ) ⊗ Φ̄
⟺ ρ^{−1}(γ_{i})ζ is an eigenvalue of γ_{i} acting on B ⊗ Φ̄
⟺ ζ ∈ ρ(γ_{i})R_{i}
⟹ ρ(γ_{i}) ∈ P_{i}.

Therefore if ρ ∈ Z_{i} and n ≥ 0, then 1 is not an eigenvalue of γ_{i}^{p^{n}} acting on (B ⊗ ρ) ⊗ Φ. It follows that for 1 ≤ i ≤ k, every n ≥ 0, and every ρ ∈ Z, we have

(B ⊗ ρ)^{γ_{i}^{p^{n}} =1} ⊗ Φ = ((B ⊗ ρ) ⊗ Φ)^{γ_{i}^{p^{n}} =1} = 0

and since B has no p-torsion we conclude that (B ⊗ ρ)^{γ_{i}^{p^{n}} =1} = 0.

Let U ⊂ Hom(Γ, O^{×}) be the set defined in (ii). We will show that U contains a countable intersection of dense open sets, so the Baire category theorem shows that U is dense. Since B is a quotient of a finite direct sum of cyclic torsion Λ-modules, it is enough to prove this when B = Λ/fΛ with a nonzero f ∈ Λ.

Suppose B = Λ/fΛ, so B ⊗ ρ ≅ Λ/Tw_{ρ^{−1}}(f)Λ. If K ⊂_{f} F ⊂ K_{∞} then

(Λ/Tw_{ρ^{−1}}(f)Λ) ⊗_{Λ} O[Gal(F/K)] is finite ⟺

ρ^{−1}χ(f) ≠ 0 for every character χ : Gal(F/K) → Φ̄^{×}. (6.1)

Let Ξ be the set of characters of finite order of Γ into Φ̄^{×}, and for χ ∈ Ξ let

Y_{χ} = {ρ ∈ Hom(Γ, O^{×}) : ρ^{−1}χ(f) ≠ 0}.

Since f ≠ 0, each Y_{χ} is open and dense in Hom(Γ, O^{×}), and (6.1) shows that U = ∩_{χ∈Ξ}Y_{χ}. Since Ξ is countable, this concludes the proof.

6.2. Twisting Cohomology Groups

For every extension L of K, write

H_{∞}^{1}(L, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F L, T).

Proposition 6.2.1. Suppose K ⊂_{f} L and ρ : Gal(LK_{∞}/K) → O^{×} is a character. The natural map on cocycles induces G_{K}-isomorphisms

(i) H_{∞}^{1}(L, T) ⊗ ρ ⟶^{∼} H_{∞}^{1}(L, T ⊗ ρ)

(ii) S_{Σ_{p}}(LK_{∞}, W) ⊗ ρ ⟶^{∼} S_{Σ_{p}}(LK_{∞}, W ⊗ ρ)

where Σ_{p} is the set of primes of K dividing p.

Proof. Let L_{∞} = LK_{∞}. Choose a sequence fields L⊂_{f} L_{1} ⊂_{f} L_{2} ⊂_{f} ⋯ ⊂ L_{∞} such that L_{∞} = ∪L_{n} and such that Gal(L_{∞}/L_{n}) is contained in the kernel of Gal(LK_{∞}/K) ⟶^{ρ} (O/p^{n}O)^{×}. Then O_{ρ}/p^{n}O_{ρ} is a trivial G_{L_{n}}-module, so the natural map on cocycles induces G_{K}-equivariant isomorphisms

H^{1}(L_{n}, T/p^{n}T) ⊗ ρ ⟶^{∼} H^{1}(L_{n}, (T/p^{n}T) ⊗ ρ).  (6.2)

Combining these isomorphisms with Lemma B.3.1 gives a sequence of isomorphisms

H_{∞}^{1}(L, T) ⊗ ρ = \underleftarrow{lim}_{n} H^{1}(L_{n}, T/p^{n}T) ⊗ ρ

⟶^{∼} \underleftarrow{lim}_{n} H^{1}(L_{n}, (T ⊗ ρ)/p^{n}(T ⊗ ρ)) = H_{∞}^{1}(L, T ⊗ ρ).

This proves (i).

Suppose w is a place of LK_{∞}. The isomorphisms (6.2), and their analogues for the completions L_{n,w}, induce the horizontal isomorphisms in the commutative diagram

\underrightarrow{lim}_{n} H^{1}(L_{n}, W) ⊗ ρ ⟶^{∼} \underrightarrow{lim}_{n} H^{1}(L_{n}, W ⊗ ρ)
res^{w}↓  ↓^{res}^{w}
\underrightarrow{lim}_{n} H^{1}(L_{n,w}, W) ⊗ ρ ⟶^{∼} \underrightarrow{lim}_{n} H^{1}(L_{n,w}, W ⊗ ρ).

(6.3)

By Lemmas B.3.3 and 1.3.5(i), if w does not divide p then

\underrightarrow{lim}_{n} H_{f}^{1}(L_{n,w}, W) ⊗ ρ = \underrightarrow{lim}_{n} H_{f}^{1}(L_{n,w}, W ⊗ ρ) = 0.

It follows that the top row of (6.3) induces the isomorphism of (ii).

Corollary 6.2.2. Suppose ρ : Γ → O^{×} is a character. Then there is an isomorphism of Λ-modules

X_{∞}(T ⊗ ρ) ≅ X_{∞}(T) ⊗ ρ,

where X_{∞}(T) = X_{∞} = Hom(S_{Σ_{p}}(K_{∞}, W^{∗}), D) and

X_{∞}(T ⊗ ρ) = Hom(S_{Σ_{p}}(K_{∞}, (W ⊗ ρ)^{∗}), D)

is the corresponding Λ-module associated to T ⊗ ρ.

Proof. The corollary follows immediately from Proposition 6.2.1(ii) applied with W replaced by W^{∗} and with L = K.

Remark 6.2.3. Note that Proposition 6.2.1 does not assert the existence of an isomorphism, or even a map, from H^{1}(L, T) to H^{1}(L, T ⊗ ρ).

6.3. Twisting Euler Systems

Definition 6.3.1. Suppose c is an Euler system for (T, K_{∞}), more specifically (in the notation of Definition 2.1.1) for (T, K, N), where K_{∞} ⊂ K and N is divisible by p and the primes where T is ramified. If K ⊂_{f} L ⊂ K then we write c_{L,∞} = {c_{LF} }_{K ⊂_{f} F ⊂K_{∞}} for the corresponding element of H_{∞}^{1}(L, T).

Suppose ρ : Gal(K/K) → O^{×} is a character which factors through a finite extension of K_{∞}. (We can always ensure this latter property by taking K_{∞} to be the compositum of all Z_{p}-extensions of K in K.) Let L be a finite extension of K in K such that, writing L_{∞} = LK_{∞},

(i) ρ factors through Gal(L_{∞}/K),

(ii) L_{∞}/K is ramified only at primes dividing N, ∞, and the conductor

of ρ. (For example, L could be a finite extension of K such that LK_{∞} is the fixed field of ker(ρ) ∩ G_{K_{∞}}.) Fix a generator ξ_{ρ} of O_{ρ}. We define a collection of cohomology classes

c^{ρ} = {c_{F}^{ρ} ∈ H^{1}(F, T ⊗ ρ) : K ⊂_{f} F ⊂ K}

as follows. If K ⊂_{f} F ⊂ K let c_{F}^{ρ} be the image of c_{F L,∞}⊗ξ_{ρ} ∈ H_{∞}^{1}(F L, T)⊗ρ under the composition

H_{∞}^{1}(F L, T) ⊗ ρ ⟶^{∼} H_{∞}^{1}(F L, T ⊗ ρ)

⟶ H^{1}(F L, T ⊗ ρ) ⟶^{Cor_{F L/F}} H^{1}(F, T ⊗ ρ)

where the first map is the isomorphism of Proposition 6.2.1(i) and the second is the natural projection from H_{∞}^{1} to H^{1}.

Remark 6.3.2. The definition of c_{F}^{ρ} is independent of our choice of L. To see this, suppose L′ is another choice satisfying the properties above. Without loss of generality we may suppose that L ⊂ L′. If K ⊂_{f} F ⊂ K, then F L′/F L is unramified outside N, ∞, and the conductor 𝔣 of ρ. The primes which divide 𝔣 but do not divide p are already ramified in F L/K, so the Euler system distribution relation shows that Cor_{F L′/F L}(c_{F L′}) = c_{F L}.

The definition of c_{F}^{ρ} does depend on the choice of ξ_{ρ}, but only up to a unit in O^{×}.

Remark 6.3.3. Let ξ_{ρ,n} denote the image in O_{ρ}/p^{n}O_{ρ} of the chosen generator ξ_{ρ} of O_{ρ}. An examination of the proof of Proposition 6.2.1(i) shows that for K ⊂_{f} F ⊂ K_{∞}, with L_{n} as in that proof, we have

c_{F L_{n}} ⊗ ξ_{ρ,n} ∈ H^{1}(F L_{n}, (T ⊗ ρ)/p^{n}(T ⊗ ρ))

and the definition of c_{F}^{ρ} easily seen to be equivalent to

c_{F}^{ρ} = lim_{n→∞} Cor_{F L_{n}/F} (c_{F L_{n}} ⊗ ξ_{ρ,n})

∈ \underleftarrow{lim}_{n} H^{1}(F, (T ⊗ ρ)/p^{n}(T ⊗ ρ)) = H^{1}(F, T ⊗ ρ).

Remark 6.3.4. When ρ is a character of finite order, this definition of c^{ρ} agrees with the one given in Definition 2.4.1. (Just take L to be the fixed field of ker(ρ).)

Theorem 6.3.5. Suppose that c is an Euler system for (T, K, N) where K_{∞} ⊂ K, and ρ : Gal(K/K) → O^{×} is a character which factors through a finite extension of K_{∞}. Then the collection of classes {c_{F}^{ρ} ∈ H^{1}(F, T ⊗ ρ)} defined above is an Euler system for (T ⊗ ρ, K, 𝔣N) where 𝔣 is the nonarchimedean, non-p part of the conductor of ρ.

Proof. Suppose K ⊂_{f} F ⊂_{f} F′ ⊂ K. We have a commutative diagram

H_{∞}^{1}(F′L, T) ⊗ ρ ⟶^{∼} H_{∞}^{1}(F′L, T ⊗ ρ) ⟶^{Cor} H^{1}(F′, T ⊗ ρ)
Cor⊗1↓  ^{Cor}↓  ^{Cor}_{F′/F} ↓
H_{∞}^{1}(F L, T) ⊗ ρ ⟶^{∼} H_{∞}^{1}(F L, T ⊗ ρ) ⟶^{Cor} H^{1}(F, T ⊗ ρ).

Since c is an Euler system, we have

Cor_{F′LK_{∞}/F LK_{∞}}(c_{F′L,∞}) = (∏_{𝔮∈S} P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}))c_{F L,∞}

where

S = {𝔮 of K : 𝔮 ramifies in F′L/K but not in F L/K, and 𝔮 ∤ N }
= {𝔮 of K : 𝔮 ramifies in F′/K but not in F/K, and 𝔮 ∤ 𝔣N },

the last equality because the conductor of L/K is divisible by 𝔣 and divides a power of 𝔣N ∞. Therefore

(Cor_{F′LK_{∞}/F LK_{∞}}(c_{F′L,∞}))⊗ξ_{ρ} = (∏_{𝔮∈S} P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})c_{F L,∞}) ⊗ ξ_{ρ}

= ∏_{𝔮∈S} P (Fr_{𝔮}^{−1}|T^{∗}; ρ(Fr_{𝔮})Fr_{𝔮}^{−1})(c_{F L,∞} ⊗ ξ_{ρ})

and so, using the diagram above,

Cor_{F′/F} (c_{F′}^{ρ}) = ∏_{𝔮∈S} P (Fr_{𝔮}^{−1}|T^{∗}; ρ(Fr_{𝔮})Fr_{𝔮}^{−1})c_{F}^{ρ} .

Since

det(1 − Fr_{𝔮}^{−1}x|(T ⊗ ρ)^{∗}) = det(1 − ρ(Fr_{𝔮})Fr_{𝔮}^{−1}x|T^{∗})
= P (Fr_{𝔮}^{−1}|T^{∗}; ρ(Fr_{𝔮})x),

this shows that c^{ρ} is an Euler system for (T ⊗ ρ, K, 𝔣N).

Lemma 6.3.6. Suppose that c is an Euler system for (T, K, N) where K_{∞} ⊂ K, and ρ, ρ′ : Gal(K/K) → O^{×} are characters which factor through a finite extension of K_{∞}. Let 𝔣_{ρ}, 𝔣_{ρ′}, 𝔣_{ρρ′} be the non-archimedean, nonp part of the conductors of ρ, ρ′, and ρρ′, respectively. Suppose that the generator ξ_{ρρ′} of O_{ρρ′} = O_{ρ} ⊗ O_{ρ′} is chosen so that ξ_{ρρ′} = ξ_{ρ} ⊗ ξ_{ρ′}.

If every prime divisor of 𝔣_{ρ}𝔣_{ρ′} divides 𝔣_{ρρ′}N, then (c^{ρ})^{ρ′} = c^{ρρ′}. In particular, if 𝔣_{ρ} | N then (c^{ρ})^{ρ^{−1}} = c.

Proof. Let L_{ρ} be a finite extension of K satisfying (i) and (ii) of Definition 6.3.1 for ρ, and similarly for L_{ρ′}.

The assumption on the conductors of ρ, ρ′, and ρρ′ ensures that the compositum L_{ρ}L_{ρ′} satisfies Definition 6.3.1(i) and (ii) for ρρ′. The lemma now follows easily from the definitions of c^{ρ}, c^{ρ′}, and c^{ρρ′} (and Remark 6.3.2).

6.4. Twisting Theorems

Recall that Γ = Gal(K_{∞}/K).

Theorem 6.4.1. If ρ : Γ → O^{×} is a character then Theorems 2.3.2, 2.3.3, and 2.3.4 for T and c are equivalent to Theorems 2.3.2, 2.3.3, and 2.3.4, respectively, for T ⊗ ρ and c^{ρ}, where c^{ρ} is the Euler system for T ⊗ ρ given by Theorem 6.3.5.

Proof. The hypotheses Hyp(K_{∞}, T), Hyp(K_{∞}, V), and Hyp(K_{∞}/K) depend only on the action of G_{K_{∞}} on T, so they are not affected by twisting by characters of Γ.

Write X_{∞}(T) = X_{∞} and let X_{∞}(T ⊗ρ) be the corresponding Λ-module associated to T ⊗ ρ, as defined in Corollary 6.2.2. That corollary says that

X_{∞}(T ⊗ ρ) ≅ X_{∞}(T) ⊗ ρ,

so by Lemma 6.1.2(i),

Tw_{ρ}(char(X_{∞}(T ⊗ ρ))) = char(X_{∞}(T)).

The argument of Lemma 6.1.2, along with Proposition 6.2.1(i), shows that

Tw_{ρ}(ind_{Λ}(c^{ρ})) = ind_{Λ}(c).

The theorem follows from these equalities.

6.5. Examples and Applications

Recall that ε_{cyc} : G_{K} → Z_{p}^{×} ⊂ O^{×} is the cyclotomic character, and let ω : G_{K} → (Z_{p}^{×})_{tors} be the Teichmüller character giving the action of G_{K} on μ_{p} (if p is odd) or on μ_{4} (if p = 2).

Tate twists. Suppose μ_{p^{∞}} ⊂ K, so ε_{cyc} factors through Gal(K/K). If T is a p-adic representation of G_{K}, then for every integer n we write T (n) for the Tate twist T ⊗ ε_{cyc}^{n}. By Theorem 6.3.5, an Euler system c for (T, K, N) gives an Euler system c^{ε_{cyc}^{n}} for (T (n), K, N), and Lemma 6.3.6 shows that (c^{ε_{cyc}^{n}})^{ε_{cyc}^{m}} = c^{ε_{cyc}^{n+m}} for every n and m.

Now take K_{∞} to be the cyclotomic Z_{p}-extension of K. Then ε_{cyc} does not necessarily factor through Gal(K_{∞}/K), but ω^{−1}ε_{cyc} does. Thus if c is an Euler system for (T, K_{∞}), then Theorem 6.4.1 shows that for every n, Theorems 2.3.2, 2.3.3, and 2.3.4 for T and c are equivalent to those same theorems for T ⊗ ω^{−n}ε_{cyc}^{n} and c^{ω^{−n}ε_{cyc}^{n}}.

Cyclotomic fields. In §3.2 and §3.4 we used cyclotomic units and Stickelberger elements, respectively, to construct Euler systems c_{cyc} for Z_{p}(1) and c_{St} for Z_{p}.

Exercise. Both c_{St}^{ε_{cyc}} and c_{cyc} are Euler systems for Z_{p}(1). Determine the relation between them.

Elliptic curves with complex multiplication. Let K be an imaginary quadratic field and K_{∞} the Z_{p}^{2}-extension of K. Suppose E is an elliptic curve defined over K with complex multiplication by the ring of integers O_{K} of K. Fix a prime 𝔭 of K above p, and let O be the completion of O_{K} at 𝔭. Let T_{𝔭}(E) denote the 𝔭-adic Tate module of E (see Example 1.1.5), which is a free rank-one O-module. Let ψ be the character

ψ : G_{K} ⟶ Aut_{O_{K}} (E_{𝔭^{∞}}) ≅ O^{×}.

Then T_{𝔭}(E) ≅ O_{ψ}.

Let c_{ell} denote the Euler system of elliptic units for O(1) over K of §3.3. The character ψε_{cyc}^{−1} factors through a finite extension of K_{∞}, so by Theorem 6.3.5 we obtain an Euler system

c_{E,𝔭} = c_{ell}^{ψε_{cyc}^{−1}}

for O(1) ⊗ ψε_{cyc}^{−1} = O_{ψ} ≅ T_{𝔭}(E). In particular we get an element

c_{E,𝔭,K} ∈ H^{1}(K, T_{𝔭}(E)).

Let V_{𝔭}(E) = T_{𝔭}(E) ⊗ K_{𝔭}. If v divides p, then as in §1.6.D we define

H_{f}^{1}(K_{v}, V_{𝔭}(E)) = image(E(K_{v}) ⊗ Q_{p} ↪ H^{1}(K_{v}, V_{𝔭}(E))).

Corollary 1.3.3 and Theorem 1.4.1 show that H^{1}(K_{v}, V_{𝔭}(E)) = 0 for all v not dividing p. Write 𝔭^{∗} for the conjugate of 𝔭 under the nontrivial automorphism of K/Q. If p splits in K, so p = 𝔭𝔭^{∗}, then one can show that

H_{f}^{1}(K_{𝔭}, V_{𝔭}(E)) = H^{1}(K_{𝔭}, V_{𝔭}(E)), H_{f}^{1}(K_{𝔭^{∗}}, V_{𝔭}(E)) = 0.

Therefore (whether or not p splits) H^{1}(K_{v}, T_{𝔭}(E)) = H_{f}^{1}(K_{v}, T_{𝔭}(E)) for all v ≠ 𝔭^{∗}, so

H^{1}(K, T_{𝔭}(E)) = S^{{𝔭^{∗}}}(K, T_{𝔭}(E)).

In particular, we have

c_{E,𝔭,K} ∈ S^{{𝔭^{∗}}}(K, T_{𝔭}(E)).

Now suppose further that the L-function L(E_{/K}, s) of E vanishes at s = 1. Then one can show (see [Ru7] §4 for the proof in the case that 𝔭 splits in K; the general case is essentially the same) that the image of c_{E,𝔭,K} in H_{s}^{1}(K_{𝔭^{∗}}, T_{𝔭}(E)) is zero, so in fact

c_{E,𝔭,K} ∈ S(K, T_{𝔭}(E)).

If we assume further that the 𝔭-part of the Tate-Shafarevich group � (E_{/K}) is finite (and this is known if E is defined over Q and ord_{s=1}L(E, s) = 1; see [Ko2] or [Ru5]) then the exact sequence

0 ⟶ E(K) ⊗ Z_{p} ⟶ S(K, T_{𝔭}(E)) ⟶ \underleftarrow{lim} � (E_{/K})_{𝔭^{n}} ⟶ 0

shows that S(K, T_{𝔭}(E)) ≅ E(K) ⊗ Z_{p}, so

c_{E,𝔭,K} ∈ E(K) ⊗ Z_{p}.

If 𝔭 splits in K, one can compute the p-adic height of c_{E,𝔭,K} in terms of the derivative of the 𝔭-adic L-function of E at s = 1. In particular one can prove the following theorem.

Theorem 6.5.1. Suppose E is defined over Q and p splits into two distinct primes in K. If ord_{s=1}L(E_{/Q}, s) = 1, then c_{E,𝔭,K} has infinite order in E(K) ⊗ Z_{p}.

See [Ru7] §9 for the details.

CHAPTER 7 Iwasawa Theory

In this chapter we use the cohomology classes constructed in Chapter 4, along with the duality results of §1.7, to prove Theorems 2.3.2, 2.3.3, and 2.3.4. The proofs follow generally along the same lines as the proof of Theorem 2.2.2 given in Chapter 5, except that where in Chapter 5 we dealt with O-modules, we must now deal with O[Gal(F/K)]-modules for K ⊂_{f} F ⊂ K_{∞}. This makes the algebra much more complicated.

In §7.1 we give the proofs of Theorems 2.3.3 and 2.3.4, assuming Theorem 2.3.2 and two propositions (Propositions 7.1.7 and 7.1.9), whose proofs will be given in the following sections.

We keep the notation of Chapter 2. In particular Γ = Gal(K_{∞}/K) and Λ = O[[Γ]]. If K ⊂_{f} F ⊂ K_{∞} and M ∈ O is nonzero, then we write Λ_{F} = O[Gal(F/K)] and

Λ_{F,M} = Λ_{F} /MΛ_{F} = (O/MO)[Gal(F/K)].

We assume throughout this chapter that we have a p-adic representation T of G_{K} and an Euler system c for (T, K_{∞}) such that

c_{K,∞} = {c_{F} }_{F} ∉ H_{∞}^{1}(K, T)_{tors}

(or else there is nothing to prove). We assume that hypotheses Hyp(K_{∞}, V) are satisfied, and we fix once and for all a τ ∈ G_{K} as in hypothesis Hyp(K_{∞}, V)(i), i.e., τ is the identity on K(1), on K_{∞}, on μ_{p^{∞}}, and on (O_{K}^{×})^{1/p^{∞}}, and dim_{Φ}(V/(τ − 1)V) = 1.

7.1. Overview

Since τ fixes μ_{p^{∞}}, we have
dim_{Φ}(V^{∗}/(τ − 1)V^{∗}) = dim_{Φ}(V^{τ=1}) = 1.

Definition 7.1.1. Fix an isomorphism

θ^{∗} : W^{∗}/(τ − 1)W^{∗} ⟶^{∼} D.

Recall that Ω = K(1)K(W, μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}). Define Ω_{∞} = K_{∞}Ω and let Ω_{∞}^{⟨τ⟩} be the fixed field of τ in Ω_{∞}.

There is a natural evaluation homomorphism

Ev^{∗} : G_{Ω_{∞}^{⟨τ⟩}} ⟶ Hom(S_{Σ_{p}}(K_{∞}, W^{∗}), D) = X_{∞},

defined as follows. For every σ ∈ G_{Ω_{∞}^{⟨τ⟩}} and class c ∈ S_{Σ_{p}}(K_{∞}, W^{∗}), we set

Ev^{∗}(σ)(c) = θ^{∗}(c(σ))

where c(σ) means any cocycle in the class c, evaluated at σ. Then c(σ) is well-defined modulo (σ − 1)W^{∗}, and σ acts on W^{∗} through Gal(Ω_{∞}/Ω_{∞}^{⟨τ⟩}) which is (topologically) generated by τ, so

(σ − 1)W^{∗} ⊂ (τ − 1)W^{∗} = ker(θ^{∗}).

Thus Ev^{∗}(σ) is well-defined, and the cocycle relation shows that Ev^{∗} is a homomorphism.

Definition 7.1.2. Define a positive integer a_{τ} by

a_{τ} = [W^{τ=1} : (W^{τ=1})_{div}] · max{|Z|, |Z^{∗}|}

where (W^{τ=1})_{div} is the maximal divisible submodule of W^{τ=1}, and Z (resp., Z^{∗}) is the maximal G_{K_{∞}}-stable submodule of (τ − 1)W (resp., (τ − 1)W^{∗}).

Lemma 7.1.3. (i) a_{τ} is finite.

(ii) If T and τ satisfy hypotheses Hyp(K_{∞}, T) then a_{τ} = 1.

Proof. Let Z be as in Definition 7.1.2. The maximal divisible submodule of Z gives rise to a G_{K_{∞}}-stable subspace V_{0} ⊂ (τ − 1)V ⫋ V. Hypothesis Hyp(K_{∞}, V)(ii) asserts that V is irreducible, so we must have V_{0} = 0. Hence Z is finite, and similarly Z^{∗} is finite. The index [W^{τ=1} : (W^{τ=1})_{div}] is finite simply because W has finite Z_{p}-corank. This proves (i).

Now suppose hypotheses Hyp(K_{∞}, T) hold. Then W_{𝔭} is an irreducible G_{K_{∞}}-module (where 𝔭 is the maximal ideal of O), and W_{𝔭} ⊄ (τ − 1)W because W_{M}/(τ − 1)W_{M} is free of rank one over O/MO for every nonzero M, so it follows that Z = 0. Similarly Z^{∗} = 0, and Proposition A.2.5 shows that W^{τ=1} = (W^{τ=1})_{div}. This proves (ii).

Recall that N is the ideal of Definition 2.1.1 corresponding to c. Consider the following two extra assumptions.

Assumption 7.1.4. For every K ⊂_{f} F ⊂ K_{∞}, both Λ_{F} /char(X_{∞})Λ_{F} and X_{∞} ⊗ Λ_{F} are finite.

Assumption 7.1.5. For every prime λ of K dividing N, the decomposition group of λ in G_{K} contains an element γ_{λ} with the property that

T^{γ_{λ}^{p^{n}} =1} = (T^{∗})^{γ_{λ}^{p^{n}} =1} = 0 for every n ≥ 0.

Remark 7.1.6. Suppose that X_{∞} is a torsion Λ-module. If ρ : Γ → O^{×} is a character, and we replace T by its twist T ⊗ρ, then Corollary 6.2.2 shows that X_{∞} is replaced by X_{∞} ⊗ ρ. Thus Lemma 6.1.3 applied to T ⊕ T^{∗} (in part (i)) and to X_{∞} ⊕ Λ/char(X_{∞}) (in part (ii)) shows that there is a ρ such that after twisting by ρ, Assumptions 7.1.4 and 7.1.5 are satisfied. Theorem 6.4.1 shows that Theorems 2.3.2, 2.3.3, and 2.3.4 for T and c are equivalent to Theorems 2.3.2, 2.3.3, and 2.3.4, respectively, for T ⊗ ρ and the twisted Euler system c^{ρ} of §6.3.

Suppose now that Theorem 2.3.2 holds (the proof will be given in §7.3), so that X_{∞} is a torsion Λ-module. Then the discussion above shows that, without loss of generality, to prove Theorems 2.3.3 and 2.3.4 we may assume that 7.1.4 and 7.1.5 are satisfied. We will assume this for the rest of this section.

As discussed in §2.3, since X_{∞} is a torsion Λ-module we can fix an injective pseudo-isomorphism

⨁_{i=1}^{r} Λ/f_{i}Λ ⟶ X_{∞},  (7.1)

where f_{1}, …, f_{r} ∈ Λ satisfy f_{i+1} | f_{i} for 1 ≤ i ≤ r − 1. The sequence of principal ideals (elementary divisors) f_{1}Λ, …, f_{r}Λ is uniquely determined by these conditions, and the characteristic ideal of X_{∞} is

char(X_{∞}) = ∏_{i=1}^{r} f_{i}Λ.  (7.2)

Since X_{∞} is a torsion Λ-module, all the f_{i} are nonzero.

Assume for the rest of this section that, in addition to hypotheses Hyp(K_{∞}, V), hypothesis Hyp(K_{∞}/K) is satisfied as well.

Proposition 7.1.7. With r as above, there are elements z_{1}, …, z_{r} ∈ X_{∞} and ideals 𝔤_{1}, …, 𝔤_{r} ⊂ Λ such that for 1 ≤ k ≤ r we have

(i) z_{k} ∈ Ev^{∗}(τG_{Ω_{∞}}), (ii) a_{τ} 𝔤_{k} ⊂ f_{k}Λ, and if k < r then 𝔤_{k} ⊂ 𝔤_{k+1}, (iii) there is a split exact sequence

0 ⟶ ∑_{i=1}^{k−1} Λz_{i} ⟶ ∑_{i=1}^{k} Λz_{i} ⟶ Λ/𝔤_{k} ⟶ 0,

(iv) a_{τ} (X_{∞}/ ∑_{i=1}^{r} Λz_{i}) is pseudo-null.

The proof of Proposition 7.1.7 will be given in §7.6. Using (7.1) it is easy to find {z_{i}}, with 𝔤_{i} = f_{i}Λ, satisfying (ii), (iii), and (iv), but condition (i) will be essential for our purposes.

Definition 7.1.8. Fix a sequence z_{1}, …, z_{r} ∈ X_{∞} as in Proposition 7.1.7 and define

Z_{∞} = ∑_{i=1}^{r} Λz_{i} ⊂ X_{∞}.

Let M denote the maximal ideal of Λ. If 0 ≤ k ≤ r, a Selmer sequence σ of length k is a k-tuple (σ_{1}, …, σ_{k}) of elements of τG_{Ω_{∞}} satisfying

Ev^{∗}(σ_{i}) − z_{i} ∈ MZ_{∞}

for 1 ≤ i ≤ k. (When k = 0, the empty sequence is a Selmer sequence.) Note that by Proposition 7.1.7(i), Selmer sequences exist, for example with all the above differences equal to zero.

Suppose M is a power of p. Let Ω_{M} = K(1)K(W_{M}, μ_{M}, (O_{K}^{×})^{1/M}), and if K ⊂_{f} F ⊂ K_{∞} let L_{F,M} ⊃ F Ω_{M} be the fixed field of the subgroup

⋂_{c∈S_{Σp} (F,W_{M}^{∗})}ker((c)_{F Ω_{M}}) ⊂ G_{F Ω_{M}} .

The restriction of S_{Σ_{p}}(F, W_{M}^{∗}) to F Ω_{M} is a finite (Lemma 1.5.7) subgroup of Hom(G_{F Ω_{M}} , W_{M}^{∗}), so L_{F,M} is a finite abelian extension of F Ω_{M}. It is straightforward to check that L_{F,M}/K is Galois and unramified outside primes above p, ∞, and primes where T is ramified.

For 0 ≤ k ≤ r we call a k-tuple (𝔮_{1}, …, 𝔮_{k}) of primes of K a Kolyvagin sequence (for F and M) if there is a Selmer sequence σ of length k such that for 1 ≤ i ≤ k,

• 𝔮_{i} does not divide the ideal N of Definition 2.1.1, and

• Fr_{𝔮_{i}} is (a conjugate of) σ_{i} on L_{F,M} (all primes not dividing N are unramified in L_{F,M}/K). If π is a Kolyvagin sequence of length k we define

𝔯(π) = ∏_{i=1}^{k} 𝔮_{i}.

By Lemma 4.1.3, 𝔯(π) belongs to the set R_{F,M} defined in Definition 4.1.1.

Let Π(k, F, M) be the set of all Kolyvagin sequences of length k for F and M. When k = 0, we make the convention that Π(k, F, M) has a single element, the empty sequence (independent of F and M). Define an ideal in Λ_{F,M}

Ψ(k, F, M) = ∑_{π∈Π(k,F,M)} ∑_{ψ} ψ(κ_{[F,𝔯(π),M]}) ⊂ Λ_{F,M}

where κ_{[F,𝔯(π),M]} is the Euler system derivative class constructed in §4.4, ⟨κ_{[F,𝔯(π),M]}⟩ is the Λ_{F,M}-submodule of H^{1}(F, W_{M}) generated by κ_{[F,𝔯(π),M]}, and the inner sum is over ψ ∈ Hom_{Λ}(⟨κ_{[F,𝔯(π),M]}⟩, Λ_{F,M}). In other words, Ψ(k, F, M) is the ideal of Λ_{F,M} generated by all homomorphic images of modules ⟨κ_{[F,𝔯(π),M]}⟩ as π runs through Π(k, F, M).

Proposition 7.1.9. There is an element h ∈ Λ such that (i) h is relatively prime to char(X_{∞}), (ii) for every K ⊂_{f} F ⊂ K_{∞} there is a power N_{F} of p such that if M is

a power of p and 0 ≤ k < r, then

a_{τ}^{5}hΨ(k, F, MN_{F})Λ_{F,M} ⊂ f_{k+1}Ψ(k + 1, F, M).

Proposition 7.1.9 is the key to the proofs of Theorems 2.3.3 and 2.3.4; it will be proved in §7.7. We now show how to use Proposition 7.1.9 to complete the proof of Theorems 2.3.3 and 2.3.4. Recall that if Σ is a set of places of K, then K_{Σ} denotes the maximal extension of K in K̄ which is unramified outside Σ.

Corollary 7.1.10. Suppose K ⊂_{f} F ⊂ K_{∞} and h ∈ Λ satisfies Proposition 7.1.9. Let Σ be a set of places of K containing all primes above p, all primes where T is ramified, and all infinite places,

If ψ ∈ Hom_{Λ}(H^{1}(K_{Σ}/F, T), Λ_{F}), then a_{τ}^{5r}h^{r}ψ(c_{F}) ∈ char(X_{∞})Λ_{F} .

Proof. Note that c_{F} ∈ H^{1}(K_{Σ}/F, T) by Corollary B.3.6.

Let N_{F} be as in Proposition 7.1.9(ii). Suppose 0 ≤ k < r and M is a power of p. Proposition 7.1.9(ii) shows that

a_{τ}^{5}hΨ(k, F, MN_{F}^{r−k})Λ_{F,M} ⊂ f_{k+1}Ψ(k + 1, F, MN_{F}^{r−k−1})Λ_{F,M},

so by induction, writing M′ = MN_{F}^{r} and using (7.2), we conclude that

a_{τ}^{5r}h^{r}Ψ(0, F, M′)Λ_{F,M} ⊂ (∏_{i=1}^{r} f_{i})Ψ(r, F, M)

⊂ (∏_{i=1}^{r} f_{i})Λ_{F,M} = char(X_{∞})Λ_{F,M}.  (7.3)

By Lemma 4.4.13(i), κ_{[F,1,M′]} is the image of c_{F} under the injection

H^{1}(K_{Σ}/F, T)/M′H^{1}(K_{Σ}/F, T) ↪ H^{1}(K_{Σ}/F, W_{M′}) ↪ H^{1}(F, W_{M′}).

Let ψ̄ denote the composition

Λ_{F,M′}κ_{[F,1,M′]} ↪ H^{1}(K_{Σ}/F, T)/M′H^{1}(K_{Σ}/F, T) ⟶^{ψ} Λ_{F,M′} → Λ_{F,M}

induced by the inverse of this injection and by ψ. By definition

ψ̄(κ_{[F,1,M′]}) ∈ Ψ(0, F, M′)Λ_{F,M},

so (7.3) shows

a_{τ}^{5r}h^{r}ψ̄(κ_{[F,1,M′]}) ∈ char(X_{∞})Λ_{F,M}.

Since this holds for every sufficiently large M, and ψ̄(κ_{[F,1,M′]}) is the reduction of ψ(c_{F}) modulo M, this completes the proof of the corollary.

Lemma 7.1.11. Suppose that G is a finite abelian group, that R is a principal ideal domain, that B is finitely generated R[G]-module with no R-torsion, and that f ∈ R[G] is not a zero-divisor. If b ∈ B is such that

{ψ(b) : ψ ∈ Hom_{R[G]}(B, R[G])} ⊂ fR[G],

then b ∈ fB.

Proof. Let B′ = Rb + fB. Since f is not a zero-divisor, we have a commutative diagram

Hom_{R[G]}(B′, fR[G]) ⟵^{f} Hom_{R[G]}(B′, R[G]) ⟶^{∼} Hom_{R}(B′, R)
↓  ↓  ↓
Hom_{R[G]}(fB, fR[G]) ⟵^{f} Hom_{R[G]}(fB, R[G]) ⟶^{∼} Hom_{R}(fB, R)

in which the horizontal maps are all isomorphisms (see for example Lemma 4.3.3 for the isomorphisms on the right).

Suppose φ̄ ∈ Hom_{R[G]}(fB, fR[G]). Since B has no R-torsion, φ̄ extends uniquely to a map φ : B → R[G], and by our assumption on b, the restriction of φ belongs to Hom_{R[G]}(B′, fR[G]). Thus all the vertical maps in the diagram above are surjective. Since B′ and fB are free R-modules, the surjectivity of the right-hand map shows that B′ = fB, which proves the lemma.

Let ind_{Λ}(c) be as in Definition 2.3.1.

Theorem 7.1.12. char(X_{∞}) divides a_{τ}^{5r}ind_{Λ}(c).

Proof. Suppose h ∈ Λ is as in Proposition 7.1.9. Let Σ be a finite set of places of K containing all primes above p, all primes where T is ramified, and all infinite places. If K ⊂_{f} F ⊂ K_{∞}, Corollary 7.1.10 and Lemma 7.1.11 applied with B = H^{1}(K_{Σ}/F, T)/H^{1}(K_{Σ}/F, T)_{tors} and b = h^{r}a_{τ}^{5r}c_{F} show (note that H^{1}(K_{Σ}/F, T) is finitely generated over Z_{p} by Proposition B.2.7) that

a_{τ}^{5r}h^{r}c_{F} ∈ char(X_{∞})(H^{1}(K_{Σ}/F, T)/H^{1}(K_{Σ}/F, T)_{tors}).

It follows from Lemma 1.2.2(ii) that if K ⊂_{f} F ⊂ K_{∞}, then H^{1}(F, T)_{tors} is annihilated by the annihilator in Λ of W^{G_{K∞}}, so

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}(H^{1}(F, T)_{tors}) ⊂ H_{∞}^{1}(K, T)_{tors}

(where the latter group is the Λ-torsion submodule). Passing to the inverse limit we deduce that

a_{τ}^{5r}h^{r}c_{K,∞} ∈ char(X_{∞})(H_{∞}^{1}(K, T)/H_{∞}^{1}(K, T)_{tors}).

Therefore if ϕ ∈ Hom_{Λ}(H_{∞}^{1}(K, T), Λ) then

a_{τ}^{5r}h^{r}ϕ(c_{K,∞}) ∈ char(X_{∞}).

Since h is relatively prime to (the principal ideal) char(X_{∞}), it follows that

a_{τ}^{5r}ϕ(c_{K,∞}) ∈ char(X_{∞}).

This completes the proof.

Proof of Theorems 2.3.3 and 2.3.4. Lemma 7.1.3(i) shows that a_{τ} is a (finite) positive integer, so Theorem 2.3.4 is immediate from Theorem 7.1.12. If in addition T and τ satisfy hypotheses Hyp(K_{∞}, T), then a_{τ} = 1 by Lemma 7.1.3(ii), and Theorem 2.3.3 follows as well.

7.2. Galois Groups and the Evaluation Map

Keep the notation of the previous section.

Definition 7.2.1. Define q_{τ} (x) = det(1 − τ^{−1}x|T^{∗})/(x − 1). Our assumptions on τ ensure that

q_{τ} (x) = det(1 − τx|T)/(x − 1) ∈ O[x]

and that, by Lemma A.2.4(ii) (applied with σ = τ^{−1}),

q_{τ} (τ^{−1}) : V/(τ − 1)V ⟶^{∼} V^{τ=1}

is an isomorphism of one-dimensional vector spaces.

The D(1)-dual of the isomorphism θ^{∗} of Definition 7.1.1 is an isomorphism

O(1) ⟶^{∼} T^{τ=1}.

The inverse of this isomorphism, together with the generator ξ of O(1) chosen in Definition 4.4.1, gives an isomorphism

θ : (W^{τ=1})_{div} ⟶^{∼} D.

Define θ̄ to be the (surjective, by Lemma A.2.4) composition

W/(τ − 1)W ⟶^{q_{τ} (τ^{−1})} (W^{τ=1})_{div} ⟶^{θ} D.

We also fix once and for all an extension of θ to W^{τ=1}

θ : W^{τ=1} ⟶ D.

This extension is not in general unique, but the difference between any two choices lies in Hom(W^{τ=1}/(W^{τ=1})_{div}, D) which is killed by a_{τ}.

Definition 7.2.2. Recall the evaluation homomorphism

Ev^{∗} : G_{Ω_{∞}^{⟨τ⟩}} ⟶ X_{∞}

of Definition 7.1.1. Similarly we define

Ev : G_{Ω_{∞}^{⟨τ⟩}} ⟶ Hom(H^{1}(K_{∞}, W), D)

by

Ev(σ)(c) = θ̄(c(σ)) = θ ∘ q_{τ} (τ^{−1})(c(σ))

for all σ ∈ G_{Ω_{∞}^{⟨τ⟩}} and c ∈ H^{1}(K_{∞}, W), where c(σ) means any cocycle representing c, evaluated at σ.

Suppose M ∈ O is nonzero and K ⊂_{f} F ⊂ K_{∞}. Let Ω_{M}^{⟨τ⟩} denote the fixed field of τ in Ω_{M} = K(1)K(W_{M}, μ_{M}, (O_{K}^{×})^{1/M})/K). In exactly the same way, we define evaluation maps

Ev_{F,M}^{∗} : G_{F Ω_{M}^{⟨τ⟩}} ⟶ Hom(S_{Σ_{p}}(F, W_{M}^{∗}), O/MO)
Ev_{F,M} : G_{F Ω_{M}^{⟨τ⟩}} ⟶ Hom(H^{1}(F, W_{M}), O/MO)

by Ev_{F,M}^{∗}(σ)(c) = θ^{∗}(c(σ)) and Ev_{F,M}(σ)(c) = θ̄(c(σ)), where we identify O/MO with M^{−1}O/O ⊂ D.

Definition 7.2.3. If η ∈ Λ, we will denote by η^{•} the image of η under the involution of Λ induced by γ ↦ γ^{−1} for γ ∈ Γ. Similarly if A is an ideal of Λ we will write A^{•} for the ideal which is the image of A under this involution.

We will use repeatedly below that if B is a Λ-module and A is an ideal of Λ which annihilates B, then A^{•} annihilates Hom(B, D).

If B is a Λ-module, Ann_{Λ}(B) will denote the annihilator in Λ of B.

Lemma 7.2.4. (i) If c ∈ H^{1}(K_{∞}, W) and Ev(γ)(c) = 0 for every γ ∈ G_{Ω_{∞}}, then a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))c = 0.

(ii) a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•} Hom(H^{1}(K_{∞}, W), D) ⊂ OEv(G_{Ω_{∞}}).

(iii) a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W^{∗}))^{•} X_{∞} ⊂ OEv^{∗}(G_{Ω_{∞}}).

Proof. Unwinding the definition of Ev, we see that the dual of Ev on G_{Ω_{∞}} is given by the composition

H^{1}(K_{∞}, W) ⟶^{res_{Ω∞}} Hom(G_{Ω_{∞}}, W)^{G_{K∞}}

⟶ Hom(G_{Ω_{∞}}, W/(τ − 1)W) ⟶^{θ̄} Hom(G_{Ω_{∞}}, D). (7.4)

The kernel of the first map is H^{1}(Ω_{∞}/K_{∞}, W). The kernel of the second map is

Hom(G_{Ω_{∞}}, W)^{G_{K∞}} ∩ Hom(G_{Ω_{∞}}, (τ − 1)W).

If ψ belongs to this intersection, then ψ(G_{Ω_{∞}}) is a G_{K_{∞}}-stable submodule of (τ − 1)W. The kernel of θ̄ is W^{q_{τ} (τ^{−1})=0}/(τ − 1)W, which has the same order as W^{τ=1}/W_{div}^{τ=1} by Proposition A.2.5 (applied with σ = τ^{−1}). Thus a_{τ} annihilates the kernel of the composition of the second and third maps.

If Ev(γ)(c) = 0 for every γ ∈ G_{Ω_{∞}}, then c maps to zero under (7.4), so this proves (i). Applying Hom_{O}( · , D) to (7.4) yields

G_{Ω_{∞}} ⊗ O ⟶^{Ev} Hom(H^{1}(K_{∞}, W), D)

and (ii) follows. The proof of (iii) is the same as the proof of (ii) (except that in that case the third map of the analogue of (7.4) is induced by θ^{∗}, which is injective).

Definition 7.2.5. Suppose K ⊂_{f} F ⊂ K_{∞} and M is a power of p. Define

R_{F,M,τ} = {𝔯 ∈ R : for every prime 𝔮 dividing 𝔯, Fr_{𝔮} belongs to the conjugacy class of τ in Gal(F Ω_{M}/K)}

where Ω_{M} is as above (and as in Definition 7.1.8). By Lemma 4.1.3, R_{F,M,τ} ⊂ R_{F,M} where R_{F,M} is the set defined in Definition 4.1.1.

Suppose 𝔮 ∈ R_{F,M,τ}. Let 𝔔 be a prime of K̄ above 𝔮 such that the corresponding Frobenius Fr_{𝔮} of 𝔮 is τ on F Ω_{M}. Recall the generator σ_{𝔮} of Gal(K(𝔮)/K) given by Definition 4.4.1, and fix a lift of σ_{𝔮} to the inertia group I_{𝔔} of 𝔔 in G_{K}. Then both Fr_{𝔮} and σ_{𝔮} belong to G_{F Ω_{M}^{⟨τ⟩}}. We define the finite evaluation maps

Ev_{𝔮,f}^{∗} = Ev_{F,M}^{∗}(Fr_{𝔮}) : S_{Σ_{p}}(F, W_{M}^{∗}) ⟶ O/MO
Ev_{𝔮,f} = Ev_{F,M}(Fr_{𝔮}) : H^{1}(F, W_{M}) ⟶ O/MO.

By Lemma 1.4.7(i) (which applies thanks to Lemma 4.1.2(i)), evaluation at σ_{𝔮} induces

H^{1}(F, W_{M}) ⟶ H^{1}(F_{𝔔}, W_{M}) ⟶ H_{s}^{1}(F_{𝔔}, W_{M}) ⟶^{∼} W_{M}^{Fr_{𝔮}=1} = W_{M}^{τ=1}

and we define the singular evaluation map

Ev_{𝔮,s} : H^{1}(F, W_{M}) ⟶ O/MO

by Ev_{𝔮,s}(c) = θ(c(σ_{𝔮})).

Note that Ev_{𝔮,f}^{∗}, Ev_{𝔮,f}, and Ev_{𝔮,s} depend on F, M, and the choice of 𝔔 (as well as the specific choice of Fr_{𝔮} and σ_{𝔮}), but we will suppress this from the notation.

Remark 7.2.6. We will usually apply Ev_{𝔮,f} and Ev_{𝔮,f}^{∗} to cohomology classes which are finite at 𝔮, so we think of these maps (via Lemma 1.4.7(i)) as measuring the finite part at 𝔮 of a cohomology class. Similarly, we view Ev_{𝔮,s} as measuring the singular part at 𝔮.

Recall that Λ_{F} = O[Gal(F/K)] and Λ_{F,M} = Λ_{F} /MΛ_{F}.

Lemma 7.2.7. Suppose that K ⊂_{f} F ⊂ K_{∞}, that M is a power of p, and that B is a Λ_{F} -module.

(i) The map

Hom_{O}(B, O/MO) ⟶ Hom_{Λ}(B, Λ_{F,M})
ψ ↦ ψ̃

defined by

ψ̃(b) = ∑_{η∈Gal(F/K)} ψ(ηb)η^{−1}

is an O-module isomorphism. (ii) If ψ ∈ Hom_{O}(B, O/MO) and σ ∈ Gal(F/K) then σ̃ψ = σ^{−1}ψ̃.

Proof. The map Hom_{Λ}(B, Λ_{F,M}) → Hom_{O}(B, O/MO) induced by composition with ∑_{η∈Gal(F/K)} a_{η}η ↦ a_{1} is a two-sided inverse of the map in (i), and (ii) is easily checked. (Note that σ acts on ψ ∈ Hom_{O}(B, O/MO) by (σψ)(b) = ψ(σ^{−1}b) for every b ∈ B, and on ψ̃ ∈ Hom_{Λ}(B, Λ_{F,M}) by (σψ̃)(b) = σ(ψ̃(b)).)

Remark 7.2.8. Note that in Lemma 7.2.7, (ii) says that the bijection of (i) is not in general a Λ_{F,M}-module homomorphism.

Definition 7.2.9. Suppose K ⊂_{f} F ⊂ K_{∞} and M is a power of p. If γ ∈ G_{F Ω_{M}^{⟨τ⟩}} and 𝔮 ∈ R_{F,M,τ} we will write

Ẽv_{F,M}(γ), Ẽv_{𝔮,f}, Ẽv_{𝔮,s} ∈ Hom_{Λ}(H^{1}(F, W_{M}), Λ_{F,M})

for the images of Ev_{F,M}(γ), Ev_{𝔮,f}, and Ev_{𝔮,s}, respectively, under the map of Lemma 7.2.7(i). Thus

Ẽv_{𝔮,f}(c) = ∑_{η∈Gal(F/K)}Ev_{𝔮,f}(ηc)η^{−1}

and similarly for Ẽv_{F,M}(γ) and Ẽv_{𝔮,s}.

The next two results, Theorems 7.2.10 and 7.2.11, are crucial for the proof of Theorem 2.3.2 and Proposition 7.1.9. They are restatements of Theorems 4.5.4 and 1.7.3(ii), respectively, in the language of these evaluation maps.

Theorem 7.2.10. Suppose that c is an Euler system, that K ⊂_{f} F ⊂ K_{∞}, that M is a power of p, that 𝔯 ∈ R_{F,M}, and that 𝔮 ∈ R_{F,M,τ} is a prime not dividing 𝔯. Let κ_{[F,𝔯,M]} be the derivative class constructed in §4.4. Then

Ẽv_{𝔮,f}(κ_{[F,𝔯,M]}) = Ẽv_{𝔮,s}(κ_{[F,𝔯𝔮,M]}).

Proof. Suppose ρ ∈ G_{K}. Theorem 4.5.4 applied to the Euler system {ρc_{F (𝔯)}} shows that, with Q_{𝔮}(x) as in Lemma 4.5.2,

Ev_{𝔮,f}(ρκ_{[F,𝔯,M]}) = θ ∘ q_{τ} (τ^{−1})((ρκ_{[F,𝔯,M]})(Fr_{𝔮}))
= θ ∘ Q_{𝔮}(Fr_{𝔮}^{−1})((ρκ_{[F,𝔯,M]})(Fr_{𝔮}))
= θ((ρκ_{[F,𝔯𝔮,M]})(σ_{𝔮}))
= Ev_{𝔮,s}(ρκ_{[F,𝔯𝔮,M]}).

(Note that one consequence of Theorem 4.5.4 is that (ρκ_{[F,𝔯𝔮,M]})(σ_{𝔮}) ∈ W_{div}^{τ=1}, so Ev_{𝔮,s}(ρκ_{[F,𝔯𝔮,M]}) does not depend on any choice made in extending θ from W_{div}^{τ=1} to W^{τ=1}.) The theorem follows immediately.

Notation. If B is a G_{K}-module, v is a place of K, and K ⊂_{f} F ⊂ K_{∞}, we will abbreviate

F_{v} = F ⊗_{K} K_{v} = ⊕_{w|v}F_{w},
H^{1}(F_{v}, B) = ⊕_{w|v}H^{1}(F_{w}, B),
H_{f}^{1}(F_{v}, B) = ⊕_{w|v}H_{f}^{1}(F_{w}, B),
c_{v} = ⊕_{w|v}c_{w} ∈ H^{1}(F_{v}, B) for every c ∈ H^{1}(F, B).

There is a natural action of Gal(F/K) on H^{1}(F_{v}, B). Concretely, every σ ∈ Gal(F/K) induces an isomorphism

H^{1}(F_{w}, B) ⟶^{∼} H^{1}(F_{σw}, B)

for every w, and summing these maps over w lying above v gives an automorphism of H^{1}(F_{v}, B); see also Corollary B.5.2. In applying Theorem 1.7.3 over the base field F instead of K, all of the maps are Gal(F/K)- homomorphisms.

Theorem 7.2.11. Suppose that K ⊂_{f} F ⊂ K_{∞}, that M is a power of p, that 𝔯𝔮 ∈ R_{F,M} and that 𝔮 is a prime in R_{F,M,τ}. Let Σ_{p𝔯} and Σ_{p𝔯𝔮} denote the sets of primes of K dividing p𝔯 and p𝔯𝔮, respectively. Then

a_{τ} Ẽv_{𝔮,s}(S^{Σ_{p𝔯𝔮}}(F, W_{M})) Ev_{𝔮,f}^{∗}|_{S_{Σp𝔯} (F,W_{M}^{∗})} = 0.

Proof. Note that Ẽv_{𝔮,s}(S^{Σ_{p𝔯𝔮}}(F, W_{M})) ⊂ Λ_{F,M} and that Ev_{𝔮,f}^{∗}|_{S_{Σp𝔯} (F,W_{M}^{∗})} is in the Λ_{F,M}-module Hom(S_{Σ_{p𝔯}}(F, W_{M}^{∗}), D).

Suppose c ∈ S_{Σ_{p𝔯}}(F, W_{M}^{∗}) and d ∈ S^{Σ_{p𝔯𝔮}}(F, W_{M}). Theorem 1.7.3(ii), applied with Σ = Σ_{p𝔯𝔮} and Σ_{0} = Σ_{p𝔯}, shows that ⟨c, d⟩_{𝔮} = 0, where ⟨ , ⟩_{𝔮} = ∑_{𝔔|𝔮} ⟨ , ⟩_{𝔮} is the sum of the local pairings of Theorem 1.4.1 at primes above 𝔮.

Let 𝔔 be the prime above 𝔮 corresponding to our choices of Fr_{𝔮} and σ_{𝔮}. Consider the diagram

H_{f}^{1}(F_{𝔔}, W_{M}^{∗}) × H_{s}^{1}(F_{𝔔}, W_{M}) ⟶^{⟨ , ⟩_{𝔔}}  O/MO
↓  ↓  ↓^{±1⊗ξ}
W_{M}^{∗}/(τ − 1)W_{M}^{∗} × (W_{M})^{τ=1} ⟶^{⟨ , ⟩_{W_{M}}} O(1)/MO(1)
θ^{∗}↓  ^{θ}↓  ↓^{a_{τ} ⊗ξ^{−1}}
O/MO  ×  O/MO  ⟶^{a_{τ}}  O/MO

where

• the upper part (including the ambiguity of sign) comes from Lemma 1.4.7 (so the upper left and upper center vertical maps are isomorphisms given by evaluation at Fr_{𝔮} and σ_{𝔮}, respectively),

• ξ is the chosen generator of Z_{p}(1) from which we defined σ_{𝔮},

• ⟨ , ⟩_{W_{M}} is induced by the natural pairing W_{M}^{∗} × W_{M} → O(1), and • the pairing on the bottom is (x, y) ↦ a_{τ} xy.

Since a_{τ} annihilates (W^{τ=1})/(W^{τ=1})_{div}, it follows from Definitions 7.1.1 and 7.2.1 of θ^{∗} and θ, respectively, that the bottom half of the diagram commutes. In other words,

a_{τ} ⟨c, d⟩_{𝔔} = ±a_{τ} θ(d(σ_{𝔮}))θ^{∗}(c(Fr_{𝔮})) = ±a_{τ} Ev_{𝔮,s}(d)Ev_{𝔮,f}^{∗}(c).

Therefore

(a_{τ} Ẽv_{𝔮,s}(d)Ev_{𝔮,f}^{∗})(c) = a_{τ} ∑_{ρ∈Gal(F/K)}Ev_{𝔮,s}(ρd)(Ev_{𝔮,f}^{∗})^{ρ^{−1}}(c)

= a_{τ} ∑_{ρ∈Gal(F/K)}Ev_{𝔮,s}(ρd)Ev_{𝔮,f}^{∗}(ρc)

= ±a_{τ} ∑_{ρ∈Gal(F/K)}⟨ρc, ρd⟩_{𝔔}

= ±a_{τ} ∑_{ρ∈Gal(F/K)}⟨c, d⟩_{𝔔^{ρ}} = ±a_{τ} ⟨c, d⟩_{𝔮} = 0.

Corollary 7.2.12. Suppose that K ⊂_{f} F ⊂ K_{∞}, that M is a power of p, that 𝔯 ∈ R_{F,M}, and that γ ∈ τG_{Ω_{∞}}. Then

a_{τ} Ẽv_{F,M}(γ)(κ_{[F,𝔯,M]}) Ev_{F,M}^{∗}(γ)|_{S_{Σp𝔯} (F,W_{M}^{∗})} = 0.

Proof. Fix a finite Galois extension L of F (μ_{M}, (O_{K}^{×})^{1/M}, W_{M}) such that the restrictions to L of κ_{[F,𝔯,M]} and of S_{Σ_{p}}(F, W_{M}^{∗}) are zero (S_{Σ_{p}}(F, W_{M}^{∗}) is finite by Lemma 1.5.7, so such an extension exists). Let N be the ideal of Definition 2.1.1 corresponding to c. Choose a prime 𝔮 of K prime to 𝔯N (and a prime 𝔔 of K̄ above 𝔮) such that Fr_{𝔮} = γ on L.

We have Ev_{F,M}^{∗}(γ) = Ev_{F,M}^{∗}(Fr_{𝔮}) = Ev_{𝔮,f}^{∗}. Since 𝔮 ∈ R_{F,M,τ}, Theorems 7.2.10 and 4.5.1 show that

Ẽv_{F,M}(γ)(κ_{[F,𝔯,M]}) = Ẽv_{𝔮,f}(κ_{[F,𝔯,M]})
= Ẽv_{𝔮,s}(κ_{[F,𝔯𝔮,M]}) ∈ Ẽv_{𝔮,s}(S^{Σ_{p𝔯𝔮}}(F, W_{M})).

Now the corollary follows from Theorem 7.2.11.

7.3. Proof of Theorem 2.3.2

In this section we will prove Theorem 2.3.2. The general idea is that if c ∉ H_{∞}^{1}(K, T)_{tors}, then we can use Corollary 7.2.12 to construct a nonzero annihilator of X_{∞}, and hence X_{∞} is Λ-torsion.

Lemma 7.3.1. Suppose K ⊂_{f} F ⊂ K_{∞} and M ∈ O is nonzero. Let res_{S,∞} denote the restriction map

S_{Σ_{p}}(F, W^{∗}) ⟶ S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} .

If v is a place of F, fix an extension w of v to K_{∞} and define

B_{v} = \begin{cases}H^{1}(K_{∞,w}/F_{v}, (W^{∗})^{G_{K∞,w}}) & if v | p, \\ H_{ur}^{1}(F_{v}, W^{∗})/H_{f}^{1}(F_{v}, W^{∗}) & if v ∤ p. \end{cases}

Then there are a submodule B of ⊕_{v|N} B_{v} and an exact sequence

B ⟶ coker(res_{S,∞}) ⟶ H^{2}(K_{∞}/F, (W^{∗})^{G_{K∞}}).

Proof. Let res_{∞} denote the restriction map H^{1}(F, W^{∗}) → H^{1}(K_{∞}, W^{∗})^{G_{F}}. The inflation-restriction exact sequence shows that the cokernel of res_{K_{∞}} is isomorphic to a subgroup of H^{2}(K_{∞}/F, (W^{∗})^{G_{K∞}}), and hence the same is true for the cokernel of

res_{K_{∞}}^{−1}(S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}}) ⟶^{res_{K∞}} S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} .

Since K_{∞}/F is unramified outside primes above p, we have

res_{K_{∞}}^{−1}(S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}}) ⊂ S^{Σ_{pN}} (F, W^{∗})

so the cokernel of the inclusion

S_{Σ_{p}}(F, W^{∗}) ↪ res_{K_{∞}}^{−1}(S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}})

injects naturally into ⊕_{v|N} B_{v}. This proves the lemma.

Lemma 7.3.2. The Λ-module X_{∞} is finitely generated.

Proof. Let J denote the augmentation ideal in Λ. Then X_{∞}/J X_{∞} = Hom(S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{K}} , D). By Nakayama’s Lemma, to prove the lemma we need only show that Hom(S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{K}} , D) is finitely generated over O.

Let res_{S,∞} denote the restriction map

S_{Σ_{p}}(K, W^{∗}) ⟶ S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{K}} .

By Lemmas 7.3.1 and 1.3.5, Hom(coker(res_{S,∞}), D) is finitely generated over O. Lemma 1.5.7(iii) shows that Hom(S_{Σ_{p}}(K, W^{∗}), D) is also finitely generated over O, and the lemma follows.

Recall that Ω_{∞} = K_{∞}(1)K(W, μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}).

Lemma 7.3.3. Suppose X_{∞} is not a torsion Λ-module. Define

J = {γ ∈ τG_{Ω_{∞}} : Ev^{∗}(γ) ∉ (X_{∞})_{tors}}.

Then the subgroup of G_{K} generated by J contains a nonempty open subgroup of G_{Ω_{∞}}.

Proof. Corollary C.2.2 (with F = K_{∞}) shows that H^{1}(Ω_{∞}/K_{∞}, W^{∗}) is a torsion Λ-module. Therefore if X_{∞} is not a torsion Λ-module, Lemma 7.2.4(iii) shows that there is a γ_{0} ∈ G_{Ω_{∞}} such that Ev^{∗}(γ_{0}) ∉ (X_{∞})_{tors}. Then either τ or τγ_{0} belongs to J, so J is nonempty.

Since X_{∞} is finitely generated by Lemma 7.3.2, the submodule (X_{∞})_{tors} is closed in X_{∞}. The map Ev^{∗} is continuous, so

J = (Ev^{∗})^{−1}(X_{∞} − (X_{∞})_{tors}) ∩ τG_{Ω_{∞}}

is open in τG_{Ω_{∞}}, and the lemma follows.

Proof of Theorem 2.3.2. Let c be the Euler system of Theorem 2.3.2. We will show, under the assumption that X_{∞} is not a torsion Λ-module, that c_{K,∞} ∈ H_{∞}^{1}(K, T)_{tors}.

Suppose that X_{∞} is not a torsion Λ-module. Choose a γ in the set J of Lemma 7.3.3, i.e., γ ∈ τG_{Ω_{∞}} such that Ev^{∗}(γ) ∉ (X_{∞})_{tors}.

Suppose K ⊂_{f} F ⊂ K_{∞} and M is a power of p. Let κ_{[F,M]} = κ_{[F,1,M]} be the derivative class constructed in §4.4. By Corollary 7.2.12 (with 𝔯 = 1, the trivial ideal),

a_{τ} Ẽv_{F,M}(γ)(κ_{[F,M]}) Ev_{F,M}^{∗}(γ) = 0.

By definition the map Ẽv_{F,M}(γ) factors through restriction to K_{∞}, and for every F ⊂_{f} F′ we have

(κ_{[F,M]})_{F′} = (Cor_{F′/F} κ_{[F′,M]})_{F′} = ∑_{ρ∈Gal(F′/F)}ρκ_{[F′,M]}.

Hence Ẽv_{F′,M}(γ)(κ_{[F′,M]}) ∈ Λ_{F′,M} restricts to Ẽv_{F,M}(γ)(κ_{[F,M]}) ∈ Λ_{F,M}. Thus \underleftarrow{lim}_{F,M} Ẽv_{F,M}(γ)(κ_{[F,M]}) ∈ Λ and

a_{τ} (\underleftarrow{lim}_{F,M} Ẽv_{F,M}(γ)(κ_{[F,M]})) Ev^{∗}(γ) = 0.

Since Ev^{∗}(γ) ∉ (X_{∞})_{tors} it follows that \underleftarrow{lim} Ẽv_{F,M}(γ)(κ_{[F,M]}) = 0. This holds for every γ ∈ J, so Lemma 7.3.3 shows that it holds for every γ in a nonempty open subgroup of G_{Ω_{∞}}. Since a nonempty open subgroup has finite index, and Λ is torsion-free, we conclude that for every F, every M, and every γ ∈ G_{Ω_{∞}}, we have Ev_{F,M}(γ)(κ_{[F,M]}) = 0. Equivalently, writing (κ_{[F,M]})_{K_{∞}} for the image of κ_{[F,M]} in H^{1}(K_{∞}, W), we have

Ev(γ)((κ_{[F,M]})_{K_{∞}}) = 0.  (7.5)

We will show that this is not compatible with the assumption that

c_{K,∞} = {c_{F} }_{F} ∉ H_{∞}^{1}(K, T)_{tors}.

Recall that Ann_{Λ}(B) denotes the annihilator of a Λ-module B. By Lemma 7.2.4(i), it follows from (7.5) that

a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))(κ_{[F,M]})_{K_{∞}} = 0.

The inflation-restriction exact sequence shows that the kernel of the restriction map H^{1}(F, W) → H^{1}(K_{∞}, W) is H^{1}(K_{∞}/F, W^{G_{K∞}}) which (since K_{∞}/K is abelian) is annihilated by Ann_{Λ}(W^{G_{K∞}}). By Lemma 1.2.2(i), the kernel of the natural map H^{1}(F, W_{M}) → H^{1}(F, W) is annihilated by m_{F} = [W^{G_{F}} : (W^{G_{F}})_{div}]. Hence for every F and M, we have

m_{F} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))Ann_{Λ}(W^{G_{K∞}})κ_{[F,M]} = 0.

By Lemma 4.4.13(i), κ_{[F,M]} is the image of c_{F} under the injection

H^{1}(F, T)/MH^{1}(F, T) ↪ H^{1}(F, W_{M}).

It follows that m_{F} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))Ann_{Λ}(W^{G_{K∞}})c_{F} is divisible in H^{1}(F, T), so by Proposition B.2.4,

m_{F} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))Ann_{Λ}(W^{G_{K∞}})c_{F} = 0.

Lemma 1.2.2(ii) shows that the kernel of multiplication by m_{F} in H^{1}(F, T) is annihilated by Ann_{Λ}(W^{G_{K∞}}), so for every K ⊂_{f} F ⊂ K_{∞},

Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))Ann_{Λ}(W^{G_{K∞}})^{2}c_{F} = 0.

But this annihilator of c_{F} is independent of F, and by Corollary C.2.2 applied with F = K_{∞}, it is nonzero as well. Thus we have found a nonzero annihilator of c_{K,∞} ∈ H_{∞}^{1}(K, T). This contradicts the assumption that c_{K,∞} ∉ H_{∞}^{1}(K, T)_{tors}, and completes the proof.

We also record the following lemma for later use.

Lemma 7.3.4. Suppose Γ ≅ Z_{p}, and either K is totally real and Leopoldt’s conjecture holds for K, or K is imaginary quadratic.

(i) If G_{K_{∞}} acts trivially on T, then X_{∞}/Ann_{Λ}(T)X_{∞} is finite.

(ii) If G_{K_{∞}} acts trivially on T (−1), then X_{∞}/Ann_{Λ}(T (−1))X_{∞} is finite.

Proof. Recall that T (−1) = T ⊗ O_{ε_{cyc}^{−1}}. We have assumed (Hyp(K_{∞}, V)) that V is an irreducible G_{K_{∞}}-representation, so the situations (i) and (ii) can only arise if rank_{O}T = 1 and T is a twist of O or O(1), respectively, by a character of Γ.

Suppose ρ is a character of Γ. If we replace T by its twist T ⊗ ρ, then Corollary 6.2.2 shows that X_{∞} is replaced by X_{∞} ⊗ ρ. Also Ann_{Λ}(T) is replaced by Tw_{ρ^{−1}}(Ann_{Λ}(T)) by Lemma 6.1.2(ii) (where Tw_{ρ} : Λ → Λ is the map of Definition 6.1.1 induced by γ ↦ ρ^{−1}(γ)γ on Γ), and similarly for Ann_{Λ}(T (−1)). It follows easily that X_{∞}/Ann_{Λ}(T)X_{∞} and X_{∞}/Ann_{Λ}(T (−1))X_{∞} remain unchanged as O-modules. Thus both assertions of the lemma are invariant under twisting by characters of Γ, so we may assume that T = O for (i) and T = O(1) for (ii). Then in both cases we are trying to show that X_{∞}/J X_{∞} is finite, where J denotes the augmentation ideal of Λ. Without loss of generality we may also suppose that O = Z_{p}.

Suppose first that T = Z_{p}(1). Then W^{∗} = Q_{p}/Z_{p} and H^{1}(K_{∞}, W^{∗}) = Hom(G_{K_{∞}}, Q_{p}/Z_{p}), so by Proposition 1.6.1 we have X_{∞} = Gal(L_{∞}/K_{∞}), where L_{∞} is the maximal everywhere unramified abelian p-extension of K_{∞} in which all primes above p split completely. A standard Iwasawa theory argument ([Iw3] §3.1) now shows that X_{∞}/J X_{∞} = Gal(L/K_{∞}) where L is the maximal abelian extension of K in L_{∞}, and that this Galois group is finitely generated.

If K is totally real and Leopoldt’s conjecture holds for K, then K has no extension with Galois group Z_{p}^{2}, so L/K_{∞} is finite. If K is imaginary quadratic then K has a unique extension with Galois group Z_{p}^{2}, but no prime above p is infinitely split in this extension, so again L/K_{∞} is finite. This proves the lemma in this case.

Now suppose T = Z_{p}, so W^{∗} = μ_{p^{∞}}. It follows directly from Lemma 7.3.1 in this case that the map

S_{Σ_{p}}(K, μ_{p^{∞}}) ⟶ S_{Σ_{p}}(K_{∞}, μ_{p^{∞}})^{G_{K}} = Hom(X_{∞}/J X_{∞}, Q_{p}/Z_{p})

has finite cokernel. Since Leopoldt’s conjecture holds for K, Corollary 1.6.5 shows that S_{Σ_{p}}(K, μ_{p^{∞}}) is finite. This completes the proof.

7.4. The Kernel and Cokernel of the Restriction Map

For the remainder of this chapter we will assume that Assumptions 7.1.4 and 7.1.5 are satisfied. Since we have now proved Theorem 2.3.2, we can do this with no loss of generality (see Remark 7.1.6).

In particular, if K ⊂_{f} F ⊂ K_{∞} and λ is a prime of F dividing N, then it follows from Assumption 7.1.5 that W^{G_{F}}, (W^{∗})^{G_{F}}, W^{G_{F_{λ}}}, and (W^{∗})^{G_{F_{λ}}} are all finite.

Definition 7.4.1. We define several ideals of Λ which will play a role in the proofs below. Recall that Ann_{Λ}(B) denotes the annihilator in Λ of a Λ-module B. Define

A_{glob} = \begin{cases}Ann_{Λ}(W^{G_{K∞}}) & if rank_{Z_{p}}Γ > 1, \\ Ann_{Λ}(W^{G_{K∞}} /(W^{G_{K∞}})_{div}) & if Γ = Z_{p}. \end{cases}

If v is a place of K, fix an extension w of v to K̄, let D_{v} be the decomposition group of v in Γ, let I_{w} be the inertia group of w in G_{K}, and let

K_{∞,w} = ∪_{K ⊂_{f} F ⊂K_{∞}}F_{w}.

Define

A_{v} = \begin{cases}Ann_{O[[D_{v}]]}(W^{G_{K∞,w}}) & if v | p and rank_{Z_{p}}D_{v} > 1, \\ Ann_{O[[D_{v}]]}(W^{G_{K∞,w}} /(W^{G_{K∞,w}})_{div}) & if v | p and D_{v} = Z_{p}, \\ Ann_{O[[D_{v}]]}(H^{1}(K_{∞,w}^{ur}/K_{∞,w}, W^{I_{v}}/(W^{I_{v}})_{div})) & if v ∤ p, \end{cases}

A_{N} = ∏_{v|N} A_{v}Λ.

We define A_{glob}^{∗}, A_{v}^{∗}, and A_{N}^{∗} in exactly the same way with W replaced by W^{∗}.

Lemma 7.4.2. The ideals A_{glob}, A_{N}, A_{glob}^{∗}, and A_{N}^{∗} defined above have height at least two in Λ.

Proof. This is clear from the definitions of these ideals.

Lemma 7.4.3. Suppose K ⊂_{f} F ⊂ K_{∞} and i ≥ 1.

(i) H^{i}(K_{∞}/F, W^{G_{K∞}}) is finite and annihilated by A_{glob}.

(ii) H^{i}(K_{∞}/F, (W^{∗})^{G_{K∞}}) is finite and annihilated by A_{glob}^{∗}.

(iii) If v is a prime of K above p and w is a prime of K_{∞} above v, then H^{i}(K_{∞,w}/F_{w}, (W^{∗})^{G_{K∞,w}}) is finite and annihilated by A_{v}^{∗}.

Proof. Let W′ = W^{G_{K∞}}, (W^{∗})^{G_{K∞}}, or (W^{∗})^{G_{K∞,w}}, and G = Gal(K_{∞}/F), Gal(K_{∞}/F), or Gal(K_{∞,w}/F_{w}), respectively. By Assumption 7.1.5, we can choose a γ ∈ G_{F_{w}} ⊂ G_{F} such that T^{γ=1} = (T^{∗})^{γ=1} = 0. Let γ̄ ∈ Γ denote the restriction of γ to K_{∞}.

Since Γ is abelian, the annihilator of W′ annihilates H^{i}(G, W′) for every i. If f(x) = det(1 − γx|T ⊕ T^{∗}) ∈ O[x], then the Cayley-Hamilton theorem shows that f(γ̄^{−1}) annihilates W′, so in particular f(γ̄^{−1}) annihilates H^{i}(G, W′). Since G acts trivially on H^{i}(G, W′), it follows that f(1) annihilates H^{i}(G, W′). Our hypothesis on γ ensures that f(1) ≠ 0, so it follows without difficulty (since G is finitely generated and W′ is co-finitely generated) that H^{i}(G, W′) is finite.

This proves the finiteness in all three cases, and the annihilation when rank_{Z_{p}}(G) > 1. Suppose now that G ≅ Z_{p}, and use the exact sequences

H^{i}(G, W′_{div}) ⟶ H^{i}(G, W′) ⟶ H^{i}(G, W′/W′_{div}) ⟶ H^{i+1}(G, W′_{div}).

If i > 1 then H^{i}(G, W′_{div}) = 0 because G has cohomological dimension 1, and if σ is a topological generator of G then (Lemma B.2.8)

H^{1}(G, W′_{div}) ≅ W′_{div}/(σ − 1)W′_{div} = 0

because W′_{div}/(σ−1)W′_{div} is a quotient of W′_{div}/(γ̄−1)W′_{div}. Thus for every i > 0 we have

H^{i}(G, W′) ≅ H^{i}(G, W′/W′_{div}),

so we see that the annihilator of W′/W′_{div} annihilates H^{i}(G, W′).

Proposition 7.4.4. Suppose K ⊂_{f} F ⊂ K_{∞} and M is a power of p. (i) The kernel of the restriction map

H^{1}(F, W) ⟶ H^{1}(K_{∞}, W)^{G_{F}}

is finite and is annihilated by A_{glob}. (ii) The kernel of the natural map

H^{1}(F, W_{M}) ⟶ H^{1}(F, W)_{M}

is finite with order bounded independently of M, and is annihilated by Ann_{Λ}(W^{G_{K∞}}).

(iii) The cokernel of the restriction map

S_{Σ_{p}}(F, W^{∗}) ⟶ S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}}

is finite and is annihilated by A_{glob}^{∗}A_{N}^{∗}. (iv) If S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} is finite, then there is a power M_{F} of p such that

if M ≥ M_{F} is a power of p, then A_{glob}^{∗}A_{N}^{∗} annihilates the cokernel of the natural map

S_{Σ_{p}}(F, W_{M}^{∗}) ⟶ S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} .

(v) The cokernel of the natural map

S_{Σ_{p}}(F, W_{M}^{∗}) ⟶ S_{Σ_{p}}(F, W^{∗})_{M}

is finite and bounded independently of M.

Proof. The inflation-restriction exact sequence shows that the kernel of the restriction map in (i) is H^{1}(K_{∞}/F, W^{G_{K∞}}), so (i) follows from Lemma 7.4.3(i). By Lemma 1.2.2(i), the kernel (ii) is W^{G_{F}} /MW^{G_{F}}, which in turn is a quotient of W^{G_{F}} /(W^{G_{F}})_{div}, and (ii) follows. Assertion (iii) is immediate from Lemmas 7.3.1, 7.4.3, and 1.3.5(iii).

Suppose further that S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} is finite. Since

S_{Σ_{p}}(F, W^{∗}) = \underrightarrow{lim} S_{Σ_{p}}(F, W_{M}^{∗}),

we can choose M_{F} so that the image of S_{Σ_{p}}(F, W_{M_{F}}^{∗}) in H^{1}(K_{∞}, W^{∗}) contains the image of res_{K_{∞}}(S_{Σ_{p}}(F, W^{∗})). With this choice (iv) follows from (iii).

By Lemma 1.5.4, the map S^{Σ_{p}}(F, W_{M}^{∗}) → S^{Σ_{p}}(F, W^{∗})_{M} is surjective. Thus the cokernel in (v) is isomorphic to a subquotient of

⊕_{w|p}ker (H^{1}(F_{w}, W_{M}^{∗}) ⟶ H^{1}(F_{w}, W^{∗})) .

For each w dividing p, Lemma 1.2.2(i) shows that the above kernel is

(W^{∗})^{G_{Fw}} /M(W^{∗})^{G_{Fw}} ,

which is a quotient of the finite group (W^{∗})^{G_{Fw}} /((W^{∗})^{G_{Fw}})_{div} and hence is bounded independently of M. This proves (v).

7.5. Galois Equivariance of the Evaluation Maps

For the proofs of Propositions 7.1.7 and 7.1.9 in the following sections, it would be convenient if G_{Ω_{∞}} were a Λ-module and Ev and Ev^{∗} were Λmodule homomorphisms. Unfortunately this makes no sense, since G_{Ω_{∞}} is not a Λ-module. We will get around this by defining an action of a subring of Λ on a quotient of G_{Ω_{∞}}, and Ev and Ev^{∗} will behave well with respect to this action.

Proposition 7.5.1. There are a subgroup Γ_{0} of finite index in Γ, characters χ, χ^{∗} : Γ_{0} → O^{×}, an abelian extension L of Ω_{∞}, and an action of Z_{p}[[Γ_{0}]] on Gal(L/Ω_{∞}) such that

(i) Ev and Ev^{∗} on G_{Ω_{∞}} factor through Gal(L/Ω_{∞}), (ii) if η ∈ Γ_{0} and γ ∈ Gal(L/Ω_{∞}) then

Ev(γ^{η}) = χ(η)η(Ev(γ)), Ev^{∗}(γ^{η}) = χ^{∗}(η)η(Ev^{∗}(γ)).

Proof. Let L be the maximal abelian p-extension of

K_{∞}(μ_{p^{∞}}, W) = K_{∞}(μ_{p^{∞}}, W^{∗}) = K_{∞}(W, W^{∗}).

Then Ω_{∞} ⊂ L, and every cocycle in H^{1}(K_{∞}, W) or in H^{1}(K_{∞}, W^{∗}) vanishes on G_{L}, so (i) is satisfied.

Consider the diagram of fields in Figure 2. By Proposition C.1.7,

[figure: K; K_{0}; K_{∞}; K(W); K_{∞}(μ_{p^{∞}}, W); Ω_{∞}; L] Figure 2

there is a finite extension K_{0} of K in K(W) ∩ K_{∞} such that the center of Gal(K(W)/K) maps onto Gal((K(W) ∩ K_{∞})/K_{0}). Define

Γ_{0} = Gal(K_{∞}/K_{0}).

Fix once and for all a set of independent topological generators {γ_{1}, …, γ_{d}} of Γ_{0}, and for 1 ≤ i ≤ d fix a lift γ̃_{i} ∈ Gal(K_{∞}(μ_{p^{∞}}, W)/K_{0}) of γ_{i} such that the restriction of γ̃_{i} to K(W) belongs to the center of Gal(K(W)/K). Since K_{∞}(μ_{p^{∞}}, W) is the compositum of K(W) with an abelian extension of K, each γ̃_{i} belongs to the center of Gal(K_{∞}(μ_{p^{∞}}, W)/K). Therefore these choices extend by multiplicativity to define a homomorphism

Γ_{0} ⟶ Gal(K_{∞}(μ_{p^{∞}}, W)/K_{0}),

whose image lies in the center of Gal(K_{∞}(μ_{p^{∞}}, W)/K), which is a section for the projection map Gal(K_{∞}(μ_{p^{∞}}, W)/K_{0}) → Γ_{0}. We will denote this map by γ ↦ γ̃, and we will use this map to define an action of Γ_{0} on Gal(L/Ω_{∞}): for γ ∈ Gal(L/Ω_{∞}) and η ∈ Γ_{0}, define

γ^{η} = η̃γη̃^{−1}.

This definition extends to give an action of Z_{p}[[Γ_{0}]] on Gal(L/Ω_{∞}). It is not canonical, since it depends on our choice of the γ̃_{i}.

By Lemma C.1.6, since V is assumed to be irreducible, every element of the center of Gal(K(W)/K) acts on W by a scalar in O^{×}. Thus the choice above defines a character

χ : Γ_{0} ⟶ O^{×}, χ(η) = η̃ ∈ Aut(W).

Similarly, if η ∈ Γ_{0} then η̃ belongs to the center of Gal(K(W^{∗})/K) so we get a second character

χ^{∗} : Γ_{0} ⟶ O^{×}, χ^{∗}(η) = η̃ ∈ Aut(W^{∗}).

Suppose that c ∈ H^{1}(K_{∞}, W), that γ ∈ Gal(L/Ω_{∞}), and that η ∈ Γ_{0}. Since Ev(γ) ∈ Hom(H^{1}(K_{∞}, W), D), we have

(ηEv(γ))(c) = Ev(γ)(η^{−1}c) = θ̄((η^{−1}c)(γ))
= θ̄(η̃^{−1}(c(γ^{η}))) = χ(η^{−1})Ev(γ^{η})(c).

In other words,

Ev(γ^{η}) = χ(η)η(Ev(γ)),

and similarly with Ev^{∗} and χ^{∗}. This proves (ii).

Recall the involution η ↦ η^{•} of Λ given by Definition 7.2.3.

Proposition 7.5.2. Suppose X′ is a Λ-submodule of X_{∞} and X_{∞}/X′ is pseudo-null. Then there is an ideal A_{0} of height at least two in Λ such that for every K ⊂_{f} F ⊂ K_{∞},

A_{0}a_{τ} Ann_{Λ}(W^{G_{K∞}})^{•}Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•}Hom(H^{1}(F, W_{M}), O/MO)
⊂ OEv_{F,M}((Ev^{∗})^{−1}(X′) ∩ G_{Ω_{∞}}).

In other words, if ψ belongs to

A_{0}a_{τ} Ann_{Λ}(W^{G_{K∞}})^{•}Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•}Hom(H^{1}(F, W_{M}), O/MO)

then there are γ_{1}, …, γ_{k} ∈ G_{Ω_{∞}} and c_{1}, …, c_{k} ∈ O such that Ev^{∗}(γ_{i}) ∈ X′ for every i and

∑_{i=1}^{k} c_{i}Ev_{F,M}(γ_{i}) = ψ.

Proof. Let Γ_{0}, L, χ, and χ^{∗} be as in Proposition 7.5.1. We define

Tw_{χ} : O[[Γ_{0}]] ⟶ O[[Γ_{0}]] by γ ↦ χ(γ)γ

and similarly for Tw_{χ^{∗}}, and then Proposition 7.5.1 shows that for every η ∈ Z_{p}[[Γ_{0}]] and γ ∈ Gal(L/Ω_{∞}),

Ev(γ^{η}) = Tw_{χ}(η)(Ev(γ)), Ev^{∗}(γ^{η}) = Tw_{χ^{∗}}(η)(Ev^{∗}(γ)).  (7.6)

Note that a pseudo-null Λ-module is also pseudo-null as a Z_{p}[[Γ_{0}]]- module, and conversely if A is an ideal of Z_{p}[[Γ_{0}]] of height at least two then AΛ is an ideal of Λ of height at least two.

Define

A = Tw_{χ^{∗}}^{−1}(Ann_{O[[Γ_{0}]]}(X_{∞}/X′)) ∩ Z_{p}[[Γ_{0}]].

Since X_{∞}/X′ is assumed to be a pseudo-null Λ-module, A is an ideal of height at least two in Z_{p}[[Γ_{0}]]. By (7.6),

Ev^{∗}(AGal(L/Ω_{∞})) = Tw_{χ^{∗}}(A)Ev^{∗}(G_{Ω_{∞}}) ⊂ X′,

and by (7.6) and Lemma 7.2.4(ii), for every K ⊂_{f} F ⊂ K_{∞},

OEv(AGal(L/Ω_{∞})) = OTw_{χ}(A)Ev(G_{Ω_{∞}})

⊃ Tw_{χ}(A)a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•}Hom(H^{1}(K_{∞}, W), D).

By Proposition 7.4.4(i) and (ii), the image of the composition

Hom(H^{1}(K_{∞}, W), D) ⟶ Hom(H^{1}(F, W), D) ⟶ Hom(H^{1}(F, W_{M}), O/MO)

contains

A_{glob}^{•}Ann_{Λ}(W^{G_{K∞}})^{•}Hom(H^{1}(F, W_{M}), O/MO).

Combining these inclusions, we see that the proposition holds with

A_{0} = A_{glob}^{•}Tw_{χ}(A),

which has height at least two by Lemma 7.4.2.

Remark 7.5.3. When Γ ≅ Z_{p} there is a simpler proof of Proposition 7.5.2, which does not rely on the noncanonical construction of Proposition 7.5.1. In that case X_{∞}/X′ is finite, so (Ev^{∗})^{−1}(X′)∩G_{Ω_{∞}} has finite index in G_{Ω_{∞}}, so by Lemma 7.2.4(ii), OEv((Ev^{∗})^{−1}(X′) ∩ G_{Ω_{∞}}) contains a subgroup of finite index (not a priori a Λ-submodule) of

a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•}Hom(H^{1}(K_{∞}, W), D).

But every subgroup of finite index contains a submodule of finite index, and hence there is a j ≥ 0 such that

M^{j}a_{τ} Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•}Hom(H^{1}(K_{∞}, W), D) ⊂ OEv((Ev^{∗})^{−1}(X′) ∩ G_{Ω_{∞}})

where we recall that M is the maximal ideal of Λ. By Proposition 7.4.4(i) and (ii), A_{glob}^{•}Ann_{Λ}(W^{G_{K∞}})^{•} annihilates the cokernel of the map

Hom(H^{1}(K_{∞}, W), D) ⟶ Hom(H^{1}(F, W_{M}), D),

so the proposition is satisfied with A_{0} = M^{j}A_{glob}^{•}

7.6. Proof of Proposition 7.1.7

Proposition 7.1.7 is very easy to prove in the following (fairly common, see the examples of Chapter 3) special case. Suppose that hypotheses Hyp(K_{∞}, T) are satisfied (so a_{τ} = 1 by Lemma 7.1.3(ii)), O = Z_{p}, and H^{1}(Ω_{∞}/K_{∞}, W^{∗}) = 0. Use (7.1) to choose a sequence z_{1}, …, z_{r} ∈ X_{∞} such that ⊕Λz_{i} ≅ ⊕Λ/f_{i}Λ. By Lemma 7.2.4(iii), under our assumptions we have

Ev^{∗}(τG_{Ω_{∞}}) = Ev^{∗}(τ) + Ev^{∗}(G_{Ω_{∞}}) = Ev^{∗}(τ) + X_{∞} = X_{∞},

so Proposition 7.1.7 holds with these z_{i} and with 𝔤_{i} = f_{i}Λ.

The rest of this section is devoted to the proof of Proposition 7.1.7 in the general case, which unfortunately is more complicated.

We say that two ideals A and B of Λ are relatively prime if A + B has height at least two.

Lemma 7.6.1. The ideal char(X_{∞}) is relatively prime to each of the ideals

Ann_{Λ}(W^{G_{K∞}}), Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W)), Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W^{∗}))^{•}.

Proof. The proofs for all three ideals are similar. If W^{G_{K∞}} is finite or if rank_{Z_{p}}(Γ) > 1 then Ann_{Λ}(W^{G_{K∞}}) has height at least two and the first assertion holds trivially. We have assumed that V is irreducible over G_{K_{∞}}, so if W^{G_{K∞}} is infinite then G_{K_{∞}} acts trivially on T. Thus (using hypothesis Hyp(K_{∞}/K)) the first assertion follows from Lemma 7.3.4(i).

The other two assertions follow similarly, using Lemma 7.3.4 and Corollary C.2.2. We sketch briefly the proof for the third ideal.

Corollary C.2.2 applied to T^{∗}, with F = K_{∞} and Ω = Ω_{∞}, gives three cases. In case (i), H^{1}(Ω_{∞}/K_{∞}, W^{∗}) is finite, so Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W^{∗})) has height at least two, and hence is relatively prime to everything. In case (ii) (resp. (iii)), G_{K} acts on T^{∗} via a character ρ of Γ (resp. ε_{cyc}ρ), and H^{1}(Ω_{∞}/K_{∞}, W^{∗}) has a subgroup C of finite index on which G_{K} acts via ρ.

Then G_{K} acts on T via ε_{cyc}ρ^{−1} (resp. ρ^{−1}), so Ann_{Λ}(C)^{•} ⊃ Ann_{Λ}(T (−1)) (resp. Ann_{Λ}(C)^{•} ⊃ Ann_{Λ}(T)). Since

Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W^{∗})) ⊃ Ann_{Λ}(C)Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W^{∗})/C)

and H^{1}(Ω_{∞}/K_{∞}, W^{∗})/C is finite, the lemma in this case follows from Lemma 7.3.4.

Lemma 7.6.2. Suppose B is a torsion Λ-module and x, y ∈ B. Suppose further that g_{x}, g_{y} ∈ Λ are such that Ann_{Λ}(x) ⊂ g_{x}Λ and Ann_{Λ}(y) ⊂ g_{y}Λ. Then there is an n ∈ Z such that

Ann_{Λ}(x + ny) ⊂ [g_{x}, g_{y}]Λ

where [g_{x}, g_{y}] denotes the least common multiple of g_{x} and g_{y}.

Proof. Suppose 𝔓 is a (height-one) prime divisor of [g_{x}, g_{y}], and define

S_{𝔓} = {n ∈ Z : Ann_{Λ}(x + ny) ⊄ 𝔓^{ord_{𝔓}[g_{x},g_{y}]}}.

Recall that 𝔭 is the maximal ideal of O. We will show that S_{𝔓} has at most one element if 𝔓 ≠ 𝔭Λ, and S_{𝔓} is contained in a congruence class modulo p if 𝔓 = 𝔭Λ. Then it will follow that Z − ∪_{𝔓}S_{𝔓} is nonempty, and every n in this set satisfies the conclusion of the lemma.

Suppose n, m ∈ S_{𝔓}, and let A = Ann_{Λ}(x + ny)∩Ann_{Λ}(x + my). Then A ⊄ 𝔓^{k}, where k = ord_{𝔓}[g_{x}, g_{y}]. But (n − m)A annihilates both y and x, so (n − m)A ⊂ 𝔓^{k} and we conclude that n − m ∈ 𝔓. If 𝔓 ≠ 𝔭Λ it follows that n = m, and if 𝔓 = 𝔭Λ then n ≡ m (mod p). This completes the proof.

Lemma 7.6.3. Suppose B is a finitely generated torsion Λ-module, and B is pseudo-isomorphic to ⊕_{i=1}^{k}Λ/h_{i}Λ, where h_{i+1} | h_{i} for 1 ≤ i < k. Suppose we are given a subring Λ_{0} of Λ such that Λ is finitely generated as a Λ_{0}-module, a Λ_{0}-submodule B_{0} ⊂ B, and an element t ∈ B such that t and B_{0} generate B over Λ. Then there are elements x_{1} ∈ t + B_{0} and x_{2}, …, x_{k} ∈ B_{0} such that

(i) Λx_{1} ≅ Λ/𝔥_{1} where 𝔥_{1} ⊂ h_{1}Λ and h_{1}Λ/𝔥_{1} is pseudo-null, (ii) if 2 ≤ j ≤ k there is a split exact sequence

0 ⟶ ∑_{i=1}^{j−1} Λx_{i} ⟶ ∑_{i=1}^{j} Λx_{i} ⟶ Λ/h_{j}Λ ⟶ 0.

If t = 0 then we can replace (i) by (i′) Λx_{1} ≅ Λ/h_{1}Λ, i.e., (ii) holds for j = 1 as well.

Proof. We will prove the lemma by induction on k.

If A is an ideal of Λ then char(Λ/A) is the unique principal ideal containing A with pseudo-null quotient. For every x ∈ B_{0} write

A_{x} = char(Λ/Ann_{Λ}(x)).

By Lemma 7.6.2 (applied successively with x = t and y running through a sequence of elements of B_{0}) we can choose x_{1} ∈ t + B_{0} such that A_{x_{1}} ⊂ A_{x} for every x ∈ t + B_{0}. Since t and B_{0} generate B over Λ, we must have A_{x_{1}} = h_{1}Λ, so (i) is satisfied. This proves the lemma when k = 1 and t ≠ 0.

If t = 0 then choose g ∈ Λ_{0}, prime (in Λ) to h_{1}, which annihilates the pseudo-null Λ-module h_{1}Λ/Ann_{Λ}(x_{1}), and replace x_{1} by gx_{1}. This element has annihilator exactly h_{1}Λ, so this completes the proof when k = 1.

If k > 1, choose x_{1} as above. Let B′ = B/Λx_{1}, let B′_{0} be the image of B_{0} in B′, and let t′ = 0. Then B′ is pseudo-isomorphic to ⊕_{i=2}^{k}Λ/h_{i}Λ, so by the induction hypothesis (in the “t = 0” case) we can choose x̄_{2}, …, x̄_{k} ∈ B′_{0} leading to split exact sequences

0 ⟶ ∑_{i=2}^{j−1} Λx̄_{i} ⟶ ∑_{i=2}^{j} Λx̄_{i} ⟶ Λ/h_{j}Λ ⟶ 0

if 2 ≤ j ≤ k.

Now for i ≥ 2 choose x_{i} to be any lift of x̄_{i} to B_{0}. We claim the lemma is satisfied with this choice of x_{1}, …, x_{k}. It will suffice to check that the exact sequences

0 ⟶ Λx_{1} ⟶ ∑_{i=1}^{j} Λx_{i} ⟶ ∑_{i=2}^{j} Λx̄_{i} ⟶ 0  (7.7)

split for 2 ≤ j ≤ k.

Let 𝔥 = Ann_{Λ}(B). Then 𝔥 ⊂ h_{1}Λ and h_{1}^{−1}𝔥 is pseudo-null. By our induction hypothesis we can choose elements ȳ_{2}, …, ȳ_{k} ∈ ∑_{i=2}^{k} Λx̄_{i} such that Λȳ_{i} ≅ Λ/h_{i}Λ for each i and ∑_{i=2}^{k} Λȳ_{i} = ∑_{i=2}^{k} Λx̄_{i}. Let y_{i} be a lift of ȳ_{i} to ∑_{i=1}^{k} Λx_{i}.

For each i we have h_{i}y_{i} ∈ Λx_{1}, say h_{i}y_{i} = c_{i}x_{1}. Then h_{i}^{−1}𝔥 annihilates c_{i}x_{1}, i.e., c_{i}𝔥 ⊂ h_{i}𝔥_{1}, and we conclude that h_{i} divides c_{i}. Now the map ȳ_{i} ↦ y_{i} − (c_{i}/h_{i})x_{1} gives a splitting of (7.7).

Proof of Proposition 7.1.7. Recall that we have a pseudo-isomorphism

⊕_{i=1}^{r}Λ/f_{i}Λ ⟶ X_{∞}.

Define a Λ-submodule

X_{0} = ΛEv^{∗}(τ) + ΛEv^{∗}(G_{Ω_{∞}}) ⊂ X_{∞}.

Then X_{∞} ⊃ X_{0} ⊃ X_{0} ∩ a_{τ} X_{∞}, and Lemmas 7.2.4(iii) and 7.6.1 show that (a_{τ} X_{∞})/(X_{0} ∩ a_{τ} X_{∞}) is pseudo-null. Thus we can find a new injective pseudo-isomorphism

⊕_{i=1}^{r}Λ/g_{i}Λ ⟶ X_{0}

with elements g_{i} ∈ Λ satisfying, for every i,

g_{i+1} | g_{i},  f_{i} | a_{τ} g_{i},  g_{i} | f_{i}.

Apply Lemma 7.6.3 with B = X_{0}, with h_{i} = g_{i}, with B_{0} = Ev^{∗}(G_{Ω_{∞}}), and with t = Ev^{∗}(τ) to produce a sequence x_{1}, …, x_{n} ∈ X_{0}. (Note that B_{0} satisfies the hypotheses of Lemma 7.6.3 with Λ_{0} = Tw_{χ^{∗}}(Z_{p}[[Γ_{0}]]), where Γ_{0} and χ^{∗} are as in Proposition 7.5.1, and Tw_{χ^{∗}} is as in the proof of Proposition 7.5.2.) Define z_{1} = x_{1} ∈ Ev^{∗}(τG_{Ω_{∞}}) and 𝔤_{1} = 𝔥_{1}. For 2 ≤ i ≤ r define

z_{i} = x_{1} + x_{i} ∈ Ev^{∗}(τG_{Ω_{∞}}), 𝔤_{i} = g_{i}Λ.

Lemma 7.6.3 shows that char(∑ Λz_{i}) = ∏ g_{i} = char(X_{0}), so X_{0}/ ∑ Λz_{i} is pseudo-null. The other conclusions of Proposition 7.1.7 for these z_{i} and 𝔤_{i} also follow immediately from Lemma 7.6.3.

Corollary 7.6.4. Suppose z_{1}, …, z_{k} and 𝔤_{1}, …, 𝔤_{k} are as in Proposition 7.1.7. If 1 ≤ k ≤ r then ∑_{i=1}^{k} Λz_{i} ≅ ⊕_{i=1}^{k}Λ/𝔤_{i} and ∑_{i=1}^{k} Λz_{i} is a direct summand of ∑_{i=1}^{r} Λz_{i}.

Proof. This follows easily by induction on k from Proposition 7.1.7(iii).

7.7. Proof of Proposition 7.1.9

In this section we will prove Proposition 7.1.9, and thereby complete the proof of Theorems 2.3.3 and 2.3.4 begun in §7.1. Keep the notation of §7.1. In particular recall that

Z_{∞} = ∑_{i=1}^{r} Λz_{i} ≅ ⨁_{i=1}^{r} Λ/𝔤_{i} ⊂ X_{∞}

where the z_{i} and 𝔤_{i} are given by Proposition 7.1.7.

If σ is a Selmer sequence of length k, as in Definition 7.1.8, define

Z_{σ} = ∑_{i=1}^{k} ΛEv^{∗}(σ_{i}) ⊂ Z_{∞}.

Lemma 7.7.1. If σ is a Selmer sequence of length k, then

Z_{σ} ≅ ⊕_{i=1}^{k}Λ/𝔤_{i}Λ

and Z_{σ} is a direct summand of Z_{∞}. If k < r and σ′ is a Selmer sequence of length k + 1 extending σ, then Z_{σ}′/Z_{σ} ≅ Λ/𝔤_{k+1}.

Proof. Let Y_{k} = ∑_{i=1}^{k} Λz_{i}. By Corollary 7.6.4, Y_{k} ≅ ⊕_{i=1}^{k}Λ/𝔤_{i} and there is a complementary submodule Y′_{k} ⊂ Z_{∞} such that Y_{k} ⊕ Y′_{k} = Z_{∞}. The image of Z_{σ} + Y′_{k} in Z_{∞}/MZ_{∞} contains the image of Y_{k} + Y′_{k} = Z_{∞}, so by Nakayama’s Lemma Z_{σ} + Y′_{k} = Z_{∞}. We will show that Z_{σ} ∩ Y′_{k} = 0, and thus Z_{∞} = Z_{σ} ⊕ Y′_{k} and

Z_{σ} ≅ Z_{∞}/Y′_{k} ≅ Y_{k} ≅ ⊕_{i=1}^{k}Λ/𝔤_{i}Λ.

If k < r and σ′ extends σ, we can repeat the argument above with k replaced by k + 1. We can choose Y′_{k+1} to be contained in Y′_{k}, and then Y′_{k}/Y′_{k+1} ≅ Λ/𝔤_{k+1} and

Z_{σ}′ ⊕ Y′_{k+1} = Z_{∞} = Z_{σ} ⊕ Y′_{k},

so

Z_{σ}′ ≅ Z_{σ} ⊕ Y′_{k}/Y′_{k+1} = Z_{σ} ⊕ Λ/𝔤_{k+1}.
It remains to show that Z_{σ} ∩ Y′_{k} = 0. For 1 ≤ i ≤ k write
Ev^{∗}(σ_{i}) = z_{i} + v_{i} + w_{i}

where v_{i} ∈ MY_{k} and w_{i} ∈ MY′_{k}. Suppose

∑_{i=1}^{k} a_{i}Ev^{∗}(σ_{i}) ∈ Y′_{k}

with a_{i} ∈ Λ; we need to show that ∑_{i=1}^{k} a_{i}Ev^{∗}(σ_{i}) = 0.

Projecting into Y_{k} we see that

∑_{i=1}^{k} a_{i}(z_{i} + v_{i}) = 0.  (7.8)

Using Proposition 7.1.7(iii) (see also Corollary 7.6.4), fix y_{1}, …, y_{k} ∈ Y_{k} so that for 1 ≤ i ≤ k we have

Y_{i} = ∑_{j=1}^{i} Λz_{j} = ⨁_{j=1}^{i} Λy_{j}

and Λy_{i} ≅ Λ/𝔤_{i}. We can rewrite (7.8) in matrix form, using these generators, as

(a_{1}, …, a_{k})A ∈ (𝔤_{1}Λ, …, 𝔤_{k}Λ)

where A is a k × k matrix with entries in Λ. Modulo M, we see that A is lower-triangular with invertible diagonal entries (since z_{i} ∈ Y_{i}, and the projection of z_{i} generates Y_{i}/Y_{i−1} = Λy_{i}, and the v_{i} vanish modulo M). Therefore A is invertible, and, since 𝔤_{i} ⊂ 𝔤_{k} for every i ≤ k, we conclude that a_{i} ∈ 𝔤_{k} for every i. But 𝔤_{k} annihilates Y′_{k} because 𝔤_{k} ⊂ 𝔤_{i} for i ≥ k, so we deduce that

∑_{i=1}^{k} a_{i}Ev^{∗}(σ_{i}) = ∑_{i=1}^{k} a_{i}w_{i} = 0.

This completes the proof of the lemma.

Lemma 7.7.2. For every Selmer sequence σ, every power M of p, and every K ⊂_{f} F ⊂ K_{∞}, the ideal Ann_{Λ}(X_{∞}/Z_{∞}) annihilates the kernel of the map

Z_{σ} ⊗ Λ_{F,M} ⟶ X_{∞} ⊗ Λ_{F,M}.

Proof. By Lemma 7.7.1, Z_{σ} is a direct summand of Z_{∞}, so Z_{σ} ⊗ Λ_{F,M} injects into Z_{∞} ⊗ Λ_{F,M}. If J_{F,M} is the kernel of the map Λ → Λ_{F,M}, then the kernel of Z_{∞} ⊗ Λ_{F,M} → X_{∞} ⊗ Λ_{F,M} is (Z_{∞} ∩ J_{F,M}X_{∞})/J_{F,M}Z_{∞}, which is annihilated by Ann_{Λ}(X_{∞}/Z_{∞}). This proves the lemma.

For the rest of this section fix a field F such that K ⊂_{f} F ⊂ K_{∞}. By Assumption 7.1.4, Λ_{F} /f_{1}Λ_{F} is finite. Fix a power of N_{F} of p such that N_{F} ≥ |Λ_{F} /f_{1}Λ_{F} | and such that N_{F} is at least as large as the integer M_{F} of Proposition 7.4.4(iv).

Let B_{0} = (A_{glob}^{∗})^{•}(A_{N}^{∗})^{•}Ann_{Λ}(X_{∞}/Z_{∞}).

Corollary 7.7.3. If σ is a Selmer sequence and M ≥ N_{F} is a power of p, then B_{0} annihilates the kernel of the natural map

Z_{σ} ⊗ Λ_{F,M} ⟶ Hom(S_{Σ_{p}}(F, W_{M}^{∗}), O/MO).

Proof. The map in question is the composition

Z_{σ} ⊗ Λ_{F,M} ⟶ X_{∞} ⊗ Λ_{F,M} ⟶ Hom(S_{Σ_{p}}(F, W_{M}^{∗}), O/MO).

It follows from Assumption 7.1.4 that S_{Σ_{p}}(K_{∞}, W^{∗})^{G_{F}} is finite, so the corollary follows from Proposition 7.4.4(iv) and Lemma 7.7.2.

If 𝔯 ∈ R, recall that Σ_{p𝔯} denotes the set of primes of K dividing p𝔯. Recall also from Definition 7.1.8 that Π(k, F, M) is the set of Kolyvagin sequences of length k for F and M, and if π = (𝔮_{1}, …, 𝔮_{k}) ∈ Π(k, F, M) then 𝔯(π) = ∏_{i=1}^{k} 𝔮_{i}.

Lemma 7.7.4. Suppose that M is a power of p, that σ is a Selmer sequence of length k, and that π is a Kolyvagin sequence corresponding to σ. Then the map of Corollary 7.7.3 factors through a surjective map

Z_{σ} ⊗ Λ_{F,M} ↠ Hom(S_{Σ_{p}}(F, W_{M}^{∗})/S_{Σ_{p𝔯(π)}}(F, W_{M}^{∗}), O/MO).

Proof. Write σ = (σ_{1}, …, σ_{k}) and π = (𝔮_{1}, …, 𝔮_{k}). The image of Z_{σ} in Hom(S_{Σ_{p}}(F, W_{M}^{∗}), O/MO) is Hom(S_{Σ_{p}}(F, W_{M}^{∗})/B, O/MO), where

B = ⋂_{{1≤i≤k \atop γ∈Gal(F/K)}} ker(Ev_{F,M}^{∗}(σ_{i})^{γ}) = ⋂_{Q of F ; Q|∏_{i} 𝔮_{i}} ker(Ev_{F,M}^{∗}(Fr_{Q})).

Each Ev_{F,M}^{∗}(Fr_{Q}) factors through a map which is injective on H_{f}^{1}(F_{Q}, W_{M}^{∗}), so the right-hand intersection is exactly S_{Σ_{p𝔯(π)}}(F, W_{M}^{∗}).

Proposition 7.7.5. Suppose that 1 ≤ k ≤ r, that M ≥ N_{F} is a power of p, and that π = (π_{1}, …, π_{k}) ∈ Π(k, F, M). Let Σ = Σ_{p𝔯(π)} and 𝔮 = 𝔮_{k}. Then with B_{0} as defined before Corollary 7.7.3,

a_{τ} B_{0}Ẽv_{𝔮,s}(S^{Σ}(F, W_{M})) ⊂ 𝔤_{k}Λ_{F,M}.

Proof. Fix M, k, and π as in the statement of the proposition. Let σ = (σ_{1}, …, σ_{k}) be a Selmer sequence corresponding to π and let σ′ = (σ_{1}, …, σ_{k−1}) and Σ′ = Σ − {𝔮}.

Consider the commutative diagram

0  0
↓  ↓
Z_{σ}′ ⊗ Λ_{F,M} ⟶ Hom(S_{Σ_{p}}(F, W_{M}^{∗})/S_{Σ′}(F, W_{M}^{∗}), O/MO)
↓  ↓
Z_{σ} ⊗ Λ_{F,M}  ⟶  Hom(S_{Σ_{p}}(F, W_{M}^{∗}), O/MO)
↓  ↓
(Z_{σ}/Z_{σ}′) ⊗ Λ_{F,M} ⟶^{j}  Hom(S_{Σ′}(F, W_{M}^{∗}), O/MO)
↓  ↓
0  0.

The left-hand column is exact by Lemma 7.7.1, and the top horizontal map is the surjection of Lemma 7.7.4. Applying the snake lemma, Corollary 7.7.3 shows that ker(j) is annihilated by B_{0}. The image of j is generated by Ev_{F,M}^{∗}(σ_{k})|_{S_{Σ′} (F,W_{M}^{∗})} = Ev_{𝔮,f}^{∗}|_{S_{Σ′} (F,W_{M}^{∗})}, and Z_{σ}/Z_{σ}′ ≅ Λ/𝔤_{k}Λ. Hence

B_{0}Ann_{Λ_{F,M}} (Ev_{𝔮,f}^{∗}|_{S_{Σ′} (F,W_{M}^{∗})}) ⊂ 𝔤_{k}Λ_{F,M}.

By Theorem 7.2.11, a_{τ} Ẽv_{𝔮,s}(S^{Σ}(F, W_{M})) annihilates Ev_{𝔮,f}^{∗}|_{S_{Σ′} (F,W_{M}^{∗})}. This proves the proposition.

Recall that we have fixed a field F. If M is a power of p and 𝔯 ∈ R_{F,M}, we will write simply κ_{[𝔯,M]} for κ_{[F,𝔯,M]}, and ⟨κ_{[𝔯,M]}⟩ for the Λ_{F,M}-submodule Λ_{F,M}κ_{[𝔯,M]} of H^{1}(F, W_{M}).

Recall also that the map Ẽv_{𝔮,s} depends on M; we now need to record that dependence. For every power M of p, instead of just Ẽv_{𝔮,s} we will write Ẽv_{𝔮,s,M} : H^{1}(F, W_{M}) → Λ_{F,M} for the singular evaluation map of Definitions 7.2.5 and 7.2.9.

Corollary 7.7.6. Suppose 1 ≤ k ≤ r and M is a power of p. Suppose further that π = (𝔮_{1}, …, 𝔮_{k}) ∈ Π(k, F, MN_{F}), and let 𝔮 = 𝔮_{k} and 𝔯 = 𝔯(π). If η ∈ a_{τ}^{2}B_{0} then

ηẼv_{𝔮,s,M}|_{⟨κ_{[𝔯,M]}⟩} ∈ f_{k}Hom_{Λ}(⟨κ_{[𝔯,M]}⟩, Λ_{F,M}).

Proof. Let M′ = MN_{F} and Σ = Σ_{p𝔯(π)}. By Propositions 7.7.5 and 7.1.7(ii),

ηẼv_{𝔮,s,M′} : S^{Σ}(F, W_{M′}) ⟶ a_{τ} 𝔤_{k}Λ_{F,M′} ⊂ f_{k}Λ_{F,M′}.

Since f_{k} | N_{F} in Λ_{F}, there is a well-defined “division by f_{k}” map

f_{k}Λ_{F,M′} ⟶ Λ_{F,M}

which sends f_{k}g to g (mod M) for every g. Let ψ′ : S^{Σ}(F, W_{M′}) → Λ_{F,M} be the composition of ηẼv_{𝔮,s,M′} with this division map.

Let ι_{N_{F} ,M′} and ι_{M′,M} be the natural maps in the exact cohomology sequence

H^{1}(F, W_{N_{F}}) ⟶^{ι_{N_{F} ,M′}} H^{1}(F, W_{M′}) ⟶^{ι_{M′,M}} H^{1}(F, W_{M}).

If we identify Λ_{F,N_{F}} with MΛ_{F,M′}, we have

Ẽv_{𝔮,s,N_{F}} = Ẽv_{𝔮,s,M′} ∘ ι_{N_{F} ,M′}.

Applying Propositions 7.7.5 and 7.1.7(ii) again we see that

ηẼv_{𝔮,s,N_{F}} (S^{Σ}(F, W_{N_{F}})) ⊂ f_{k}Λ_{F,N_{F}} ,

and it follows that ψ′ ∘ ι_{N_{F} ,M′} = 0. Therefore ψ′ factors through ι_{M′,M}, i.e.,

ψ′ = ψ ∘ ι_{M′,M} where ψ ∈ Hom_{Λ}(ι_{M′,M}(S^{Σ}(F, W_{M′})), Λ_{F,M}).

Using Lemma 4.4.13(iii) and Theorem 4.5.1, we also have a diagram

[figure: κ_{[𝔯,M′]}; ∈; S^{Σ}(F, W_{M′}); ηẼv_{𝔮,s,M′}; ι_{M′,M}; f_{k}ψ′; Λ_{F,M′}; κ_{[𝔯,M]}; ∈; S^{Σ}(F, W_{M}); ηẼv_{𝔮,s,M}; Λ_{F,M}]

It follows that

f_{k}ψ(κ_{[𝔯,M]}) = f_{k}ψ′(κ_{[𝔯,M′]}) = ηẼv_{𝔮,s}(κ_{[𝔯,M]}),

and so ηẼv_{𝔮,s,M} = f_{k}ψ on ⟨κ_{[𝔯,M]}⟩.

The following is a precise version of Proposition 7.1.9. Define

B = a_{τ}^{4}A_{0}^{•}B_{0}Ann_{Λ}(W^{G_{K∞}})Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))

where A_{0} is the ideal of Proposition 7.5.2 applied with

X′ = {x ∈ X_{∞} : a_{τ} x ∈ MZ_{∞}}

(so by Proposition 7.1.7(iv), X_{∞}/X′ is pseudo-null) and B_{0} is as defined before Corollary 7.7.3.

Proposition 7.7.7. If M ≥ N_{F} is a power of p and 0 ≤ k < r, then

BΨ(k, F, N_{F} M)Λ_{F,M} ⊂ f_{k+1}Ψ(k + 1, F, M).

Proof. Let M′ = N_{F} M. Fix a Kolyvagin sequence π ∈ Π(k, F, M′), let 𝔯 = 𝔯(π), and fix ψ : ⟨κ_{[𝔯,M′]}⟩ → Λ_{F,M′}. We need to show that

Bψ(κ_{[𝔯,M′]})Λ_{F,M} ⊂ f_{k+1}Ψ(k + 1, F, M).

The idea of the proof is as follows. Ideally, we would like to find γ ∈ τG_{Ω_{∞}} such that

(a) Ev^{∗}(γ) ∈ z_{k+1} + MZ_{∞},

(b) Ẽv_{F,M′}(γ) = ψ on ⟨κ_{[𝔯,M′]}⟩, and choose a prime 𝔮 whose Frobenius on a suitable extension of F is γ. If we can do this then (a) says we can use 𝔮 to extend π to a Kolyvagin sequence of length k + 1, (b) combined with Theorem 7.2.10 shows that ψ(κ_{[𝔯,M′]}) = Ẽv_{𝔮,s,(}κ_{[𝔯𝔮,M′]}), and Corollary 7.7.6 shows that the restriction of Ẽv_{𝔮,s} to ⟨κ_{[𝔯𝔮,M]}⟩ is (almost) divisible by f_{k+1}.

Unfortunately, conditions (a) and (b) on γ may not be independent, and it may not be possible to satisfy them simultaneously. Instead, we will use Proposition 7.5.2 to find a finite set of elements {γ_{i}} such that Ev^{∗}(γ_{i}) ∈ MZ_{∞} and such that, instead of (b), a “small multiple” of ψ is a linear combination of the Ẽv_{F,M′}(γ_{i}).

We now return to the proof. Let ψ_{0} ∈ Hom(⟨κ_{[𝔯,M′]}⟩, O/M′O) be the homomorphism corresponding to ψ under the isomorphism of Lemma 7.2.7(i). If

η ∈ A_{0}a_{τ}^{2}Ann_{Λ}(W^{G_{K∞}})^{•}Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W))^{•},

then by Proposition 7.5.2 (applied with X′ as defined just before the statement of this proposition) there are γ_{1}, …, γ_{j} ∈ G_{Ω_{∞}} and c_{1}, …, c_{j} ∈ O such that Ev^{∗}(γ_{i}) ∈ (MZ_{∞}) for every i and

∑_{i=1}^{j} c_{i}Ev_{F,M′}(γ_{i})|_{⟨κ_{[𝔯,M′]}⟩} = ηψ_{0}.  (7.9)

Fix i such that 1 ≤ i ≤ j. Let σ be a Selmer sequence corresponding to π. Choose δ ∈ τG_{Ω_{∞}} such that Ev^{∗}(δ) = z_{k+1} (by Proposition 7.1.7(i)), and define two Selmer sequences σ′ and σ′′ of length k + 1 extending σ by σ′_{k+1} = δ and σ′′_{k+1} = δγ_{i}. (These are Selmer sequences because Ev^{∗}(δ) = z_{k+1} and Ev^{∗}(γ_{i}) ∈ MZ_{∞}.) Fix primes 𝔮′, 𝔮′′ of K such that, for an appropriate choice of primes above 𝔮′ and 𝔮′′, we have

Fr_{𝔮′} = σ′_{k+1}, Fr_{𝔮′′} = σ′′_{k+1} on L

where L is a finite Galois extension of F containing F (W_{M′}, μ_{M′}, (O_{K}^{×})^{1/M′}) and such that the restriction to L of every element of the finite groups (see Lemma 1.5.7) S_{Σ_{p}}(F, W_{M′}^{∗}) and S^{Σ_{p𝔯}}(F, W_{M′}) is zero.

We define two Kolyvagin sequences π′, π′′ ∈ Π(k + 1, F, M′) extending π by setting 𝔮′_{k+1} = 𝔮′ and 𝔮′′_{k+1} = 𝔮′′. By Corollary 7.7.6, if η′ ∈ a_{τ}^{2}B_{0} we can choose

ψ′ ∈ Hom_{Λ}(⟨κ_{[𝔯𝔮′,M]}⟩, Λ_{F,M}), ψ′′ ∈ Hom_{Λ}(⟨κ_{[𝔯𝔮′′,M]}⟩, Λ_{F,M})

so that

f_{k+1}ψ′(κ_{[𝔯𝔮′,M]}) = η′Ẽv_{𝔮′,s,M}(κ_{[𝔯𝔮′,M]}),
f_{k+1}ψ′′(κ_{[𝔯𝔮′′,M]}) = η′Ẽv_{𝔮′′,s,M}(κ_{[𝔯𝔮′′,M]}).

Therefore, using Theorem 7.2.10 for the third equality below,

η′Ẽv_{F,M′}(γ_{i})(κ_{[𝔯,M′]})
= η′Ẽv_{F,M′}(σ′′_{k+1})(κ_{[𝔯,M′]}) − η′Ẽv_{F,M′}(σ′_{k+1})(κ_{[𝔯,M′]})
= η′Ẽv_{F,M′}(Fr_{𝔮′′})(κ_{[𝔯,M′]}) − η′Ẽv_{F,M′}(Fr_{𝔮′})(κ_{[𝔯,M′]})
= η′Ẽv_{𝔮′′,s,M′}(κ_{[𝔯𝔮′′,M′]}) − η′Ẽv_{𝔮′,s,M′}(κ_{[𝔯𝔮′,M′]})
≡ f_{k+1}(ψ′′(κ_{[𝔯𝔮′′,M]}) − ψ′(κ_{[𝔯𝔮′,M]})) (mod M)
∈ f_{k+1}Ψ(k + 1, F, M).

By (7.9) and Lemma 7.2.7(ii),

∑_{i=1}^{j} c_{i}Ẽv_{F,M′}(γ_{i}) = η̃ψ_{0} = η^{•}ψ,

so we conclude that

η^{•}η′ψ(κ_{[𝔯,M′]})Λ_{F,M} ⊂ f_{k+1}Ψ(k + 1, F, M).

As η and η′ vary, the products η^{•}η′ generate B, and the Proposition is proved.

Proof of Proposition 7.1.9. Observe that

• A_{0}, A_{glob}^{∗}, and A_{N}^{∗} have height at least two by Lemma 7.4.2 and Proposition 7.5.2,

• Ann_{Λ}(H^{1}(Ω_{∞}/K_{∞}, W)) and Ann_{Λ}(W^{G_{K∞}}) are prime to char(X_{∞}) by Lemma 7.6.1, and

• Ann_{Λ}(X_{∞}/Z_{∞}) contains the product of a_{τ} and an ideal of height

at least two by Proposition 7.1.7(iv). An ideal of height at least two necessarily contains an element relatively prime to char(X_{∞}) (since char(X_{∞}) ≠ 0 by Theorem 2.3.2), so the ideal B defined before the statement of Proposition 7.7.7 contains the product of a_{τ}^{5} and an element h of Λ prime to char(X_{∞}). Now Proposition 7.1.9 follows from Proposition 7.7.7.

CHAPTER 8 Euler Systems and p-adic L-functions

So far we have discussed at length how an Euler system for a p-adic representation T of G_{K} controls the Selmer groups S(K, W^{∗}) and S(K_{∞}, W^{∗}). This raises several natural questions which we have not yet touched on.

• Except for the examples in Chapter 3, we have not discussed at all how to produce Euler systems. For which representations do (nontrivial) Euler systems exist?

• If there is a nontrivial Euler system c for T, then there are infinitely many such (for example, we can act on c by elements of O[[G_{K}]]). Is there a “best” Euler system?

• Conjecturally, Selmer groups should be related to L-functions and their special values. Is there an Euler system related to an Lfunction attached to T?

In this chapter we will sketch a picture which gives a highly conjectural partial answer to these questions, by describing a fundamental connection between Euler systems and (p-adic) L-functions. This general picture will rest on several layers of conjectures, but nonetheless there are several known examples (such as the ones in Chapter 3) where the connection is proved.

The connection is made via the work of Perrin-Riou [PR2], [PR4]. Briefly, for certain p-adic representations T of G_{Q}, and subject to some vast but plausible conjectures, Perrin-Riou shows how to view the p-adic L-functions attached to twists of T by characters of conductor m as elements in H_{∞}^{1}(Q(μ_{m}), T) (or more precisely, in the tensor product of H_{∞}^{1}(Q(μ_{m}), T) with the field of fractions of Λ). As we will see below in §8.3, these cohomology classes (if they exist) satisfy the distribution relation defining an Euler system for T. In other words, Perrin-Riou’s conjectural elements form an Euler system, and since they arise from padic L-functions, Theorems 2.2.10 and 2.3.8 then relate the Selmer groups S(Q, W^{∗}) and S(Q_{∞}, W^{∗}) to L-values.

164 8. EULER SYSTEMS AND p-ADIC L-FUNCTIONS

8.1. The Setting For this chapter we will assume

• K = Q, i.e., T is a p-adic representation of G_{Q},

• the scalar ring O is Z_{p}. The first assumption is not too serious a restriction, as in general one could consider the induced representation Ind_{K/Q}T. The second is completely unimportant, and is made only for notational convenience.

Following Perrin-Riou [PR4], we will also make the more serious assumptions that V = T ⊗ Q_{p} is the p-adic realization of a “motivic structure” in the sense of [FPR] Chapter III, that T corresponds to an integral structure on this motive, and that the representation V is crystalline at p.

We let D(V) denote Fontaine’s filtered vector space attached to V, i.e.,

D(V) = (V ⊗_{Q_{p}} B_{cris})^{G_{Qp}} .

By definition,

V is crystalline ⟺ dim_{Q_{p}}D(V) = dim_{Q_{p}} V.

Suppose F is an abelian extension of Q, unramified at p. Then F has [F : Q] distinct embeddings into B_{cris} and we also define

D_{F} (V) = D(⊕_{F ↪B_{cris}}V) ≅ D(Ind_{F/Q}V)

where G_{Q} acts on ⊕_{F ↪B_{cris}}V by acting both on V and on the set of embeddings.

Suppose E is a finite extension of Q_{p}, with ring of integers O_{E}, and χ : Gal(F/Q) → E^{×} is a character. Let O_{χ} denote a free rank-one O_{E}- module (with a fixed generator) on which Gal(F/Q) acts via χ, and write T ⊗ χ for T ⊗ O_{χ} and V ⊗ χ for V ⊗ O_{χ}. Let

ϵ_{χ} = ∑_{γ∈Gal(F/Q)}χ(γ)γ^{−1} ∈ O_{E}[Gal(F/Q)].

Lemma 8.1.1. (i) There is a natural identification

D_{F} (V) ≅ F ⊗_{Q} D(V).

(ii) Each choice of embedding F ↪ B_{cris} induces an isomorphism

D(V ⊗ χ) ≅ ϵ_{χ^{−1}}(E ⊗_{Q_{p}} D_{F} (V))

where we let Gal(F/Q) act on D_{F} (V) via its action on F in (i).

Proof. We have

D_{F} (V) = (⊕_{j:F ↪B_{cris}}V ⊗ B_{cris})^{G_{Qp}} ,

so there is a natural embedding

F ⊗_{Q} D(V) ↪ D_{F} (V) given by α ⊗ d ↦ ⊕_{j}(j(α)d).  (8.1)

In general

dim_{Q_{p}} D_{F} (V) ≤ dim_{Q_{p}}(Ind_{F/Q}V) = dim_{Q_{p}}(F ⊗_{Q} D(V)),

so we conclude that equality must hold (i.e., Ind_{F/Q}V is crystalline), and the map (8.1) is an isomorphism. This proves (i).

For (ii), let (E ⊗V ⊗B_{cris})^{χ^{−1}} be the subspace of E ⊗_{Q_{p}} V ⊗_{Q_{p}} B_{cris} on which G_{Q_{p}} (acting on V and B_{cris}, not on E) acts via χ^{−1}. An embedding j : F ↪ B_{cris} induces an embedding E ⊗ F ↪ E ⊗ B_{cris}, and hence (using (i)) an isomorphism

ϵ_{χ^{−1}}(E ⊗ D_{F} (V)) = ϵ_{χ^{−1}}(E ⊗ F) ⊗ D(V) ⟶^{∼} (E ⊗ V ⊗ B_{cris})^{χ^{−1}}.

Our fixed generator of O_{χ} induces an isomorphism from (E ⊗V ⊗B_{cris})^{χ^{−1}} to (V ⊗ χ ⊗ B_{cris})^{G_{Qp}} = D(V ⊗ χ). This proves (ii).

As in Chapter 3 we let Q_{∞} = ∪Q_{n} denote the cyclotomic Z_{p}-extension of Q, let Γ = Gal(Q_{∞}/Q), and let Λ = Z_{p}[[Γ]] be the Iwasawa algebra. Let H be the extended Iwasawa algebra defined by Perrin-Riou in [PR2] §1, i.e., if we identify Λ with a power series ring Z_{p}[[X]] in the usual way, and we let Q_{p}[[X]]_{r} ⊂ Q_{p}[[X]] denote the Q_{p}-vector space of power series which converge on the open unit ball in \overline{Q_{p}} with growth

sup_{|X|<ρ} |f(X)| = o( sup_{|X|<ρ} | log(1 + X)|^{r})

as ρ → 1^{−}, then H is the Λ-algebra

H = Λ ⊗_{Z_{p}[[X]]} (\underrightarrow{lim}_{r} Q_{p}[[X]]_{r}).

We let K be the field of fractions of H.

Suppose F is an abelian extension of Q unramified at p. In [PR2] (see also [PR4] §1.2) Perrin-Riou constructs^{1} what she calls a “logarithme élargi”, a Z_{p}[[Gal(F Q_{∞}/Q)]]-module homomorphism

⨁_{v|p} \underleftarrow{lim}_{n} H^{1}((F Q_{n})_{v}, T) ⟶ K ⊗ D_{F} (V).

This is a generalization of work of Coleman [Co], who defined such a map in the case where T = Z_{p}(1). Composing Perrin-Riou’s map with the local restriction maps we obtain a Z_{p}[[Gal(F Q_{∞}/Q)]]-module homomorphism

L_{F} : H_{∞}^{1}(F, T) = \underleftarrow{lim}_{n} H^{1}(F Q_{n}, T) ⟶ K ⊗ D_{F} (V)

1Perrin-Riou’s construction only deals with odd primes p. We will implicitly assume as part of the conjecture below that her construction can be extended to p = 2 to produce a map with similar properties.

which will be crucial in what follows. If F′ ⊂ F then there is a commutative diagram

\begin{matrix}H_{∞}^{1}(F, T) & ⟶^{L_{F}} & K ⊗ D_{F} (V) \\ ^{Cor}_{F/F′} ↓ & & ↓^{Tr}_{F/F′} \\ H_{∞}^{1}(F′, T) & ⟶^{L_{F′}} & K ⊗ D_{F′}(V). \end{matrix}  (8.2)

8.2. Perrin-Riou’s p-adic L-function and Related Conjectures

Let d = d(V) = dim_{Q_{p}}(V),

d_{+} = d_{+}(V) = dim_{Q_{p}}(V^{+}) = dim_{Q_{p}}(V^{c=1})

where c is a complex conjugation in G_{Q}, and let

d_{−} = d_{−}(V) = dim_{Q_{p}}(V^{−}) = dim_{Q_{p}}(V^{c=−1}) = d − d_{+}.

Let ω : G_{Q} → (Z_{p}^{×})_{tors} be the Teichmüller character giving the action of G_{Q} on μ_{p} (if p is odd) or on μ_{4} (if p = 2), and define

⟨ε⟩ = ω^{−1}ε_{cyc} : G_{Q} ↠ Γ ⟶^{∼} \begin{cases}1 + pZ_{p} & if p is odd, \\ 1 + 4Z_{2} & if p = 2. \end{cases}

Fix embeddings \overline{Q} ↪ C and \overline{Q} ↪ \overline{Q_{p}}.

Suppose that E is a finite extension of Q_{p} and χ : G_{Q} → E^{×} is an even character of finite order, unramified at p.

Conjecture 8.2.1 (Perrin-Riou [PR4] §4.2). Under the assumptions on T at the beginning of §8.1, if r ∈ Z^{+} is divisible by the conductor of χ then there is a p-adic L-function

L_{r}^{(p)}(T ⊗ χ) ∈ K ⊗ ∧_{E}^{d_{+}}D(V^{∗} ⊗ χ^{−1}).

See [PR4] §4.2 for the properties defining this p-adic L-function (when p > 2). For our purposes we only say loosely that L_{r}^{(p)}(T ⊗ χ) is defined so that for characters ρ of Γ of finite order and for sufficiently large positive integers k,

⟨ε⟩^{k}ρ(L_{r}^{(p)}(T ⊗ χ))

= (p-Euler factor) × \frac{L_{r}(V ⊗ χω^{k}ρ^{−1}, −k)}{(archimedean period)} × (p-adic period).

Here L_{r}(V ⊗ χω^{k}ρ^{−1}, s) is the (conjectural) complex L-function of the representation V ⊗χω^{k}ρ^{−1} with Euler factors at primes dividing r removed,

8.2. PERRIN-RIOU’S p-ADIC L-FUNCTION AND RELATED CONJECTURES 167

which has an Euler product expansion

∏_{ℓ∤r} l_{ℓ}(V ⊗ χω^{k}ρ^{−1}, s)^{−1}.  (8.3)

For primes ℓ ≠ p where V is unramified, the Euler factor at ℓ is defined by

l_{ℓ}(V ⊗ χω^{k}ρ^{−1}, s) = det(1 − Fr_{ℓ}^{−1}x|V ⊗ χω^{k}ρ^{−1})|_{x=ℓ^{−s}},

so in particular

l_{ℓ}(V ⊗ χω^{k}ρ^{−1}, −k) = ⟨ε⟩^{k}ρ(det(1 − Fr_{ℓ}^{−1}x|V)|_{x=χ^{−1}(ℓ)Fr_{ℓ}}).

Hence for such ℓ, writing P (Fr_{ℓ}^{−1}|T; x) = det(1 − Fr_{ℓ}^{−1}x|T) as in §2.1, we have

L_{rℓ}^{(p)}(T ⊗ χ) = P (Fr_{ℓ}^{−1}|T; χ^{−1}(ℓ)Fr_{ℓ})L_{r}^{(p)}(T ⊗ χ).  (8.4)

The following statement is in the spirit of the conjectures of Perrin-Riou in [PR4] §4.4, but stronger. In fact it is so strong that this formulation is certainly not true in general (see Remark 8.2.5 below). However, one can hope that it is “almost” true.

For r ∈ Z^{+} write Δ_{r} = Gal(Q(μ_{r})^{+}/Q), where Q(μ_{m})^{+} is the real subfield of Q(μ_{m}), and

Λ_{r} = Λ ⊗ Z_{p}[Δ_{r}] = Z_{p}[[Gal(Q_{∞}(μ_{r})^{+}/Q)]].

The involution γ ↦ γ^{−1} of Γ induces involutions of K and hence of K⊗∧_{E}^{d_{−}}D(V ⊗χ^{−1}), for example. We denote this last involution by f ↦ f^{ι}.

Wishful Thinking 8.2.2. Suppose r ∈ Z^{+} is prime to p. Then there is an element ξ_{r} ∈ ∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T) such that for every finite extension E of Q_{p} and every character χ : Δ_{r} → E^{×},

ϵ_{χ}^{⊗d_{−}}(L_{Q(μ_{r})^{+}}^{⊗d_{−}}(ξ_{r})) = L_{r}^{(p)}(T^{∗} ⊗ χ)^{ι}.

Remark 8.2.3. In the equality above, since d_{−}(V) = d_{+}(V^{∗}), the righthand side lies in K ⊗ ∧_{E}^{d_{−}}D(V ⊗ χ^{−1}). On the left,

L_{Q(μ_{r})^{+}}^{⊗d_{−}} : ∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T) ⟶ K ⊗ ∧_{Q_{p}[Δ_{r}]}^{d_{−}}D_{Q(μ_{r})}^{+}(V)

is the map induced by L_{Q(μ_{r})}^{+}. Recalling that ϵ_{χ} = ∑ χ(γ)γ^{−1}, we also have a map

ϵ_{χ} : D_{Q(μ_{r})}^{+}(V) ⟶ D(V ⊗ χ^{−1})

from Lemma 8.1.1(ii) (our chosen embedding \overline{Q} ↪ \overline{Q_{p}} gives an embedding Q(μ_{r}) ↪ B_{cris}) which induces a map

ϵ_{χ}^{⊗d_{−}} : K ⊗ ∧_{Q_{p}[Δ_{r}]}^{d_{−}}D_{Q(μ_{r})}^{+}(V) ⟶ K ⊗ ∧_{E}^{d_{−}}D(V ⊗ χ^{−1}).

Note that this makes sense even if d_{−} = 0, in which case ϵ_{χ}^{⊗0} is the projection from K⊗Q_{p}[Δ_{r}] to K⊗E induced by χ. Thus the optimistic equality of 8.2.2 above is an identity between two elements of K ⊗ ∧_{E}^{d_{−}}D(V ⊗ χ^{−1}).

Remark 8.2.4. The statement above is a strengthening and “extrapolation” (by introducing the level r) of the conjectures of Perrin-Riou in §4.4 of [PR4]. We have also rephrased the conjecture in terms of L_{r}^{(p)}(T^{∗} ⊗ χ) instead of L_{r}^{(p)}(T ⊗ χ^{−1}) by using the functional equation [PR4] §4.3.2, because it simplifies the formulas below.

Remark 8.2.5. One reason that the optimistic statement 8.2.2 cannot be true in general is that it asserts that the p-adic L-functions should all be “integral” in a strong sense. But the L-values can have denominators, coming from W^{G_{Q∞(μ_{r})^{+}}} where W = T ⊗ (Q_{p}/Z_{p}). Inspired by the theorem of Deligne and Ribet [DR], by Stark’s conjecture [T5] (where this denominator has been extensively studied), and by Perrin-Riou’s [PR4] Conjecture 4.4.2 (and Lemme 1.3.3), one is led to the following slightly more modest assertion. We optimistically call it a conjecture in the hope that at least some similar statement is true.

Conjecture 8.2.6. Suppose that r ∈ Z^{+} is prime to p, that d_{−} = 1, and that α ∈ Z_{p}[[G_{Q}]] annihilates W^{G_{Q∞(μ_{r})^{+}}} .

Then there is an element ξ_{r} = ξ_{r}^{(α)} ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T) such that for every finite extension E of Q_{p} and every character χ : Δ_{r} → E^{×},

ϵ_{χ}L_{Q(μ_{r})}^{+}(ξ_{r}) = χ(α)L_{r}^{(p)}(T^{∗} ⊗ χ)^{ι},

where χ(α) denotes the image of α under the composition

Z_{p}[[G_{Q}]] ↠ Λ_{r} ≅ Λ ⊗ Z_{p}[Δ_{r}] ⟶^{1⊗χ} Λ ⊗ E ⟶ K ⊗ E.

Note that if T is unramified at every prime dividing r, then

T^{G_{Q∞(μ_{r})}} = T^{G_{Q∞}} and W^{G_{Q∞(μ_{r})}} = W^{G_{Q∞}}  (8.5)

exactly as in Lemma 4.2.5(i), since Gal(Q_{∞}(μ_{r})/Q_{∞}) is generated by inertia groups which act trivially on T^{G_{Q∞}} and W^{G_{Q∞}}.

8.3. Connection with Euler Systems when d_{−} = 1

Suppose that T is as above, that d_{−} = 1, that Conjectures 8.2.1 and 8.2.6 hold, and that the weak Leopoldt conjecture (see [PR4] §1.3) holds for T^{∗}. For technical reasons we also assume that T^{G_{Q∞}} = 0. Let N be the product of all rational primes where T is ramified.

8.3. CONNECTION WITH EULER SYSTEMS WHEN d_{−} = 1 169

Fix an element α ∈ Z_{p}[[G_{Q}]] which annihilates W^{G_{Q∞}}. By (8.5), α annihilates W^{G_{Q∞(μ_{r})^{+}}} for every r ∈ Z^{+} prime to Np. For such r, let

ξ_{r} = {ξ_{n,r}} ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T), with ξ_{n,r} ∈ H^{1}(Q_{n}(μ_{r})^{+}, T),

be an element satisfying the conclusion of Conjecture 8.2.6.

Proposition 8.3.1. With hypotheses and notation as above, suppose r is prime to Np and ℓ is a prime not dividing Nrp. Then for every n ≥ 0 we have

Cor_{Q_{n}(μ_{rℓ})^{+}/Q_{n}(μ_{r})^{+}}ξ_{n,rℓ} = P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})ξ_{n,r}

where P (Fr_{ℓ}^{−1}|T^{∗}; x) = det(1 − Fr_{ℓ}^{−1}x|T^{∗}) ∈ Z_{p}[x].

Proof. Suppose that E is large enough so that all characters of Δ_{r} into \overline{Q_{p}}^{×} take values in E^{×}, and that χ : Δ_{r} → E^{×} is a (necessarily even) character. Write χ′ for the character Δ_{rℓ} ↠ Δ_{r} ⟶^{χ} E^{×}. Using (8.2) and the definition of ξ_{rℓ} we see that

ϵ_{χ}L_{Q(μ_{r})}^{+}(Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}ξ_{rℓ}) = ϵ_{χ}Tr ∘ L_{Q(μ_{rℓ})}^{+}(ξ_{rℓ})

= ϵ_{χ′}L_{Q(μ_{rℓ})}^{+}(ξ_{rℓ}) = χ(α)L_{rℓ}^{(p)}(T^{∗} ⊗ χ)^{ι}.

On the other hand, the definition of ξ_{r} shows that

ϵ_{χ}L_{Q(μ_{r})}^{+}(ξ_{r}) = χ(α)L_{r}^{(p)}(T^{∗} ⊗ χ)^{ι}.

Replacing T by T^{∗} in (8.4) and applying the involution ι yields

L_{rℓ}^{(p)}(T^{∗} ⊗ χ)^{ι} = P (Fr_{ℓ}^{−1}|T^{∗}; χ^{−1}(ℓ)Fr_{ℓ}^{−1})L_{r}^{(p)}(T^{∗} ⊗ χ)^{ι}.

Combining these equalities shows that

ϵ_{χ}L_{Q(μ_{r})}^{+}(Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}ξ_{rℓ}) = ϵ_{χ}L_{Q(μ_{r})}^{+}(P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})ξ_{r})

for every χ, and therefore since ∑_{χ} ϵ_{χ} = [Q(μ_{r})^{+} : Q] ∈ E[Δ_{r}]^{×}, we conclude that

L_{Q(μ_{r})}^{+}(Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}ξ_{rℓ}) = L_{Q(μ_{r})}^{+}(P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})ξ_{r}).

It remains only to show that, under our hypotheses, L_{Q(μ_{r})}^{+} is injective. Recall that L_{Q(μ_{r})}^{+} is the composition

H_{∞}^{1}(Q(μ_{r})^{+}, T) ⟶ ⨁_{v|p} \underleftarrow{lim}_{n} H^{1}(Q_{n}(μ_{r})_{v}^{+}, T) ⟶ K ⊗ D_{Q(μ_{r})}^{+}(V). (8.6)

The weak Leopoldt conjecture, which we have assumed, implies that ([PR4] (1.4.2) and Corollary B.3.6) the restriction map

H_{∞}^{1}(Q(μ_{r})^{+}, T) ⟶ ⨁_{q|Np} \underleftarrow{lim}_{n} ⊕_{v|q} H^{1}(Q_{n}(μ_{r})_{v}^{+}, T)

is injective. By Proposition A.2.3 of [PR4], \underleftarrow{lim} ⊕_{v|q} H^{1}(Q_{n}(μ_{r})_{v}^{+}, T) is a torsion Λ-module if q ≠ p. Therefore the kernel of the first map of (8.6) is a torsion Λ-module, and the definition of the second map ([PR4] §1.2.5) shows that its kernel is torsion as well. But by [PR4] Lemme 1.3.3, the Λ-torsion submodule of H_{∞}^{1}(Q(μ_{r})^{+}, T) is T^{G_{Q∞(μ_{r})^{+}}} , which is T^{G_{Q∞}} by (8.5), and by our hypothesis this is zero. Thus L_{Q(μ_{r})}^{+} is injective and the proposition follows.

Corollary 8.3.2. With notation as above, the collection

{ξ_{n,r} ∈ H^{1}(Q_{n}(μ_{r})^{+}, T) : n ≥ 0 and r prime to Np}

defines an Euler system for (T, Q_{∞}(μ′)^{+}, Np) in the sense of Definition 2.1.1 and Remark 2.1.3, where μ′ is the group of all roots of unity of order prime to Np.

Proof. This is immediate from Proposition 8.3.1.

Remark 8.3.3. There is another way to think about the existence of Euler systems when d_{−} = 1, in terms of complex L-functions. Namely, the Euler product (8.3) for L(V^{∗}, s) converges (conjecturally), and hence is nonzero, if s is a sufficiently large positive integer. This allows us to read off the value of ord_{s=−k}L(V, s) for large positive integers k in terms of the Γ-factors in the functional equation relating L(V, s) and L(V^{∗}, s). Subject to standard conjectures, one can show in this way that

ord_{s=0}L(V ⊗ ⟨ε⟩^{−k}ρ, s) = d_{−}

for all sufficiently large positive integers k and all characters ρ of finite order of Gal(Q_{∞}/Q).

Fix one such k. The Beilinson and Bloch-Kato conjectures then predict that the leading term in the Taylor expansion of L(V ⊗ ⟨ε⟩^{−k}ρ, s) at 0 can be expressed in terms of, among other things, a d_{−} × d_{−} regulator. When d_{−} = 1, this predicts the existence of certain special elements, and one can hope that these elements produce an Euler system for T ⊗ ⟨ε⟩^{−k}.

By Theorem 6.3.5, an Euler system for T ⊗ ⟨ε⟩^{−k} can then be twisted to produce an Euler system for T.

Remark 8.3.4. In the next section we consider the example T = Z_{p}(1), which has d = d^{−} = 1. Other interesting examples are when T is the Tate module of a modular elliptic curve (as in §3.5), which has d^{−} = d^{+} = 1, or when T is the symmetric square of the Tate module of an elliptic curve (as in §3.6), which has d^{−} = 1 and d^{+} = 2.

8.4. Example: Cyclotomic Units

In this section we discuss the example T = Z_{p}(1). The results in this section were all worked out by Perrin-Riou in [PR3]. The main ideas of the computation go back to Iwasawa.

We suppose for this section that p > 2. We will show that Conjecture 8.2.6 is true for Z_{p}(1), and in the process we will show that the Euler system of cyclotomic units discussed in §3.2 arises in the way described in the previous section. Note that d_{−}(Q_{p}(1)) = d(Q_{p}(1)) = 1 and d_{+}(Q_{p}(1)) = 0.

For every r ∈ Z^{+} prime to p, and n ≥ 0, let

c̃_{p^{n}r} = N_{Q(μ_{rp^{n+1}} )/Q_{n}(μ_{r})^{+}}(ζ_{rp^{n+1}} − 1) ∈ (Q_{n}(μ_{r})^{+})^{×}

⊂ H^{1}(Q_{n}(μ_{r})^{+}, Z_{p}(1)),

the Euler system of §3.2, and

c̃_{r,∞} = {c̃_{p^{n}r}}_{n} ∈ H_{∞}^{1}(Q(μ_{r})^{+}, Z_{p}(1)).

Let

u_{r}(X) = ζ_{r}(1 + X)^{r^{−1}} − 1 ∈ (Z[μ_{r}] ⊗ Z_{p})[[X]]

and

h_{r}(X) = ∏_{β∈μ_{p−1}⊂Z_{p}^{×}}u_{r}((1 + X)^{β} − 1)ū_{r}((1 + X)^{β} − 1)

where ū_{r}(X) = ζ_{r}^{−1}(1 + X)^{r^{−1}} − 1. Then h_{r} is the “Coleman power series” attached to c̃_{r,∞}, i.e., for every n ≥ 0,

h_{r}^{Fr_{p}^{−n−1}}(ζ_{p^{n+1}} − 1) = c̃_{p^{n}r}.

The p-adic L-functions L_{r}^{(p)}(Z_{p} ⊗ χ) that arise below are the Kubota-Leopoldt p-adic L-functions, so their existence does not rely on any conjectures. The following proposition is essentially due to Iwasawa and Coleman, but we present it in the language of Perrin-Riou, following [PR3].

Proposition 8.4.1. Suppose r ≥ 1 and E is a finite extension of Q_{p}. For every character χ : Δ_{r} → E^{×}, we have

ϵ_{χ}L_{Q(μ_{r})}^{+}(c̃_{r,∞}) = 2L_{r}^{(p)}(Z_{p} ⊗ χ)^{ι}.

Proof. Suppose first that r > 1. By [PR3] §1.8, §3.1 (or [PR2] §4.1.3) and [Iw2],

L_{Q(μ_{r})}^{+}(c̃_{r,∞}) ∈ Λ ⊗ D_{Q(μ_{r})}^{+}(Q_{p}(1)) = Q(μ_{r})^{+} ⊗ Λ ⊗ D(Q_{p}(1)), (8.7)
L_{r}^{(p)}(Z_{p} ⊗ χ) ∈ Λ ⊗ D(Q_{p}(1) ⊗ χ^{−1}) = ϵ_{χ}(Λ ⊗ D_{Q(μ_{r})}^{+}(Q_{p}(1))), (8.8)

where the equalities use Lemma 8.1.1. Let e_{−1} denote the canonical generator of the one-dimensional vector space D(Q_{p}(1)), and define

H_{r}(X) = log h_{r}(X) − \frac{1}{p} log h_{r}^{Fr_{p}}((1 + X)^{p} − 1).

From the definition in [PR4] §1.2.5 (see also [PR3] §1.3 and §3.1.4), we see that

L_{Q(μ_{r})}^{+}(c̃_{r,∞}) = F_{r}e_{−1}

where F_{r} ∈ Q(μ_{r})^{+} ⊗ Λ is such that for every k ≥ 1,

⟨ε⟩^{k}(F_{r}) = (D^{k}H_{r})(ζ_{p} − 1)

where D is the derivation (1 + X)\frac{d}{dX}. Thus if χ : Δ_{r} → E^{×} is a character, then

ϵ_{χ}L_{Q(μ_{r})}^{+}(c̃_{r,∞}) = F_{r,χ}e_{−1}

where F_{r,χ} ∈ Q(μ_{r})^{+} ⊗ Λ ⊗ E is such that for every k ≥ 1,

⟨ε⟩^{k}(F_{r,χ}) = ∑_{γ∈Δ_{r}} χ^{−1}(γ)(D^{k}H_{r}^{γ})(ζ_{p} − 1).

Therefore by Lemma D.2.2, if k ≥ 2 we have

⟨ε⟩^{k}(F_{r,χ}) = 2Γ(k)(−2πi)^{−k}L(χ^{−1}ω^{k}, k) × \begin{cases}−χ(p)p^{k} & if (p − 1) ∤ k, \\ 1 − p^{k−1}χ(p) & if (p − 1) | k, \end{cases}

so by the formulas in [PR4] §4.2 and §4.3.3 we see that for k ≥ 2,

⟨ε⟩^{k}(ϵ_{χ}L_{Q(μ_{r})}^{+}(c̃_{r,∞})) = ⟨ε⟩^{k}(F_{r,χ}e_{−1})

= 2⟨ε⟩^{−k}(L_{r}^{(p)}(Z_{p} ⊗ χ)) = 2⟨ε⟩^{k}(L_{r}^{(p)}(Z_{p} ⊗ χ)^{ι})

(the Gauss sums which appear in the formulas of [PR4] and [PR3] are not present here because we never identified Q[Gal(Q(μ_{r})/Q)] with Q(μ_{r}) as in [PR3] §1.8). By (8.7) and (8.8), these equalities suffice to prove the proposition when r > 1. A similar computation shows that for every σ ∈ G_{Q},

L_{Q(μ_{r})}^{+}((σ − 1)c̃_{1,∞}) = 2(σ − 1)L_{1}^{(p)}(Z_{p})^{ι}.

Corollary 8.4.2. Conjecture 8.2.6 holds for T = Z_{p}(1).

Proof. We have assumed that p > 2. Therefore μ_{p^{∞}}^{G_{Q∞}} = {1}, and for every α ∈ Z_{p}[[G_{Q}]], Proposition 8.4.1 shows that

ξ_{r} = \frac{1}{2}αc̃_{r,∞} ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T)

satisfies Conjecture 8.2.6.

8.5. CONNECTION WITH EULER SYSTEMS WHEN d_{−} > 1 173

8.5. Connection with Euler Systems when d_{−} > 1

Suppose now that T is such that d_{−} is greater than 1, and suppose that some version of the assertion 8.2.2 is true, i.e., suppose there is an integer N divisible by all primes where T is ramified, and an element α ∈ Z_{p}[[G_{Q}]] such that for every integer r prime to Np, there is an element

ξ_{r} ∈ ∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T)

satisfying

ϵ_{χ}^{⊗d_{−}}L_{Q(μ_{r})^{+}}^{⊗d_{−}}(ξ_{r}) = χ(α)L_{r}^{(p)}(T^{∗} ⊗ χ)^{ι}

for every character χ of Δ_{r}. We also suppose again that the weak Leopoldt conjecture holds for T^{∗}. In this section we will adapt an idea from [Ru8] §6 to construct Euler systems (elements in H_{∞}^{1}(Q(μ_{r})^{+}, T)) from the elements ξ_{r} ∈ ∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T).

Lemma 8.5.1. With hypotheses and notation as above, suppose r is prime to Np and ℓ is a prime not dividing Nrp. Then

Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}^{⊗d_{−}}(ξ_{rℓ}) − P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})(ξ_{r})

belongs to the Λ-torsion submodule of ∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T), where

P (Fr_{ℓ}^{−1}|T^{∗}; x) = det(1 − Fr_{ℓ}^{−1}x|T^{∗}) ∈ Z_{p}[x]

and

Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}^{⊗d_{−}} : ∧^{d_{−}}H_{∞}^{1}(Q(μ_{rℓ})^{+}, T) ⟶ ∧^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T)

is the map induced by corestriction.

Proof. Exactly as in the proof of Proposition 8.3.1, we deduce that

L_{Q(μ_{r})^{+}}^{⊗d_{−}}(Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}^{⊗d_{−}}(ξ_{rℓ})) = L_{Q(μ_{r})^{+}}^{⊗d_{−}}(P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})(ξ_{r})).

Also as in the proof of Proposition 8.3.1, the kernel of L_{Q(μ_{r})}^{+} is a torsion Λ-module, and so the kernel of L_{Q(μ_{r})^{+}}^{⊗d_{−}} is torsion as well.

Suppose that r is prime to Np, and φ ∈ Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r}). Then φ induces a Λ_{r}-module homomorphism from ∧_{Λ_{r}}^{k}H_{∞}^{1}(Q(μ_{r})^{+}, T) to ∧_{Λ_{r}}^{k−1}H_{∞}^{1}(Q(μ_{r})^{+}, T) for all k ≥ 1 by the usual formula

c_{1} ∧ ⋯ ∧ c_{k} ↦ ∑_{i=1}^{k}(−1)^{i+1}φ(c_{i})c_{1} ∧ ⋯ ∧ c_{i−1} ∧ c_{i+1} ⋯ ∧ c_{k}.

Iterating this construction d_{−} − 1 times gives a map

∧_{Λ_{r}}^{d_{−}−1} Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r})

⟶ Hom(∧_{Λ_{r}}^{d_{−}}H_{∞}^{1}(Q(μ_{r})^{+}, T), H_{∞}^{1}(Q(μ_{r})^{+}, T)). (8.9)

If s | r then there is a natural map

N_{r/s} : Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r}) ⟶ Hom_{Λ_{s}}(H_{∞}^{1}(Q(μ_{s})^{+}, T), Λ_{s})

induced by restriction H_{∞}^{1}(Q(μ_{s})^{+}, T) → H_{∞}^{1}(Q(μ_{r})^{+}, T) and the identification Λ_{s} ≅ Λ_{r}^{Gal(Q(μ_{r})^{+}/Q(μ_{s})^{+})}.

Proposition 8.5.2. With notation as above, suppose that T^{G_{Q∞}} = 0 and that

𝔖 = \{𝔖_{r}\} ∈ \underleftarrow{lim}_{r} ∧_{Λ_{r}}^{d_{−}−1} Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r}),

inverse limit with respect to the maps N_{r/s}^{⊗d_{−}−1}. If r is prime to Np then 𝔖_{r}(ξ_{r}) ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T), and if further ℓ is a prime not dividing Nrp then

Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}(𝔖_{rℓ}(ξ_{rℓ})) = P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})(𝔖_{r}(ξ_{r})).

In other words, if we write 𝔖_{r}(ξ_{r}) = {ξ_{n,r}}_{n} then the collection

{ξ_{n,r} ∈ H^{1}(Q_{n}(μ_{r})^{+}, T)}

is an Euler system for T (Definition 2.1.1 and Remark 2.1.3).

Proof. The proof is identical to that of Proposition 6.2 and Corollary 6.3 of [Ru8]. Using (8.9) it is clear that 𝔖_{r}(ξ_{r}) ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T). Suppose that s | r. If ϕ ∈ Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r}) and c ∈ H_{∞}^{1}(Q(μ_{r})^{+}, T), then ϕ(c) ∈ Λ_{r}, and its image under the restriction map Λ_{r} → Λ_{s} is (N_{r/s}ϕ)(Cor_{Q(μ_{r})^{+}/Q(μ_{s})^{+}}(c)). From this it follows without difficulty that

Cor_{Q(μ_{r})^{+}/Q(μ_{s})^{+}}(𝔖_{r}(ξ_{r})) = 𝔖_{s}(Cor_{Q(μ_{r})^{+}/Q(μ_{s})^{+}}^{⊗d_{−}}(ξ_{s})).

Combining this equality with Lemma 8.5.1 shows that

Cor_{Q(μ_{rℓ})^{+}/Q(μ_{r})^{+}}(𝔖_{rℓ}(ξ_{rℓ})) − P (Fr_{ℓ}^{−1}|T^{∗}; Fr_{ℓ}^{−1})𝔖_{r}(ξ_{r})

belongs to the Λ-torsion submodule of H_{∞}^{1}(Q(μ_{r})^{+}, T). But by [PR4] (Lemme 1.3.3) and (8.5) this torsion submodule is T^{G_{Q∞}}, which we have assumed to be zero.

Remark 8.5.3. Of course, Proposition 8.5.2 is only useful if we know something about the size of \underleftarrow{lim} ∧_{Λ_{r}}^{d_{−}−1} Hom_{Λ_{r}}(H_{∞}^{1}(Q(μ_{r})^{+}, T), Λ_{r}), and in particular that it is nonzero. See [Ru8] §6 for an example.

CHAPTER 9 Variants

In this chapter we discuss several alternatives and extensions to the definition of Euler systems we gave in Chapter 2.

9.1. Rigidity

It is tempting to remove from the definition of an Euler system the requirement that the field K (over whose subfields the Euler system classes are defined) contain a Z_{p}-extension of K. After all, the proofs of the Theorems of §2.2 only use the derivative classes κ_{[K,𝔯,M]} and not the κ_{[F,𝔯,M]} for larger extensions F of K in K_{∞}. However, our proofs of the properties of the derivative classes κ_{[K,𝔯,M]} very much used the fact that the Euler system class c_{K(𝔯)} is a “universal norm” in the extension K_{∞}(𝔯)/K(𝔯).

In fact, some such assumption is needed, as the following example shows. Suppose K has class number one, N is an ideal of K divisible by p and all primes where T is ramified, and T has the property that P (Fr_{𝔮}^{−1}|T^{∗}; 1) = 0 for every 𝔮 not dividing N. (For example, if T is the symmetric square of the Tate module of an elliptic curve as in §3.6 then T has this property.) Suppose further that K is the maximal abelian extension of K unramified outside N (so K does not contain a Z_{p}-extension of K) and c satisfies the distribution relation in Definition 2.1.1 of an Euler system for (T, K, N). Then in Definition 2.1.1, the only equations connecting c_{K} with the other c_{F} are of the form

Cor_{F/K}c_{F} = ∏_{𝔮∈Σ(F/K)}P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})c_{K} = ∏_{𝔮∈Σ(F/K)}P (Fr_{𝔮}^{−1}|T^{∗}; 1)c_{K}.

If F ≠ K then the set Σ(F/K) of primes ramifying in F/K is nonempty, so the right-hand side will always be zero. In other words c_{K} does not appear in any nontrivial Euler system relations, so we can replace c_{K} by any element at all in H^{1}(K, T) without disturbing the distribution relation of Definition 2.1.1. For example, the collection defined by c_{F} = 0 for F ≠ K, with c_{K} arbitrary, satisfies those relations. Since there are examples satisfying the conditions above with non-trivial Selmer groups, in this situation one cannot hope for a theorem like Theorem 2.2.2 (or Theorem 4.5.4), in which the conclusion depends in an essential way on c_{K}.

However, there are other possible ways to ensure the “rigidity” of an Euler system. Let R be the set of squarefree ideals of K prime to N, as in Chapter 4. In Definition 2.1.1, we can replace condition (ii) by

(ii′) at least one of the conditions (a), (b), (c) below is satisfied:

(a) K contains a Z_{p}^{d}-extension of K in which no finite prime splits completely,

(b) c_{K(𝔯)} ∈ S^{Σ_{p}}(K(𝔯), T) for every 𝔯 ∈ R, and there is a γ ∈ G_{K} such that T^{γ=1} = 0 and γ = 1 on K(1)K(μ_{p^{∞}}, (O_{K}^{×})^{1/p^{∞}}),

(c) c_{K(𝔯)} ∈ S^{Σ_{p}}(K(𝔯), T) for every 𝔯 ∈ R, the collection {c_{K(𝔯)}} satisfies the congruence of Corollary 4.8.1, and for every prime 𝔮 ∈ R and every power n of p, Fr_{𝔮}^{n} − 1 is injective on T.

Condition (ii′)(a) is condition (ii) of the original definition.

Under this more general definition, Theorems 2.2.2, 2.2.3, and 2.2.10 all hold, with the same conclusions as before, under the additional mild assumption that T^{G_{K(1)}} = 0. We indicate very briefly how to adapt the proofs in Chapters 4 and 5 to cover this expanded definition.

The idea is that there is a power m of p, independent of M, such that one can still construct the derivative classes κ_{[K,𝔯,M]}, and prove the local properties of §4.5, under the assumption 𝔯 ∈ R_{K,Mm} rather than 𝔯 ∈ R_{K,M}. This additional assumption does not interfere with the proofs of the theorems of Chapter 2.

Construction of the derivative classes. Since we assumed T^{G_{K(1)}} = 0, Lemma 4.2.5(i) shows that T^{G_{K(𝔯)}} = 0 for every 𝔯. Thus if we replace 𝕎_{M} by 𝕋 = Maps(G_{K}, T) in Proposition 4.4.5 we get a short exact sequence

0 ⟶ 𝕋^{G_{F (𝔯)}} ⟶ (𝕋/T)^{G_{F (𝔯)}} ⟶^{δ_{F (𝔯)}} H^{1}(F (𝔯), T) ⟶ 0.

Now as in Proposition 4.4.8, but using the exact sequence above instead of Proposition 4.4.7, we can find a map d : X_{F (𝔯)} → (𝕋/T)^{G_{F (𝔯)}} lifting c. Projecting this map to (𝕎_{M}/W_{M})^{G_{F (𝔯)}} we can proceed exactly as in Definition 4.4.10 to define κ_{[F,𝔯,M]}.

Analogue of Theorem 4.5.1. We will use Corollary 4.6.5 instead of Theorem 4.5.1. Corollary 4.6.5 follows directly from Proposition 4.6.1, which is included as part of (ii′)(b) and (ii′)(c). (In §4.6 we used assumption (ii′)(a) and Corollary B.3.5 to prove Proposition 4.6.1.)

Analogue of Theorem 4.5.4. Theorem 4.5.4 follows directly from Lemma 4.7.3, so it will suffice to prove a form of that lemma. Suppose first that (ii′)(b) holds with an element γ ∈ G_{K}. Fix 𝔯𝔮 ∈ R, a power M of p, and a power M′ of p divisible by MP (γ|T; 1). By the definition of γ, we have P (γ|T; 1) ≠ 0. Let n = |μ_{p^{∞}} ∩ K|. Choose a prime 𝔩 of K such that

(a) Fr_{𝔩} = γ on K(1)K(W_{M′}, μ_{nM′}, (O_{K}^{×})^{1/(nM′)}),

(b) Fr_{𝔩} = 1 on K(𝔯𝔮),

(c) Fr_{𝔩} ≠ 1 on K(λ^{1/(np)}) where λO_{K} = 𝔮^{h} with h equal to the order

of 𝔮 in the ideal class group of K. (Exercise: show that these conditions can be satisfied simultaneously.) One can imitate the proof of Lemma 4.7.3 by using the extensions K(𝔩)/K in place of the finite extensions of K in K_{∞}. Condition (a) and the definition of γ ensure that nM′ | [K(𝔩) : K(1)]. Condition (c) ensures that the decomposition group of 𝔮 has index dividing n in Gal(K(𝔩)/K), and therefore has order at least M′. The key point is that although c_{K(𝔯)} and c_{K(𝔯𝔮)} are not “universal norms” from K(𝔯𝔩) and K(𝔯𝔮𝔩) (as they would be from K_{∞}(𝔯) and K_{∞}(𝔯𝔮)), the Euler system distribution relation shows that P (Fr_{𝔩}^{−1}|T^{∗}; Fr_{𝔩}^{−1})c_{K(𝔯)} is a norm from K(𝔯𝔩) and similarly with 𝔯 replaced by 𝔯𝔮. Conditions (a) and (b) imply that in O[Gal(K(𝔯𝔮)/K)],

P (Fr_{𝔩}^{−1}|T^{∗}; Fr_{𝔩}^{−1}) = P (Fr_{𝔩}^{−1}|T^{∗}; 1) ≡ P (γ^{−1}|T^{∗}; 1) (mod M)
= P (γ|T; 1).

Now imitating the proof of Lemma 4.7.3 one can show that, with notation as in the statement of that lemma, if 𝔯𝔮 ∈ R_{K,M′} then

P (γ|T; 1)(N_{𝔮}γd̂(x_{F (𝔯𝔮)}) − P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1})γd̂(x_{F (𝔯)})) = 0 ∈ W_{M′}

This suffices to prove that κ_{[F,𝔯,M]} and κ_{[F,𝔯𝔮,M]} satisfy the equality of Theorem 4.5.4.

Now suppose (ii′)(c) holds. In §4.8 we used Lemma 4.7.3 to prove the congruence of Corollary 4.8.1. Under the assumptions (ii′)(c) we can just reverse the argument to prove Lemma 4.7.3, and then Theorem 4.5.4.

Example 9.1.1 (cyclotomic units revisited). With this expanded definition, we can redefine the cyclotomic unit Euler system of §3.2. Namely, for every m > 1 prime to p define

c̃_{m} = (ζ_{m} − 1)(ζ_{m}^{−1} − 1) ∈ (Q(μ_{m})^{+})^{×} ⊂ H^{1}(Q(μ_{m})^{+}, Z_{p}(1))

and set c̃_{1} = 1. This collection is not an Euler system, even under our expanded Definition 2.1.1. (If it were, then for every prime ℓ ≠ p, the class c̃_{Q(μ_{ℓ})} would belong to S^{Σ_{p}}(Q(μ_{ℓ}), Z_{p}(1)), but this is not the case because c̃_{Q(μ_{ℓ})} ∉ H_{f}^{1}(Q(μ_{ℓ})_{ℓ}, Z_{p}(1)).)

However, suppose χ : G_{Q} → O^{×} is a nontrivial character of finite order, and its conductor f is prime to p. Then we can twist c̃ by χ^{−1} as in Definition 2.4.1, and with the modified definition above, the collection c = c̃^{χ^{−1}} is an Euler system for (Z_{p}(1)⊗χ^{−1}, Q^{ab,p}, fp), where Q^{ab,p} is the maximal abelian extension of Q unramified outside p. Namely, although condition (ii′)(a) does not hold, (ii′)(b) (with γ ∈ G_{Q(μ_{p^{∞}})} and χ(γ) ≠ 1) and (ii′)(c) (see Example 4.8.2) both do hold. With this Euler system we can remove one of the hypotheses from Theorem 3.2.3 and Corollary 3.2.4. With notation as in §3.2 (so L is the field cut out by χ), we have the following theorem.

Theorem 9.1.2. Suppose p > 2 and χ is a nontrivial even character of conductor prime to p. Then

|A_{L}^{χ}| = [E_{L}^{χ} : C_{L,χ}].

Sketch of proof. If χ(p) ≠ 1 this is Corollary 3.2.4. So we may assume that the conductor of χ is prime to p and use the Euler system constructed above. For this Euler system, c_{1} generates C_{L,χ}, so exactly as in the proof of Theorem 3.2.3 we deduce from Theorem 2.2.2 that

|S_{Σ_{p}}(Q, (Q_{p}/Z_{p}) ⊗ χ)| divides [E_{L}^{χ} : C_{L,χ}].

Unfortunately, Proposition 1.6.1 shows that

S_{Σ_{p}}(Q, (Q_{p}/Z_{p}) ⊗ χ) = Hom(A_{L}^{χ}/P, D)

where P is the subgroup of A_{L}^{χ} generated by the classes of primes of L above p. This is not quite what we need.

To complete the proof, we observe that the derivative classes κ_{[K,r,M]} attached to our Euler system all lie in S^{Σ_{r}}(Q, μ_{M} ⊗ χ^{−1}), not just in S^{Σ_{rp}}(Q, μ_{M} ⊗ χ^{−1}) as Theorem 4.5.1 shows in the general case. (This follows from the fact that c_{Q(μ_{r})} ∈ S(Q(μ_{r}), Z_{p}(1) ⊗ χ^{−1}) for every r. See for example [Ru3] Proposition 2.4.) Therefore we can repeat the proof of Theorem 2.2.2, but using Σ_{0} = ∅ and Σ = Σ_{r} in Theorem 1.7.3 instead of Σ_{0} = {p} and Σ = Σ_{rp}, to conclude that

|A_{L}^{χ}| = |S(Q, (Q_{p}/Z_{p}) ⊗ χ)| divides [E_{L}^{χ} : C_{L,χ}].

Now the equality of the theorem follows from the analytic class number formula exactly as in Corollary 3.2.4.

9.2. Finite Primes Splitting Completely in K_{∞}/K

Definition 2.1.1 of an Euler system requires a Z_{p}^{d}-extension K_{∞}/K, with K_{∞} ⊂ K, such that no finite prime splits completely in K_{∞}/K.

In fact, the assumption that no prime splits completely is unnecessarily strong. We can remove this hypothesis if we assume instead that (∗) for every prime 𝔮 of K which splits completely in K_{∞}/K, and for

every finite extension F of K in K, we have (c_{F})_{𝔮} ∈ H_{ur}^{1}(F_{𝔮}, T). The set of primes which split completely in K_{∞}/K has density zero, so such primes do not interfere with our Tchebotarev arguments. Using this fact and (∗), the proofs in Chapters 4 through 7 work without significant modification in this more general setting.

9.3. Euler Systems of Finite Depth

Definition 9.3.1. Fix a nonzero M ∈ O. An Euler system for W_{M} (or an Euler system of depth M) is a collection of cohomology classes satisfying all the properties of Definition 2.1.1 except that instead of c_{F} ∈ H^{1}(F, T) we require c_{F} ∈ H^{1}(F, W_{M}). An Euler system in the sense of Definition 2.1.1 can be viewed as an Euler system of infinite depth.

Remark 9.3.2. For this definition we could replace W_{M} by a free O/MOmodule of finite rank with an action of G_{K}; it is not necessary that it can be written as T/MT for some T.

The construction of the derivative classes κ_{[F,𝔯,M]} in Chapter 4 only used the images of the classes c_{L} (for various L) in H^{1}(L, W_{M}). Thus if c is an Euler system for W_{M} then we can define the classes κ_{[F,𝔯,M]} exactly as in Chapter 4.

The proof of Theorem 4.5.4 also only used the images of the Euler system classes in H^{1}(L, W_{M}), so that theorem still holds for the derivative classes of an Euler system for W_{M}. However, the proof of Theorem 4.5.1 used the images of the Euler system classes in H^{1}(L, T), so that proof breaks down in this setting. However, as discussed in §9.1 above (and see Remark 4.6.4), we can still prove a weaker version of Theorem 4.5.1, and this will suffice for some applications.

For example, the proofs in Chapters 4 and 5 will prove the following theorem. Keep the setting and notation of Chapter 2 (so in particular, for simplicity, W_{M} = T/MT).

Theorem 9.3.3. Suppose M ∈ O is nonzero and c is an Euler system for W_{M}. Suppose that Hypotheses Hyp(K, T) hold, that the error terms 𝔫_{W} and 𝔫_{W}^{∗} of Theorem 2.2.2 are both zero, and that W_{M}^{G_{K}} = 0. Let

𝔞_{1} = ⋂_{primes 𝔮 of K 𝔮∤p}Ann_{O}(W^{I_{𝔮}}/(W^{I_{𝔮}})_{div})

and let 𝔞_{2} ⊂ O be the annihilator of 𝔞_{1}c_{K} in H^{1}(K, W_{M}). Then

(M/𝔞_{2})𝔞_{1}S_{Σ_{p}}(K, W_{M}^{∗}) = 0.

In particular if 𝔞_{1}c_{K} ≠ 0 then S_{Σ_{p}}(K, W^{∗}) is finite.

Remark 9.3.4. The ideal 𝔞_{1} of Theorem 9.3.3 is finite, since W^{I}/(W^{I})_{div} is finite for all 𝔮 and is zero if T is unramified at 𝔮. See the proof of Corollary 4.6.5.

One could reformulate Theorem 9.3.3 for a general G_{K}-module W̄ which is free of finite rank over O/MO, i.e., one which does not come from a “T ”, but one would have to redefine the Selmer group since our definition depends on T, not just on W_{M}.

9.4. Anticyclotomic Euler Systems

The “Euler system of Heegner points”, one of Kolyvagin’s original Euler systems, is not an Euler system under our Definition 2.1.1. If one tries to make the definition fit with K = Q, the problem is that the cohomology classes (Heegner points) are not defined over abelian extensions of Q, but rather over abelian extensions of an imaginary quadratic field which are “anticyclotomic” (and hence not abelian) over Q. On the other hand, if one tries to make the definition fit by taking K to be an appropriate imaginary quadratic field, then the problem is that the Heegner points are not defined over large enough abelian extensions of K, but only over those which are anticyclotomic over Q.

We will not discuss Heegner points in any detail (see instead [Ko2], [Ru2], or [Gro2]), but in this section we propose an expanded definition of Euler systems that will include “anticyclotomic” Euler systems such as Heegner points as examples.

Fix a number field K and a p-adic representation T of G_{K} as in §2.1. Suppose d is a positive integer dividing p − 1, and χ : G_{K} → Z_{p}^{×} is a character of order d. Let K′ = K̄^{ker(χ)} be the cyclic extension of degree d of K cut out by χ.

For every prime 𝔮 of K not dividing p let K′(𝔮)_{χ} denote the maximal p-extension of K′ inside the ray class field of K′ modulo 𝔮, such that Gal(K′/K) acts on Gal(K′(𝔮)_{χ}/K′) via the character χ. Similarly, let K′(1)_{χ} denote the χ-part of the maximal unramified p-extension of K′.

Now suppose K′ is an (infinite) abelian p-extension of K′ and N is an ideal of K divisible by p, the conductor of χ, and all primes where T is ramified, and such that K′ contains K′(𝔮)_{χ} for every prime 𝔮 of K not dividing N.

Definition 9.4.1. A collection of cohomology classes

c = {c_{F} ∈ H^{1}(F, T) : K′ ⊂_{f} F ⊂ K′}

is a χ-anticyclotomic Euler system for (T, K′, N) (or simply for T) if (i) whenever K′ ⊂_{f} F ⊂_{f} F′ ⊂ K′, then

Cor_{F′/F} (c_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}P (Fr_{𝔮}^{−1}|T^{∗}; Fr_{𝔮}^{−1}))c_{F}

where Σ(F′/F) is the set of primes of K not dividing N which ramify in F′ but not in F, as always Fr_{𝔮} is a Frobenius of 𝔮 in G_{K}, and P (Fr_{𝔮}^{−1}|T^{∗}; x) = det(1 − Fr_{𝔮}^{−1}x|T^{∗}) ∈ O[x],

(ii) at least one of the following analogues of the hypotheses (ii′) of §9.1

holds:

(a) K′ contains a Z_{p}^{d}-extension K′_{∞} of K′ such that no finite prime splits completely in K′_{∞}/K′, and Gal(K′/K) acts on Gal(K′_{∞}/K′) via χ,

(b) c_{K′(𝔯)_{χ}} ∈ S^{Σ_{p}}(K′(𝔯)_{χ}, T) for every 𝔯, and there is a γ ∈ G_{K} such that ε_{cyc}(γ) = χ(γ), and T^{γ=1} = 0, and γ^{d} = 1 on K′(1)_{χ}(μ_{p^{∞}}, (O_{K′}^{×})^{1/p^{∞}}),

(c) c_{K(𝔯)} ∈ S^{Σ_{p}}(K(𝔯), T) for every 𝔯, the classes {c_{F} } satisfy the appropriate analogue of the congruence of Corollary 4.8.1, and for every 𝔮 not dividing N and every power n of p, Fr_{𝔮}^{n}−1 is injective on T.

Remark 9.4.2. If d = 1, then χ is trivial, K′ = K, and thus a χ-anticyclo-tomic Euler system for T is the same as an Euler system for T in the sense of Definition 2.1.1 (or §9.1).

Suppose K = Q and χ is an odd quadratic character, so d = 2. Then K′ is an imaginary quadratic field and K′ is an anticyclotomic p-extension of K′. If T is the Tate module of a modular elliptic curve, and we make the additional assumption that χ(q) = 1 for every prime q dividing the conductor of χ, then the Heegner points in anticyclotomic extensions of K′ give a χ-anticyclotomic Euler system for T. (One must modify the Heegner points slightly, as in §9.6 below, to obtain the correct distribution relation.) Note that in this situation we could take K′ to contain the anticyclotomic Z_{p}-extension K′_{∞} of K′. However, all rational primes which are inert in K′ split completely in K′_{∞}/K′ so condition (ii)(a) of Definition 9.4.1 fails. However, both (ii)(b) and (ii)(c) hold.

Let

Ω′ = K′(1)_{χ}(μ_{p^{∞}}, (O_{K′}^{×})^{1/p^{∞}}, W).

For every i ∈ Z let

ind_{O}(c, χ^{i}) = sup{n : c_{K′}^{χ^{i}} ∈ 𝔭^{n}H^{1}(K′, T) + H^{1}(K′, T)_{tors}} ≤ ∞,

where c_{K′}^{χ^{i}} denotes the projection of c_{K′} into the subgroup H^{1}(K′, T)^{χ^{i}} of H^{1}(K′, T) on which Gal(K′/K) acts via χ^{i}. In this setting of anticyclotomic Euler systems one can prove the following theorem.

Theorem 9.4.3. Suppose c is a χ-anticyclotomic Euler system for T. Suppose further that H^{1}(Ω′/K′, W) = H^{1}(Ω′/K′, W^{∗}) = 0, that T ⊗ 𝕜 is an irreducible 𝕜[G_{K′}]-module, and that there is a τ ∈ G_{K} such that

• ε_{cyc}(τ) = χ(τ),

• τ^{d} is the identity on K′(1)_{χ}(μ_{p^{∞}}, (O_{K′}^{×})^{1/p^{∞}}),
• if ζ ∈ μ_{d} ⊂ O^{×} then T/(τ − ζ)T is free of rank one over O.

Then for every i,

𝔭^{ind^{O}(c,χ^{i})}S_{Σ_{p}}(K′, W^{∗})^{χ^{1−i}} = 0.

Remark 9.4.4. In the setting of the Heegner point Euler system mentioned in Remark 9.4.2, K′ is an imaginary quadratic field, and we can take τ to be a complex conjugation. Theorem 9.4.3 then says that the “minus part” of the Heegner point in E(K′) controls the “plus part” of the Selmer group of E over K′, and vice versa.

Sketch of proof. Given a χ-anticyclotomic Euler system and a power M of p, one can proceed exactly as in §4.4 to define derivative classes

κ_{[K′,𝔯,M]} ∈ H^{1}(K′, W_{M})

for every 𝔯 ∈ R_{K′,M,τ}, where R_{K′,M,τ} is the set of squarefree ideals of K divisible only by primes 𝔮 such that 𝔮 ∤ N and such that the Frobenius of 𝔮 in Gal(K′(1)_{χ}(μ_{M}, (O_{K′}^{×})^{1/M}, W_{M})/K is (conjugate to) τ. These classes satisfy analogues of Theorems 4.5.1 and 4.5.4, and can be used along with global duality (Theorem 1.7.3) to bound the appropriate Selmer group.

The main difference between the case of trivial χ (i.e., Theorem 2.2.2) and nontrivial χ is the way the powers of χ appear in the statement of Theorem 9.4.3. This is due to the “anticyclotomic” version of Theorem 4.5.4, which states that for 𝔯𝔮 ∈ R_{K′,M,τ}, we have

loc_{𝔮}^{s}(κ_{[K′,𝔯𝔮,M]}) = ϕ_{𝔮}^{fs}(κ_{[K′,𝔯,M]})

where

ϕ_{𝔮}^{fs} : H_{f}^{1}(K′_{𝔮}, W_{M}) ⟶ H_{s}^{1}(K′_{𝔮}, W_{M}).

As usual we write H_{f}^{1}(K′_{𝔮}, W_{M}) = ⊕_{v|𝔮}H_{f}^{1}(K′_{v}, W_{M}) and similarly for H_{s}^{1}(K′_{𝔮}, W_{M}), so that both are Gal(K′/K)-modules. However, ϕ_{𝔮}^{fs} is not Gal(K′/K)-equivariant; for 𝔮 ∈ R_{K′,M,τ}, one can show that

ϕ_{𝔮}^{fs}(H_{f}^{1}(K′_{𝔮}, W_{M})^{χ^{i}}) ⊂ H_{s}^{1}(K′_{𝔮}, W_{M})^{χ^{i−1}}.

Thus, taking 𝔯 = 1 and letting 𝔮 vary, we obtain a large collection of classes in H^{1}(K′, W_{M})^{χ^{i−1}}, ramified at only one prime of K not dividing p, whose ramification is expressed in terms of c_{K′}^{χ^{i}}, and these classes can be used to annihilate classes in S_{Σ_{p}}(K′, W^{∗})^{χ^{1−i}}. This is how Theorem 9.4.3 is proved.

Remark 9.4.5. To prove an analogue of Theorem 2.2.2 and bound the order of the various components of S_{Σ_{p}}(K′, W^{∗}), we would need to proceed by induction as in Chapter 5. Unfortunately this is not at all straightforward, because at each step of the induction we move to a different component. We will not attempt to formulate, much less prove, such a statement here.

In the case of Kolyvagin’s Euler system of Heegner points, the induction succeeds by using the fact that T^{∗} ≅ T. When d > 2 there is no obvious property to take the place of this self-duality. Also, if d = 2, then χ takes values ±1, so if L is an abelian extension of K′ it makes sense to ask if Gal(K′/K) acts on Gal(L/K′) via χ. When d > 2, this only makes sense when L/K′ is a p-extension. This is sufficient to discuss and work with Euler systems, but it raises the question of whether one should expect χ-anticyclotomic Euler systems with d > 2 to exist.

9.5. Additional Local Conditions

Inspired both by work on Stark’s conjectures (see for example [Gro1] or [Ru6]) and by the connection between Euler systems and L-functions (see Chapter 8), we now allow the imposition of additional local conditions on Euler system cohomology classes.

Suppose Σ and Σ′ are disjoint finite sets of places of K. If A is T, W, W_{M}, T^{∗}, W^{∗} or W_{M}^{∗}, define

S_{Σ′}^{Σ}(K, A) = ker (S^{Σ}(K, A) → ⊕_{v∈Σ′}H^{1}(K_{v}, A))

and similarly with K replaced by a finite extension. For example, S_{Σ′}^{Σ}(K, T) consists of all classes c ∈ H^{1}(K, T) satisfying the local conditions

• c_{v} ∈ H_{f}^{1}(K_{v}, T) if v ∉ Σ ∪ Σ′,

• c_{v} = 0 if v ∈ Σ′,

• no restriction for v ∈ Σ.

Definition 9.5.1. Suppose c is an Euler system for (T, K, N), and Σ is a finite set of primes of K not dividing p. We say c is trivial at Σ if c_{F} ∈ S_{Σ}^{Σ_{p}}(F, T) for every F.

If an Euler system is trivial at Σ, we can use it to bound the Selmer group S_{Σ_{p}}^{Σ}(K, W^{∗}). The proof will be the same as the original case where Σ is empty, once we have the following strengthening of Theorem 4.5.1.

Theorem 9.5.2. Let Σ be a finite set of primes of K not dividing p. If c is an Euler system for T, trivial at Σ, then the derivative classes κ_{[F,𝔯,M]} constructed in §4.4 satisfy

κ_{[F,𝔯,M]} ∈ S_{Σ}^{Σ_{p𝔯}}(F, W_{M}).

Proof. By Theorem 4.5.1, we only need to show that (κ_{[F,𝔯,M]})_{𝔮} = 0 if 𝔮 ∈ Σ. The proof is similar to that of Theorem 4.5.1 in §4.6. We use the notation of that proof.

Fix a lift d : X_{F (𝔯)} → 𝕎_{M}/W_{M} of c as in Proposition 4.4.8 and write d_{𝔮} for the image of d in Hom(X_{F (𝔯)}, 𝕎_{M}/Ind_{D}^{G_{K}} (W_{M})) in the diagram of Lemma 4.6.7. Then d_{𝔮} is a lift of c in the sense of Proposition 4.6.8, but so is the zero map, since (c_{F (𝔯)})_{𝔮} = 0. Therefore the uniqueness portion of Proposition 4.6.8 shows that d_{𝔮} ∈ image(Hom(X_{F (𝔯)}, 𝕎_{M}^{G_{F (𝔯)}})), and from this it follows without difficulty, as in the proof of Theorem 4.5.1, that (κ_{[F,𝔯,M]})_{𝔮} = 0.

The following analogue of Theorem 2.2.2 (using the same notation) is an example of the kind of bound that comes from using an Euler system which is trivial at Σ.

Theorem 9.5.3. Suppose that p > 2 and that T satisfies Hyp(K, T). Let Σ be a finite set of primes of K not dividing p. If c is an Euler system for T, trivial at Σ, then

ℓ_{O}(S_{Σ_{p}}^{Σ}(K, W^{∗})) ≤ ind_{O}(c) + 𝔫_{W} + 𝔫_{W}^{∗}

where

𝔫_{W} = ℓ_{O}(H^{1}(Ω/K, W) ∩ S_{Σ}^{Σ_{p}}(K, W)),
𝔫_{W}^{∗} = ℓ_{O}(H^{1}(Ω/K, W^{∗}) ∩ S_{Σ_{p}}(K, W^{∗})).

Proof. The proof is identical to that of Theorem 2.2.2, using Theorem 9.5.2 instead of Theorem 4.5.1.

Remarks 9.5.4. There are similar analogues of the other theorems of Chapter 2, bounding S_{Σ_{p}}^{Σ}(K, W^{∗}) and S_{Σ_{p}}^{Σ}(K_{∞}, W^{∗}).

By taking Σ to be large, we can ensure that the error term 𝔫_{W} in Theorem 9.5.3 is small.

In the spirit of Chapter 8, if we think of Euler systems as corresponding to p-adic L-functions, then an Euler system which is trivial at Σ corresponds to a p-adic L-function with modified Euler factors at primes in Σ. As in [Gro1] §1 (where our Σ is denoted T), these Euler factors can be used to remove denominators from the original p-adic L-function (see Remark 8.2.5 and Conjecture 8.2.6).

9.6. Varying the Euler Factors

It may happen that one has a collection of cohomology classes satisfying distribution relations different from the ones in Definition 2.1.1. Under certain conditions one can modify the given classes to obtain an Euler system.

Return again to the setting of §2.1. We fix a number field K and a p-adic representation T of G_{K}. Suppose K is an abelian extension of K and N is an ideal of K divisible by p and all primes where T is ramified. If K ⊂_{f} F ⊂_{f} F′ ⊂ K, let Σ(F′/F) denote the set of primes of K not dividing N which ramify in F′/K but not in F/K.

Lemma 9.6.1. Suppose {f_{𝔮} ∈ O[x] : 𝔮 ∤ N } and {g_{𝔮} ∈ O[x] : 𝔮 ∤ N } are two collections of polynomials such that f_{𝔮}(x) ≡ g_{𝔮}(x) (mod N(𝔮) − 1) for every 𝔮, and suppose {c̃_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K} is a collection of cohomology classes such that if K ⊂_{f} F ⊂_{f} F′ ⊂ K, then

Cor_{F′/F} (c̃_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}f_{𝔮}(Fr_{𝔮}^{−1}))c̃_{F} .

Then there is a collection of classes {c_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K} satisfying the following properties.

(i) If K ⊂_{f} F ⊂_{f} F′ ⊂ K, then

Cor_{F′/F} (c_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}g_{𝔮}(Fr_{𝔮}^{−1}))c_{F} .

(ii) If K ⊂_{f} F ⊂ K and F/K is unramified outside N, then

c_{F} = c̃_{F} .

(iii) Suppose K ⊂_{f} F ⊂ K and χ is a character of Gal(F/K) of conductor 𝔣. If every prime which ramifies in F/K divides N 𝔣, then

∑_{γ∈Gal(F/K)}χ(γ)γc_{F} = ∑_{γ∈Gal(F/K)}χ(γ)γc̃_{F} .

Proof. If K ⊂_{f} F ⊂ K let Σ(F) = Σ(F/K), and if S is a finite set of primes of K let F_{S} be the largest extension of K in F which is unramified outside S and N. If 𝔮 ∤ N let d_{𝔮} = g_{𝔮}(Fr_{𝔮}^{−1}) − f_{𝔮}(Fr_{𝔮}^{−1}). For every F define

c_{F} = ∑_{S⊂Σ(F)} \frac{∏_{𝔮∈Σ(F)−S} d_{𝔮}}{[F : F_{S}]} ( ∏_{𝔮∈S−Σ(F_{S})}f_{𝔮}(Fr_{𝔮}^{−1}))c̃_{F_{S}}.

(Let I_{𝔮}(F/K) denote the inertia group of 𝔮 in Gal(F/K). Then Gal(F/F_{S}) is generated by {I_{𝔮} : 𝔮 ∈ Σ(F) − S}, and |I_{𝔮}| divides (N(𝔮) − 1) in O, so [F : F_{S}] divides ∏_{𝔮∈Σ(F)−S}(N(𝔮)−1). Since d_{𝔮} ∈ (N(𝔮)−1)O[Gal(F/K)], the fractions above belong to O[Gal(F/K)].)

With this definition, (ii) is clear. Assertion (iii) (of which (ii) is a special case) also holds, because if every prime which ramifies in F/K divides N 𝔣, and if S is a proper subset of Σ(F), then χ does not factor through Gal(F_{S}/K) and so ∑_{γ∈Gal(F/K)} χ(γ)γc̃_{F_{S}} = 0.

For (i), observe that for every S, we have F′_{S} ∩ F = F_{S}. Thus, using the diagram

[figure: F_{S}; F′_{S}; F; F′_{S}F; F′]

we see that

Cor_{F′/F} (c̃_{F′_{S}}) = Cor_{F′_{S}F/F} Cor_{F′/F′_{S}F} (c̃_{F′_{S}}) = [F′ : F′_{S}F]Cor_{F′_{S}F/F} (c̃_{F′_{S}})

= \frac{[F′ : F]}{[F′_{S} : F_{S}]}Cor_{F′_{S}/F_{S}}(c̃_{F′_{S}}) = \frac{[F′ : F]}{[F′_{S} : F_{S}]}( ∏_{𝔮∈Σ(F′_{S}/F_{S})}f_{𝔮}(Fr_{𝔮}^{−1}))c̃_{F_{S}},

and so Cor_{F′/F} (c_{F′}) = ∑_{S⊂Σ(F′)} a_{S}c̃_{F_{S}} where

a_{S} = \frac{∏_{𝔮∈Σ(F′)−S} d_{𝔮}}{[F′ : F′_{S}]} ( ∏_{𝔮∈S−Σ(F′_{S})}f_{𝔮}(Fr_{𝔮}^{−1}))\frac{[F′ : F]}{[F′_{S} : F_{S}]}( ∏_{𝔮∈Σ(F′_{S}/F_{S})}f_{𝔮}(Fr_{𝔮}^{−1}))

= \frac{∏_{𝔮∈Σ(F′)−S} d_{𝔮}}{[F : F_{S}]} ∏_{𝔮∈S−Σ(F_{S})}f_{𝔮}(Fr_{𝔮}^{−1}).

Since F_{S} = F_{S∩Σ(F)}, we can group together those sets S which have the same intersection with Σ(F), and we get a new expression

Cor_{F′/F} (c_{F′}) = ∑_{S⊂Σ(F)} b_{S}c̃_{F_{S}}

where

b_{S} = ∑_{S′⊂Σ(F′/F)} \frac{∏_{𝔮∈Σ(F′)−S−S′} d_{𝔮}}{[F : F_{S}]} ∏_{𝔮∈S∪S′−Σ(F_{S})}f_{𝔮}(Fr_{𝔮}^{−1})

= \frac{∏_{𝔮∈Σ(F)−S} d_{𝔮}}{[F : F_{S}]} ∏_{𝔮∈S−Σ(F_{S})}f_{𝔮}(Fr_{𝔮}^{−1})

× ∑_{S′⊂Σ(F′/F)}(  ∏_{𝔮∈Σ(F′/F)−S′}d_{𝔮}) ∏_{𝔮∈S′} f_{𝔮}(Fr_{𝔮}^{−1}).

Since

∑_{S′⊂Σ(F′/F)}(  ∏_{𝔮∈Σ(F′/F)−S′}d_{𝔮}) ∏_{𝔮∈S′} f_{𝔮}(Fr_{𝔮}^{−1}) = ∏_{𝔮∈Σ(F′/F)}(d_{𝔮} + f_{𝔮}(Fr_{𝔮}^{−1}))

= ∏_{𝔮∈Σ(F′/F)}g_{𝔮}(Fr_{𝔮}^{−1}),

we conclude that Cor_{F′/F} (c_{F′}) = ∏_{𝔮∈Σ(F′/F)} g_{𝔮}(Fr_{𝔮}^{−1})c_{F} as desired.

Example 9.6.2. Suppose that K = Q, that f_{q}(x) = 1 − x, and that g_{q}(x) = 1−q^{−1}x. Then f_{q}(x) ≡ g_{q}(x) (mod (q −1)Z_{p}) for every q ≠ p. By applying Lemma 9.6.1 with these data to the collection {c̃′_{F} ∈ H^{1}(F, Z_{p})} constructed in Definition 3.4.2, we obtain an Euler system for Z_{p}.

Lemma 9.6.3. Suppose {f_{𝔮}(x) ∈ O[x, x^{−1}] : 𝔮 ∤ N } is a collection of polynomials, {u_{𝔮} ∈ O^{×} : 𝔮 ∤ N } is a collection of units, d ∈ Z, and

{c̃_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K}

is a collection of cohomology classes such that if K ⊂_{f} F ⊂_{f} F′ ⊂ K then

Cor_{F′/F} (c̃_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}f_{𝔮}(Fr_{𝔮}))c̃_{F} .

For each 𝔮 define g_{𝔮}(x) = u_{𝔮}x^{d}f_{𝔮}(x^{−1}) ∈ O[x, x^{−1}]. Then there is a collection of classes

{c_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K}

such that (i) for all F and F′ as above,

Cor_{F′/F} (c_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}g_{𝔮}(Fr_{𝔮}^{−1}))c_{F} ,

(ii) for every finite extension F of K unramified outside N,

c_{F} = c̃_{F} .

Proof. For every F define

c_{F} = ( ∏_{𝔮∈Σ(F/K)}u_{𝔮}Fr_{𝔮}^{−d})c̃_{F}

where we fix some Frobenius Fr_{𝔮} ∈ Gal(K^{ab}/K) (previously we always had Fr_{𝔮} acting through an extension unramified at 𝔮). Then it is easy to check that this collection has the desired properties.

Let P (Fr_{𝔮}^{−1}|T; x) = det(1 − Fr_{𝔮}^{−1}x|T).

Corollary 9.6.4. Suppose {c̃_{F} ∈ H^{1}(F, T) : K ⊂_{f} F ⊂ K} is a collection of cohomology classes such that if K ⊂_{f} F ⊂_{f} F′ ⊂ K, then

Cor_{F′/F} (c̃_{F′}) = ( ∏_{𝔮∈Σ(F′/F)}P (Fr_{𝔮}^{−1}|T; Fr_{𝔮}))c̃_{F} .

Then there is an Euler system {c_{F} } for (T, K, N) such that for every finite extension F of K unramified outside N,

c_{F} = c̃_{F} .

Proof. This will follow directly from the previous two lemmas. For every 𝔮 we have

P (Fr_{𝔮}^{−1}|T; x^{−1}) = det(1 − Fr_{𝔮}^{−1}x^{−1}|T) = det(1 − N(𝔮)^{−1}Fr_{𝔮}x^{−1}|T^{∗})
= (−N(𝔮))^{−d} det(Fr_{𝔮}|T^{∗})x^{−d} det(1 − N(𝔮)Fr_{𝔮}^{−1}x|T^{∗})

where d = rank_{O}T. Thus if we first apply Lemma 9.6.3 with

f_{𝔮} = P (Fr_{𝔮}^{−1}|T; x), u_{𝔮} = (−N(𝔮))^{d} det(Fr_{𝔮}|T^{∗})^{−1},

and then apply Lemma 9.6.1 with

f_{𝔮} = P (Fr_{𝔮}^{−1}|T^{∗}; N(𝔮)x), g_{𝔮} = P (Fr_{𝔮}^{−1}|T^{∗}; x),

we obtain the desired Euler system.

Remark 9.6.5. If one has a collection of cohomology classes satisfying the “wrong” distribution relation, one can either modify the classes as we did above, to get an Euler system, or else one can keep the given cohomology classes and modify the proofs in Chapters 4 through 7 instead.

APPENDIX A Linear Algebra

Suppose for this appendix that O is a discrete valuation ring. Let ℓ_{O}(B) denote the length of an O-module B.

A.1. Herbrand Quotients

Suppose α, β ∈ O[x] are nonzero.

Definition A.1.1. If S is an O[x]-module and αβS = 0, then

αS ⊂ S^{β=0}, βS ⊂ S^{α=0},

and we define the (additive) Herbrand quotient

h(S) = ℓ_{O}(S^{β=0}/αS) − ℓ_{O}(S^{α=0}/βS)

if both lengths are finite.

Example A.1.2. If S = O[x]/αβO[x] then

S^{β=0} = αS = αO[x]/αβO[x], S^{α=0} = βS = βO[x]/αβO[x],

so h(S) = 0.

Proposition A.1.3. (i) If S is an O[x]/αβO[x]-module and ℓ_{O}(S) is

finite, then h(S) = 0. (ii) If 0 → S′ → S → S′′ → 0 is an exact sequence of O[x]/αβO[x]-

modules and two of the three Herbrand quotients exist, then the third exists and

h(S) = h(S′) + h(S′′).

Proof. This is a standard fact about Herbrand quotients, see for example [Se3] §VIII.4. If α = (x^{n} − 1)/(x − 1) and β = x − 1, and if G is a cyclic group of order n with a generator which acts on S as multiplication by x, then

Ĥ^{0}(G, S) = S^{β=0}/αS and Ĥ^{1}(G, S) = S^{α=0}/βS.

For completeness we sketch a proof in our more general setting.

Assertion (i) follows from the exact sequences

0 ⟶ S^{α=0} ⟶ S ⟶^{α} αS ⟶ 0,

0 ⟶ S^{β=0}/αS ⟶ S/αS ⟶^{β} S^{α=0} ⟶ S^{α=0}/βS ⟶ 0.

For (ii), multiplication by β induces a snake lemma exact sequence

0 → S′^{β=0} → S^{β=0} → S′′^{β=0} ⟶^{ψ} S′/βS′ → S/βS → S′′/βS′′ → 0.

This gives rise to a commutative diagram

0 ⟶ coker(ψ) ⟶ S/βS ⟶ S′′/βS′′ ⟶ 0
α↓  ^{α}↓  ^{α}↓
0 ⟶ S′^{β=0} ⟶ S^{β=0} ⟶ ker(ψ) ⟶ 0.

Applying the snake lemma again gives an exact sequence

0 ⟶ A ⟶ S^{α=0}/βS ⟶ S′′^{α=0}/βS′′

⟶ S′^{β=0}/αS′ ⟶ S^{β=0}/αS ⟶ B ⟶ 0

where A and B are defined by the exact sequence

0 ⟶ B ⟶ S′′^{β=0}/αS′′ ⟶^{ψ} S′^{α=0}/βS′ ⟶ A ⟶ 0.

Assertion (ii) follows from these two exact sequences.

Lemma A.1.4. Suppose αβ = ∏_{i=1}^{k} ρ_{i} with ρ_{i} ∈ O[x], and suppose further that ρ_{i} is relatively prime to β if 2 ≤ i ≤ k. Then

h(⊕_{i}O[x]/ρ_{i}O[x]) = 0.

Proof. For 1 ≤ i ≤ k let S_{i} = O[x]/ρ_{i}O[x]. If ρ_{i} is relatively prime to β (and therefore divides α), we see that S_{i}^{β=0} = αS_{i} = 0 and S_{i}^{α=0} = S_{i}. Thus for i ≥ 2

h(S_{i}) = −ℓ_{O}(S_{i}/βS_{i}) = −ℓ_{O}(O[x]/(β, ρ_{i}))

which is finite. By Proposition A.1.3 and Example A.1.2 we conclude that the Herbrand quotient h(S_{1}) exists as well, and that

h(⊕_{i}O[x]/ρ_{i}O[x]) = ∑_{i} h(S_{i}) = h(O[x]/αβ) = 0.

A.2. p-ADIC REPRESENTATIONS 191

A.2. p-adic Representations

Let T be a free O-module of finite rank, and let σ be an O-linear automorphism of T. Let p(x) = det(1 − σ^{−1}x|T) ∈ O[x], and suppose further that p(1) = 0 (i.e., det(1 − σ|T) = 0). Then there is a unique polynomial q(x) ∈ O[x] such that

p(x) = (1 − x)q(x).

The Cayley-Hamilton theorem shows that p(σ) = 0, so T is an O[x]/p(x)- module, with x acting via σ. Thus we are in the setting of §A.1, with α = q(x) and β = x − 1.

Let Φ denote the field of fractions of O and let V = T ⊗ Φ.

Lemma A.2.1. Suppose that T ≅ ⊕_{i}O[x]/f_{i}(x)O[x] with f_{i}(x) ∈ O[x], and that dim_{Φ}(V/(σ − 1)V) = 1. Then the Herbrand quotient h(T) = 0.

Proof. We have p(x) = ∏_{i} f_{i}(x). Since dim_{Φ}(V/(σ − 1)V) = 1, exactly one of the f_{i}(x) (say, f_{1}) is divisible by x − 1. Thus we can apply Lemma A.1.4 to conclude that h(T) = 0.

Lemma A.2.2. There is an O[σ]-submodule S of T such that (i) ℓ_{O}(T/S) is finite, (ii) S ≅ ⊕_{i}O[x]/f_{i}(x)O[x] with f_{1}(x), …, f_{k}(x) ∈ O[x].

Proof. Since the polynomial ring Φ[x] is a principal ideal domain, V is a direct sum of cyclic Φ[σ]-modules, and the lemma follows easily.

Proposition A.2.3. If dim_{Φ}(V/(σ − 1)V) = 1 then h(T) = 0.

Proof. This is immediate from Proposition A.1.3 and Lemmas A.2.1 and A.2.2.

Lemma A.2.4. Suppose dim_{Φ}(V/(σ − 1)V) = 1. Then (i) V^{q(σ)=0} = (σ − 1)V and V^{σ=1} = q(σ)V, (ii) the map V/(σ − 1)V ⟶^{q(σ)} V^{σ=1} is an isomorphism.

Proof. Viewing V as a Φ[x]-module with x acting via σ, there is an isomorphism

V ≅ ⨁_{i} Φ[x]/f_{i}(x)^{e_{i}}Φ[x]

where the f_{i}(x) ∈ Φ[x] are irreducible, f_{i}(0) = 1, and

∏_{i} f_{i}(x)^{e_{i}} = p(x) = (1 − x)q(x).

Since dim_{Φ}(V/(σ − 1)V) = 1, precisely one of the f_{i} is 1 − x. Both assertions follow easily from this.

Proposition A.2.5. Suppose dim_{Φ}(V/(σ − 1)V) = 1, and let W = V/T. Then the lengths of the following O-modules are finite and equal:

(i) T^{σ=1}/q(σ)T  (iv) W^{σ=1}/q(σ)W
(ii) T^{q(σ)=0}/(σ − 1)T  (v) W^{q(σ)=0}/(σ − 1)W
(iii) (T/(σ − 1)T)_{tors}  (vi) W^{σ=1}/W_{div}^{σ=1}

where W_{div}^{σ=1} denotes the maximal divisible O-submodule of W^{σ=1}.

Proof. Proposition A.2.3 says that h(T) = 0, so (i) and (ii) have the same (finite) length. Similarly, by Lemma A.2.4(i) we have h(V) = 0, so Proposition A.1.3(ii) shows that h(W) = 0 as well. Thus (iv) and (v) have the same length.

Lemma A.2.4(i) shows that V^{q(σ)=0}/(σ − 1)V = 0, and it follows that T^{q(σ)=0}/(σ − 1)T is a torsion O-module. Since T/T^{q(σ)=0} is torsion-free we have

(T/(σ − 1)T)_{tors} = T^{q(σ)=0}/(σ − 1)T

and so (ii) and (iii) are isomorphic. It follows similarly from Lemma A.2.4(i) that q(σ)W = W_{div}^{σ=1} and (iv) is isomorphic to (vi).

It remains to compare (i) with (v). Consider the diagram

\xymatrix{ & T/(σ − 1)T \ar[r] \ar[d]_{q(σ)} & V/(σ − 1)V \ar[r] \ar[d]_{q(σ)} & W/(σ − 1)W \ar[r] \ar[d]_{q(σ)} & 0 \\ 0 \ar[r] & T^{σ=1} \ar[r] & V^{σ=1} \ar[r] & W^{σ=1}.}

By Lemma A.2.4(ii), the center vertical map is an isomorphism, so the snake lemma gives (i) ≅ (v).

For the next two corollaries let W = V/T, and if M ∈ O let W_{M} denote the kernel of multiplication by M on W.

Corollary A.2.6. Suppose dim_{Φ}(V/(σ − 1)V) = 1, and let b denote the common length of the modules in Proposition A.2.5. Then the kernel and cokernel of the map

W_{M}/(σ − 1)W_{M} ⟶^{q(σ)} W_{M}^{σ=1}

both have length at most 2b.

Proof. Consider the diagram

\begin{matrix}W/(σ − 1)W & ⟶_{ϕ}^{q(σ)} & W^{σ=1} \\ ↑ & & ↑ \\ W_{M}/(σ − 1)W_{M} & ⟶_{ϕ_{M}}^{q(σ)} & W_{M}^{σ=1}. \end{matrix}  (A.1)

The kernel and cokernel of ϕ are (v) and (iv) of Proposition A.2.5, respectively, and therefore both have length b. Multiplying the exact sequence

0 ⟶ W_{M} ⟶ W ⟶^{M} W ⟶ 0

by σ − 1 yields a snake lemma exact sequence

W^{σ=1} ⟶^{M} W^{σ=1} ⟶ W_{M}/(σ − 1)W_{M} ⟶ W/(σ − 1)W.

Therefore the kernel of the left vertical map of (A.1) is W^{σ=1}/M(W^{σ=1}), which is a quotient of the module (vi) of Proposition A.2.5, and hence has length at most b. Thus we conclude that ℓ_{O}(ker(ϕ_{M})) ≤ 2b. The exact sequence

0 ⟶ W_{M}^{σ=1} ⟶ W_{M} ⟶^{σ−1} W_{M} ⟶ W_{M}/(σ − 1)W_{M} ⟶ 0

shows that ℓ_{O}(W_{M}/(σ − 1)W_{M}) = ℓ_{O}(W_{M}^{σ=1}), so

ℓ_{O}(coker(ϕ_{M})) = ℓ_{O}(ker(ϕ_{M})) ≤ 2b

as well.

Corollary A.2.7. Suppose

(a) τ is an O-linear automorphism of W_{M} such that W_{M}/(τ − 1)W_{M} is free of rank one over O/MO,

(b) Q(x) ∈ (O/MO)[x] is such that (1 − x)Q(x) = det(1 − τ^{−1}x|W_{M}). Then the map

W_{M}/(τ − 1)W_{M} ⟶^{Q(τ)} W_{M}^{τ=1}

is an isomorphism.

Proof. We will show that there is an automorphism σ of T such that (i) σ induces τ on W_{M}, (ii) T/(σ − 1)T is free of rank one over O.

Once we have done this, we will apply Proposition A.2.5 and Corollary A.2.6 with this σ. Condition (ii) shows that the module (T/(σ − 1)T)_{tors} of Proposition A.2.5(iii) is zero, so the integer b of Corollary A.2.6 is zero. It follows from condition (i) (together with the fact that 1 − x is not a zerodivisor in (O/MO)[x]) that q(σ) reduces to Q(τ) on W_{M}, so this corollary will follow from Corollary A.2.6.

It remains to find a σ satisfying (i) and (ii). Since W_{M}/(τ − 1)W_{M} is free of rank one over O/MO, it follows that W_{M}^{τ=1} is free of rank one over O/MO as well. Therefore we can choose a basis {w_{1}, …, w_{d}} of W_{M} such that τw_{1} = w_{1}, where d = rank_{O}T.

For each i fix t_{i} ∈ T which reduces to w_{i}. By Nakayama’s Lemma {t_{1}, t_{2}, …, t_{d}} is an O-basis of T, and we define σ on this basis by lifting the action of τ on the w_{i}, and requiring that σ(t_{1}) = t_{1}. Then σ is an automorphism because det(σ) ≡ det(τ) (mod M), and clearly (i) is satisfied. Further rank_{O}T/(σ − 1)T ≥ 1, and since (T/(σ − 1)T) ⊗ (O/MO) = W_{M}/(τ − 1)W_{M} is a cyclic O-module, we can apply Nakayama’s Lemma again to deduce (ii).

APPENDIX B Continuous Cohomology and Inverse Limits

Notation. If G and T are topological groups then Hom(G, T) will always denote the group of continuous homomorphisms from G to T. We denote by Maps(G, T) the topological group of continuous functions (not necessarily homomorphisms) from G to T, with the compact-open topology.

B.1. Preliminaries

Since we will use it repeatedly, we record without proof the following well-known result.

Proposition B.1.1. (i) Suppose {A_{n}}, {B_{n}}, and {C_{n}} are inverse systems of topological groups and there are exact sequences

0 ⟶ A_{n} ⟶ B_{n} ⟶ C_{n} ⟶ 0

for every n, compatible with the maps of the inverse systems. If the A_{n} are compact, then the induced sequence

0 ⟶ \underleftarrow{lim}_{n} A_{n} ⟶ \underleftarrow{lim}_{n} B_{n} ⟶ \underleftarrow{lim}_{n} C_{n} ⟶ 0

is exact. (ii) If O is a discrete valuation ring with fraction field Φ and {A_{n}} is

an inverse system of finite O-modules, then the canonical map

\underrightarrow{lim}_{n} Hom(A_{n}, Φ/O) ⟶ Hom(\underleftarrow{lim}_{n} A_{n}, Φ/O)

is an isomorphism.

B.2. Continuous Cohomology

For this section suppose G is a profinite group and T is a topological G-module, i.e., an abelian topological group with a continuous action of G.

Definition B.2.1. Following Tate [T4], we define the continuous cohomology groups H^{i}(G, T) as follows. Let C^{i}(G, T) = Maps(G^{i}, T). For every i ≥ 0 there is a coboundary map d_{i} : C^{i}(G, T) → C^{i+1}(G, T) defined in the usual way (see for example [Se3] §VII.3), and we set

H^{i}(G, T) = ker(d_{i})/image(d_{i−1}).

If 0 → T′ → T → T′′ → 0 is an exact sequence and if there is a continuous section (again a map of sets, not necessarily a homomorphism) from T′′ → T, then

0 ⟶ C^{i}(G, T′) ⟶ C^{i}(G, T) ⟶ C^{i}(G, T′′) ⟶ 0

is exact for every i and there is a long exact sequence

⋯ ⟶ H^{i}(G, T′) ⟶ H^{i}(G, T) ⟶ H^{i}(G, T′′) ⟶ H^{i+1}(G, T′) ⟶ · · · .

Remark B.2.2. Note that if T′′ is topologically discrete, as is assumed in the more “classical” formulations of profinite group cohomology, then there is always a continuous section T′′ → T. This is the case whenever T′ is open in T. Also, if T is a finitely generated Z_{p}-module or a finite-dimensional Q_{p}-vector space with the usual topology, then there is a continuous section. These are the only situations in which we will use these cohomology groups.

For the situations of interest to us, the following propositions will allow us to work with the cohomology groups H^{i}(G, T) exactly as if T were discrete. The first two are due to Tate [T4]; see also Jannsen [J].

Proposition B.2.3 ([T4] Corollary 2.2, [J] §2). Suppose i > 0 and T = \underleftarrow{lim} T_{n} where each T_{n} is a finite G-module. If H^{i−1}(G, T_{n}) is finite for every n then

H^{i}(G, T) = \underleftarrow{lim}_{n} H^{i}(G, T_{n}).

Proposition B.2.4 ([T4] Proposition 2.3). If T is a finitely generated Z_{p}-module and i ≥ 0, then H^{i}(G, T) has no divisible elements and the natural map

H^{i}(G, T) ⊗ Q_{p} ⟶ H^{i}(G, T ⊗ Q_{p})

is an isomorphism.

Proposition B.2.5. Suppose H is a closed, normal subgroup of G. (i) There is an inflation-restriction exact sequence

0 ⟶ H^{1}(G/H, T^{H}) ⟶ H^{1}(G, T) ⟶ H^{1}(H, T).

(ii) Suppose further that p is a prime, and for every G-module (resp. Hmodule) S of finite, p-power order, H^{1}(G, S) and H^{2}(G, S) (resp. H^{1}(H, S)) is finite. If T is discrete, or if T is a finitely generated Z_{p}-module, or if T is a finite-dimensional Q_{p}-vector space, then there is a Hochschild-Serre exact sequence extending the sequence in (i)

0 ⟶ H^{1}(G/H, T^{H}) ⟶ H^{1}(G, T) ⟶ H^{1}(H, T)^{G/H}

⟶ H^{2}(G/H, T^{H}) ⟶ H^{2}(G, T).

Proof. If T is discrete both assertions are standard. The proof of (i) in general is identical to the proof in this classical case.

Suppose T is a finitely generated Z_{p}-module. Then for every n ≥ 0, the quotient T/p^{n}T is discrete, so there is a Hochschild-Serre exact sequence for T/p^{n}T. Our hypotheses ensure that all the terms in this sequence are finite, and so taking the inverse limit over n and applying Proposition B.2.3 gives the exact sequence of (ii) for T.

If T is a finite-dimensional Q_{p}-vector space, choose a G-stable Z_{p}- lattice T_{0} ⊂ T. Then by the previous case we have a Hochschild-Serre exact sequence for T_{0}, and tensoring with Q_{p} and using Proposition B.2.4 gives the desired exact sequence for T.

Remark B.2.6. To apply Proposition B.2.5(ii) we need to know when a group G has the property that H^{i}(G, S) is finite for every i and every Gmodule S of finite p-power order. For example, this is true whenever the pro-p-part of G is (topologically) finitely generated.

We also have the following well-known result. In the most important case i = 1 it follows easily from class field theory (see for example [Se2] Propositions II.14 and III.8). We say that a Z_{p}-module is co-finitely generated if its Pontryagin dual is finitely generated.

Proposition B.2.7. Suppose that one of the following conditions holds.

(i) K is a global field, K_{S} is a (possibly infinite) Galois extension of K unramified outside a finite set of places of K, and G = Gal(K_{S}/K),

(ii) K is a local field and G = G_{K}, or (iii) K is a local field of residue characteristic different from p and G is

the inertia group in G_{K}. If T is a G-module which is finite (resp. finitely generated over Z_{p}, resp. co-finitely generated over Z_{p}, resp. finite-dimensional over Q_{p}) and i ≥ 0, then H^{i}(G, T) is finite (resp. finitely generated over Z_{p}, resp. co-finitely generated over Z_{p}, resp. finite-dimensional over Q_{p}).

Let Ẑ denote the profinite completion of Z.

Lemma B.2.8. Suppose G is a topologically cyclic profinite group with infinite pro-p-part (for example, G ≅ Ẑ or G ≅ Z_{p}), and γ is a topological generator of G. Suppose T is a Z_{p}[G_{K}]-module which is either a finitely generated Z_{p}-module, or a finite-dimensional Q_{p}-vector space, or a discrete torsion Z_{p}-module. Then

H^{1}(G, T) ≅ T/(γ − 1)T

with an isomorphism induced by evaluating cocycles at γ.

Proof. It is easy to see that evaluating cocycles at γ induces a well-defined, injective map H^{1}(G, T) → T/(γ − 1)T. It remains only to show that this map is surjective.

Using direct limits, inverse limits (Proposition B.2.3), and/or tensoring with Q_{p} (Proposition B.2.4), we can reduce this lemma to the case where T is finite. When T is finite, the lemma is well-known, see for example [Se3] §XIII.1.

B.3. Inverse Limits

For this section suppose that K is a field, p is a rational prime, and T is a Z_{p}[G_{K}]-module which is finitely generated over Z_{p}.

As usual, we write K ⊂_{f} F to indicate that F is a finite extension of K. If K_{∞} is an infinite extension of K and {C_{F} : K ⊂_{f} F ⊂ K_{∞}} is an inverse system of abelian groups, we will write \{c_{F} \} for a typical element of \underleftarrow{lim} C_{F} with c_{F} ∈ C_{F}.

Lemma B.3.1. If K ⊂_{f} F_{1} ⊂_{f} F_{2} ⊂_{f} ⋯ and ∪_{n=1}^{∞}F_{n} = K_{∞} then

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T) = \underleftarrow{lim}_{n} H^{1}(F_{n}, T/p^{n}T).

Proof. By Proposition B.2.3 we have

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T) = \underleftarrow{lim}_{n} H^{1}(F_{n}, T) = \underleftarrow{lim}_{n} \underleftarrow{lim}_{m} H^{1}(F_{n}, T/p^{m}T)

= \underleftarrow{lim}_{n} H^{1}(F_{n}, T/p^{n}T).

Lemma B.3.2. Suppose K_{∞} is an infinite p-extension of K. Then

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}T^{G_{F}} = 0

where the maps in the inverse system are given by the norm maps

N_{F′/F} : T^{G_{F′}} ⟶ T^{G_{F}}

for K ⊂_{f} F ⊂_{f} F′ ⊂ K_{∞}.

Proof. Define a submodule T_{0} of T by

T_{0} = ∪_{K ⊂_{f} F ⊂K_{∞}}T^{G_{F}} .

Then T_{0} is finitely generated over Z_{p} since T is, so we must have T_{0} = T^{G_{F_{0}}} for some finite extension F_{0} of K in K_{∞}. Therefore

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}T^{G_{F}} = \underleftarrow{lim}_{F_{0} ⊂_{f} F ⊂K∞}T^{G_{F}} = \underleftarrow{lim}_{F_{0} ⊂_{f} F ⊂K∞}T_{0}

where the norm maps N_{F′/F} in the right-hand inverse system are multiplication by [F′ : F]. Since T_{0} is finitely generated over Z_{p}, and for every F the index [F′ : F] is divisible by arbitrarily large powers of p as F′ varies, this inverse limit is zero.

If K is a finite extension of Q_{ℓ} for some ℓ, let H_{ur}^{1}(K, T) denote the subgroup of unramified classes in H^{1}(K, T), as defined in Definition 1.3.1.

Lemma B.3.3. Suppose T is a discrete Z_{p}-module, K is a finite extension of Q_{ℓ} for some ℓ ≠ p, and K_{∞} is an extension of K containing the unique Z_{p}-extension of K. Then

\underrightarrow{lim}_{K ⊂_{f} F ⊂K∞} H_{ur}^{1}(F, T) = 0.

Proof. Without loss of generality we may assume that K_{∞} is the Z_{p}- extension of K, and then the general case will follow immediately. In particular K_{∞}/K is unramified.

If K ⊂_{f} F ⊂ K_{∞} then H_{ur}^{1}(F, T) = H^{1}(K^{ur}/F, T^{I}) (Lemma 1.3.2(i)) where I ⊂ G_{K_{∞}} is the inertia group in G_{K}. Thus (since T is discrete)

\underrightarrow{lim}_{K ⊂_{f} F ⊂K∞} H_{ur}^{1}(F, T) = H^{1}(K^{ur}/K_{∞}, T^{I}).

But the pro-p-part of Gal(K^{ur}/K_{∞}) is trivial, so this is zero.

Proposition B.3.4. Suppose K is a finite extension of Q_{ℓ} for some ℓ ≠ p, and K_{∞} is the unique Z_{p}-extension of K. If \{c_{F} \} ∈ \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T) then

c_{F} ∈ H_{ur}^{1}(F, T) for every F.

Proof. Let I ⊂ G_{K} denote the inertia group. Since ℓ ≠ p, the extension K_{∞}/K is unramified, so I is also the inertia group in G_{F} for every F ⊂ K_{∞}. Thus for K ⊂_{f} F ⊂ K_{∞} we have an exact sequence

0 ⟶ H_{ur}^{1}(F, T) ⟶ H^{1}(F, T) ⟶ H^{1}(I, T)^{G_{F}} .

Since ℓ ≠ p, Proposition B.2.7(iii) shows that H^{1}(I, T) is finitely generated over Z_{p}. Therefore taking inverse limits with respect to F and applying Lemma B.3.2 to the G_{K}-module H^{1}(I, T) shows that

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H_{ur}^{1}(F, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T)

which proves the proposition.

For the next two corollaries, suppose that K is a number field and K_{∞} is an abelian extension of K satisfying

Gal(K_{∞}/K) ≅ Z_{p}^{d} for some d ≥ 1.

Corollary B.3.5. Suppose

\{c_{F} \} ∈ \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T).

If K ⊂_{f} F ⊂ K_{∞}, and v is a prime of F not dividing p whose decomposition group in Gal(K_{∞}/K) is infinite, then (c_{F})_{v} ∈ H_{ur}^{1}(F_{v}, T).

Proof. Let w be a prime of K_{∞} above v. Since the decomposition group of v in Gal(K_{∞}/K) is infinite, if K ⊂_{f} F ⊂ K_{∞} we can find F ⊂_{f} F′ ⊂ F_{∞} ⊂ K_{∞} such that Gal(F_{∞}/F′) ≅ Z_{p} and w is undecomposed in F_{∞}/F′. Thus Proposition B.3.4 applied to the classes {(c_{L})_{w} : F′ ⊂_{f} L ⊂ F_{∞}} shows that (c_{F′})_{w} ∈ H_{ur}^{1}(F′_{w}, T). Since this holds for all w above v, and Cor_{F′/F} (c_{F′}) = c_{F}, we deduce that (c_{F})_{v} ∈ H_{ur}^{1}(F_{v}, T).

Corollary B.3.6. If S is a set of places of K containing all primes where T is ramified, all primes dividing p, all primes whose decomposition group in Gal(K_{∞}/K) is finite, and all infinite places, then

\underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T) = \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(K_{S}/F, T)

where K_{S} is the maximal extension of K unramified outside S.

Proof. Suppose that

\{c_{F} \} ∈ \underleftarrow{lim}_{K ⊂_{f} F ⊂K∞}H^{1}(F, T).

Fix a field F such that K ⊂_{f} F ⊂ K_{∞}. Let I ⊂ G_{K} be an inertia group of a prime 𝔮 of K not in S. Since F/K is unramified at 𝔮, we see that I is also an inertia group of a prime Q of F above 𝔮, so by Corollary B.3.5 the restriction of c_{F} is zero in H^{1}(I, T) = Hom(I, T). It follows that every cocycle representing c_{F} factors through Gal(K_{S}/F), which proves the corollary.

B.4. Induced Modules

Again we suppose that G is a profinite group, and now H is a closed subgroup of G and T is a discrete H-module (not necessarily a G-module).

Definition B.4.1. Define the induced module Ind_{H}(T) = Ind_{H}^{G}(T) by

Ind_{H}(T) = {f ∈ Maps(G, T) : f(ηγ) = ηf(γ) for every γ ∈ G, η ∈ H}.

We let G act on Ind_{H}(T) by

(gf)(γ) = f(γg) for g, γ ∈ G.

Since T is discrete, Ind_{H}(T) a discrete G-module.

If H = {1}, then Ind_{H}(T) is simply Maps(G, T). If H′ is a closed subgroup of H then there is a natural inclusion Ind_{H}(T) ⊂ Ind_{H′}(T). If T is a G-module then evaluation at 1 induces an isomorphism Ind_{G}(T) ⟶^{∼} T, and so there is a natural (continuous) inclusion T ↪ Ind_{H}(T), which sends t ∈ T to the map γ ↦ γt.

Proposition B.4.2. Suppose T, G, and H are as above, and Γ is an open subgroup of G. For every i ≥ 0 there is an isomorphism

H^{i}(Γ, Ind_{H}(T)) ≅ ⨁_{g∈H\G/Γ} H^{i}(gΓg^{−1} ∩ H, T).

Proof. First suppose i = 0. Fix a set S ⊂ G of double coset representatives for H\G/Γ. If f ∈ Ind_{H}(T)^{Γ} then for every s ∈ S,

f(hsγ) = h(f(s)) for every h ∈ H and γ ∈ Γ.  (B.1)

In particular if h ∈ sΓs^{−1} ∩ H, then h(f(s)) = f(s), so f(s) ∈ T^{(sΓs^{−1}∩H)}. Conversely, if for every s ∈ S we have an element f(s) ∈ T^{sΓs^{−1}∩H}, then we can use (B.1) to define an element f ∈ Ind_{H}(T)^{Γ}. This proves the proposition when i = 0.

Now suppose i ≥ 1. The functor T ⇝ Ind_{H}(T) is exact on the category of discrete H-modules, so the proposition for T with i ≥ 1 follows from the case i = 0 and the Leray spectral sequence comparing the functors

A ⇝ ⨁_{g∈H\G/Γ}A^{gΓg^{−1}∩H}, A ⇝ Ind_{H}(A), B ⇝ B^{Γ}

(see for example [Sh] pp. 50–51).

Remark B.4.3. When Γ = G, Proposition B.4.2 is Shapiro’s Lemma (see for example the exercise on p. 125, §VII.5 of [Se3]).

Corollary B.4.4. With T, G, and H as above, for every open subgroup Γ of G there is an exact sequence

0 ⟶ Ind_{H}(T)^{Γ} ⟶ Ind_{{1}}(T)^{Γ}

⟶ (Ind_{{1}}(T)/Ind_{H}(T))^{Γ} ⟶ H^{1}(Γ, Ind_{H}(T)) ⟶ 0

Proof. By Proposition B.4.2 with H = {1}, we have H^{1}(Γ, Ind_{{1}}(T)) = 0. Thus the long exact Γ-cohomology sequence of the canonical exact sequence

0 ⟶ Ind_{H}(T) ⟶ Ind_{{1}}(T) ⟶ Ind_{{1}}(T)/Ind_{H}(T) ⟶ 0

gives the exact sequence of the corollary.

Proposition B.4.5. Suppose K is a field, F is a finite extension of K, and T is a discrete G_{K}-module. Let 𝕋 = Ind_{{1}}^{G_{K}}(T). Then there is a commutative diagram with exact rows

\xymatrix{0 \ar[r] & T^{G_{K}} \ar[r] \ar@{^{(}->}[d] & 𝕋^{G_{K}} \ar[r] \ar@{^{(}->}[d] & (𝕋/T)^{G_{K}} \ar[r] \ar@{^{(}->}[d] & H^{1}(K, T) \ar[r] \ar[d]^{_{Res_{F}}} & 0 \\ 0 \ar[r] & T^{G_{F}} \ar[r] \ar[d]^{N_{F/K}} & 𝕋^{G_{F}} \ar[r] \ar[d]^{N_{F/K}} & (𝕋/T)^{G_{F}} \ar[r] \ar[d]^{N_{F/K}} & H^{1}(F, T) \ar[r] \ar[d]^{_{Cor_{F/K}}} & 0 \\ 0 \ar[r] & T^{G_{K}} \ar[r] & 𝕋^{G_{K}} \ar[r] & (𝕋/T)^{G_{K}} \ar[r] & H^{1}(K, T) \ar[r] & 0.}

Proof. The horizontal sequences are the exact sequences of Corollary B.4.4 applied with H = G_{K} and Γ = G_{K} or G_{F}. The commutativity of the lower right square is essentially the definition of the corestriction map, and the rest of the commutativity is clear.

B.5. Semilocal Galois Cohomology

Suppose for this section that K is a number field, that 𝔮 is a prime of K, that F is a finite extension of K, and that S is the set of primes of F above 𝔮. For every prime Q ∈ S fix a prime 𝔔 of K̄ above Q and let I_{Q} ⊂ D_{Q} ⊂ G_{K} denote the inertia group and decomposition group of 𝔔. Fix a prime Q_{0} ∈ S and write D = D_{Q_{0}} and I = I_{Q_{0}}. For every Q ∈ S fix

g_{Q} ∈ G_{K} such that 𝔔 = g−1_{Q}𝔔_{0}. Then D_{Q} = g−1_{Q}Dg_{Q}.

Let T be a discrete G_{K}-module, and let T′ ⊂ T be a subset which is a D-submodule, i.e., D sends T′ into itself. For every Q ∈ S we let T′_{Q} = g−1_{Q}T′, and then T′_{Q} is a D_{Q}-module.

Proposition B.5.1. With notation as above, if i ≥ 0 then there is a canonical isomorphism

H^{i}(F, Ind_{D}^{G_{K}} (T′)) ≅ ⨁_{Q∈S} H^{i}(F_{Q}, T′_{Q}).

Proof. The map

\begin{matrix}D\G_{K}/G_{F} & ⟶ & S \\ DgG_{F} & ↦ & g^{−1}Q_{0} \end{matrix}

is a bijection. Applying Proposition B.4.2 with G = G_{K}, with H = D, and with Γ = G_{F} yields

H^{i}(F, Ind_{D}^{G_{K}} (T′)) ≅ ⨁_{Q∈S} H^{i}(g_{Q}G_{F} g−1_{Q} ∩ D, T′)

≅ ⨁_{Q∈S} H^{i}(G_{F} ∩ D_{Q}, T′_{Q}) = ⨁_{Q∈S} H^{i}(F_{Q}, T′_{Q}).

An examination of the proof of Proposition B.4.2 shows that this isomorphism is independent of the choices of the representatives g_{Q}.

Corollary B.5.2. With notation as above, there are canonical isomorphisms

H^{i}(F, Ind_{D}^{G_{K}} (T)) ≅ ⨁_{Q∈S} H^{i}(F_{Q}, T),
H^{i}(F, Ind_{D}^{G_{K}} (T^{I})) ≅ ⨁_{Q∈S} H^{i}(F_{Q}, T^{I_{Q}}).

Proof. Apply Proposition B.5.1 with T′ = T and with T′ = T^{I}.

Corollary B.5.3. Suppose F is a finite Galois extension of K and T is a finitely generated Z_{p}-module with a continuous action of G_{K}. Let V = T ⊗ Q_{p}.

(i) If [F : K] is prime to p, then the restriction map induces an isomorphism

H^{1}(K_{𝔮}, T) ≅ (⊕_{Q|𝔮}H^{1}(F_{Q}, T))^{Gal(F/K)} .

(ii) The restriction map induces an isomorphism

H^{1}(K_{𝔮}, V) ≅ (⊕_{Q|𝔮}H^{1}(F_{Q}, V))^{Gal(F/K)} .

Proof. For every n ≥ 0 we have a commutative diagram

H^{1}(K, Ind_{D}^{G_{K}} (T/p^{n}T))  ⟶^{∼}  H^{1}(K_{𝔮}, T/p^{n}T)
Res_{F} ↓  ↓^{⊕Res_{Q}}
H^{1}(F, Ind_{D}^{G_{K}} (T/p^{n}T))^{Gal(F/K)} ⟶^{∼} (⊕_{Q∈S}H^{1}(F_{Q}, T/p^{n}T))^{Gal(F/K)}

where the vertical maps are restriction maps and the horizontal maps are the isomorphisms of Corollary B.5.2. The inflation-restriction sequence shows that the left-hand vertical map has kernel and cokernel annihilated by [F : K], and hence the right-hand map does as well. Taking the inverse limit of the right-hand maps and applying Proposition B.2.3 shows that the restriction map

H^{1}(K_{𝔮}, T) ⟶ (⊕_{Q∈S}H^{1}(F_{Q}, T))^{Gal(F/K)}

has kernel and cokernel annihilated by [F : K]. This proves (i), and (ii) follows by tensoring with Q_{p} and using Proposition B.2.4.

APPENDIX C Cohomology of p-adic Analytic Groups

Suppose that F is a number field and W is a G_{F} -module. When analyzing a Selmer group S(F, W) as defined in §1.5, one frequently wants to restrict to an extension Ω of F such that the action of G_{Ω} on W is trivial, and then study the image of S(F, W) under the restriction map

H^{1}(F, W) ⟶ H^{1}(Ω, W) = Hom(G_{Ω}, W).

In this appendix we will show (Corollary C.2.2) how to control the kernel of this restriction map.

C.1. Irreducible Actions of Compact Groups

Theorem C.1.1. Suppose V is a finite-dimensional Q_{p}-vector space, and G is a compact subgroup of GL(V) which acts irreducibly on V. Then H^{1}(G, V) = 0.

The proof will be divided into a series of lemmas. For this section we fix a finite-dimensional Q_{p}-vector space V and a compact subgroup G of GL(V) which acts irreducibly on V, as in the statement of Theorem C.1.1. Let Z denote the center of G.

Lemma C.1.2. If g ∈ Z and g ≠ 1, then g − 1 is invertible on V.

Proof. Let V_{1} = ker(g −1). Since g is in the center of G, the subspace V_{1} is stable under G. Since g ≠ 1 we have V_{1} ≠ V. Hence by our irreducibility assumption, V_{1} = 0.

Lemma C.1.3. If Z ≠ {1} then H^{1}(G, V) = 0.

Proof. Suppose that g ∈ Z and g ≠ 1. Let B be the closed subgroup of Z generated by g. We have an inflation-restriction exact sequence

0 ⟶ H^{1}(G/B, V^{B}) ⟶ H^{1}(G, V) ⟶ H^{1}(B, V).

By Lemma C.1.2 we have V^{B} = 0, and

H^{1}(B, V) ⊂ V/(g − 1)V = 0.

206 C. COHOMOLOGY OF p-ADIC ANALYTIC GROUPS

Lemma C.1.4. Suppose U is an open normal subgroup of G. Then V is completely reducible as a representation of U.

Proof. Let V_{0} denote the subspace of V generated by all irreducible Usubspaces of V. Since U is normal in G, we see that V_{0} is stable under G. Clearly V_{0} ≠ 0, so the irreducibility hypothesis for G implies that V_{0} = V. It follows easily that V is a direct sum of a finite collection of irreducible U-subspaces.

For a general reference for the background material on p-adic Lie groups, Lie algebras, and their cohomology which we use below, see [Laz] or [Bo].

Proposition C.1.5. Lie(G) is reductive.

Proof. It follows from Lemma C.1.4 that the representation of Lie(G) on V is semisimple, and it is clearly also faithful. By [Bo] §I.6.4 Proposition 5, it follows that Lie(G) is reductive.

Proof of Theorem C.1.1. The compact subgroup G of GL(V) is a profinite p-analytic group in the sense of [Laz] §III.3.2. Therefore by Lazard’s Théorème [Laz] V.2.4.10, for every sufficiently small open normal subgroup U of G,

H^{1}(G, V) = H^{1}(U, V)^{G} = H^{1}(Lie(G), V)^{G}.

If the center of Lie(G) is zero then (since Lie(G) is reductive by Proposition C.1.5) Lie(G) is semisimple, and in that case (see [Bo] Exercise 1(b), §I.6) H^{1}(Lie(G), V) = 0. If the center of Lie(G) is not zero then every sufficiently small open normal subgroup U of G has nontrivial center, and then Lemmas C.1.3 and C.1.4 together show that H^{1}(U, V) = 0. Thus in either case we can conclude that H^{1}(G, V) = 0.

Recall that Z is the center of G.

Lemma C.1.6. Suppose that O is the ring of integers of a finite extension Φ of Q_{p}, that V is a Φ-vector space, and that G acts Φ-linearly on V. If G contains an element g such that dim_{Φ}(V/(g − 1)V) = 1, then Z acts on V via scalars in O^{×}.

Proof. The one-dimensional subspace ker(g − 1) of V is preserved by Z. Let χ : Z → Aut(ker(g − 1)) ≅ Φ^{×} be the character determined by this action. Since Z is compact, χ(Z) ⊂ O^{×}. Let

V_{χ} = {v ∈ V : zv = χ(z)v for every z ∈ Z}.

Then V_{χ} is nonzero and stable under G, so the irreducibility of V implies that V_{χ} = V.

Proposition C.1.7. Suppose that A is an abelian quotient of G. Then the projection of Z to A has finite cokernel.

Proof. Let π : G ↠ A be the projection map. Since A is compact, it is finitely generated.

By Proposition C.1.5, G is reductive. It follows easily that the induced map of Lie algebras sends the center of Lie(G) onto Lie(A). Therefore [A : π(Z_{U})] is finite, where Z_{U} is the center of a sufficiently small open normal subgroup U of G.

The finite group G/U acts on Z_{U} by conjugation, and we define

N(z) = ∑_{g∈G/U} z^{g}

(writing Z_{U} as an additive group). Clearly N(Z_{U}) ⊂ Z, and also (since ker(π) contains all commutators) π(N(z)) = π([G : U]z) for every z ∈ Z_{U}. Therefore π(Z) contains [G : U]π(Z_{U}). This completes the proof.

C.2. Application to Galois Representations

For this section fix a (possibly infinite) Galois extension F/K of fields of characteristic different from p, and a subgroup B of K^{×}. (In our applications K will be a number field, F will be a (possibly infinite) abelian extension of K, and B will be O_{K}^{×}.) Suppose that O is the ring of integers of a finite extension Φ of Q_{p}, and V is a finite-dimensional Φ-vector space with a continuous Φ-linear action of G_{K} such that V is irreducible over G_{F}. Let Ω = F (μ_{p^{∞}}, B^{1/p^{∞}}, V), the smallest extension of F whose absolute Galois group acts trivially on μ_{p^{∞}}, on B^{1/p^{∞}}, and on V. The result we will need for our applications to Selmer groups is the following.

Theorem C.2.1. One of the following three situations holds.

(i) H^{1}(Ω/F, V) = 0.

(ii) G_{K} acts on V via a character ρ of Gal(F/K), and Gal(F/K) acts

on H^{1}(Ω/F, V) via ρ. (iii) B is infinite, G_{K} acts on V via ε_{cyc}ρ where ε_{cyc} is the cyclotomic

character and ρ is a character of Gal(F/K), and Gal(F/K) acts on H^{1}(Ω/F, V) via ρ.

Proof. Let Ω_{V} = F (V), the smallest extension of F such that G_{Ω_{V}} acts trivially on V (so Ω_{V} = F̄^{H} where H = ker(G_{F} → Aut(V)), and Ω_{V} is necessarily Galois over F). Define D = Gal(Ω_{V} /F) and Ω_{V,μ} = Ω_{V} (μ_{p^{∞}}).

We have a diagram

[figure: K(μ_{p^{∞}}); Ω_{V,μ}; Ω; K; F; D; Ω_{V}]

The inflation-restriction exact sequence gives

H^{1}(D, V) ⟶ H^{1}(Ω/F, V) ⟶ H^{1}(Ω/Ω_{V} , V)^{D}.

The map D → Aut(V) is injective by the definition of Ω_{V}, so D is isomorphic to a compact subgroup of GL(V). We have assumed that D acts irreducibly on V, so Theorem C.1.1 shows that H^{1}(D, V) = 0. Thus we have an injection

H^{1}(Ω/F, V) ↪ H^{1}(Ω/Ω_{V} , V)^{D} = Hom(Gal(Ω/Ω_{V}), V)^{D}.

If Hom(Gal(Ω/Ω_{V}), V)^{D} = 0 then (i) holds. We consider two cases.

Case I: Ω_{V,μ} ≠ Ω_{V}. In this case Gal(Ω_{V,μ}/Ω_{V}) acts on Gal(Ω/Ω_{V,μ}) via the (nontrivial) cyclotomic character. Let Ω_{ab} denote the maximal abelian extension of Ω_{V} in Ω. Then Gal(Ω_{V,μ}/Ω_{V}) acts on Gal(Ω_{ab}/Ω_{V,μ}) trivially and via the cyclotomic character. We deduce from this that Gal(Ω_{ab}/Ω_{V,μ}) is killed by |μ_{p^{∞}} ∩ Ω_{V} |, which is finite since Ω_{V,μ} ≠ Ω_{V}. Hence Hom(Gal(Ω_{ab}/Ω_{V,μ}), V) = 0, so

Hom(Gal(Ω/Ω_{V}), V)^{D} = Hom(Gal(Ω_{ab}/Ω_{V}), V)^{D}

= Hom(Gal(Ω_{V,μ}/Ω_{V}), V)^{D} = Hom(Gal(Ω_{V,μ}/Ω_{V}), V^{D})

since D (and in fact all of Gal(Ω_{V} /K)) acts trivially on Gal(Ω_{V,μ}/Ω_{V}). Since D acts irreducibly on V, either V^{D} = 0 or V is one-dimensional with trivial action of G_{F}. Therefore (i) or (ii) is satisfied in this case.

Case II: Ω_{V,μ} = Ω_{V}. In this case μ_{p^{∞}} ⊂ Ω_{V}, so Gal(Ω/Ω_{V}) is abelian and Gal(Ω_{V} /K) acts on Gal(Ω/Ω_{V}) via the cyclotomic character. Thus

Hom(Gal(Ω/Ω_{V}), V)^{D} = Hom(Gal(Ω/Ω_{V}), V^{ε_{cyc}})

where V^{ε_{cyc}} denotes the subspace of V on which D (and hence G_{F}) acts via ε_{cyc}. Again, since D acts irreducibly on V, either V^{ε_{cyc}} = 0 or V is one-dimensional with G_{F} acting via ε_{cyc}. Therefore (i) or (iii) is satisfied in this case.

Corollary C.2.2. Suppose T is a finitely generated O-submodule of V which is stable under G_{K}, and let W = V/T. Then one of the following three situations holds.

(i) H^{1}(Ω/F, W) is finite.

(ii) G_{K} acts on T via a character ρ of Gal(F/K), and H^{1}(Ω/F, W) has

a subgroup of finite index on which Gal(F/K) acts via ρ. (iii) B is infinite, G_{K} acts on T via ε_{cyc}ρ where ε_{cyc} is the cyclotomic

character and ρ is a character of Gal(F/K), and H^{1}(Ω/F, W) has a subgroup of finite index on which Gal(F/K) acts via ρ.

Proof. From Proposition B.2.4 and the exact sequence

H^{1}(Ω/F, V) → H^{1}(Ω/F, W) → H^{2}(Ω/F, T) → H^{2}(Ω/F, V)

we see that the cokernel of H^{1}(Ω/F, V) → H^{1}(Ω/F, W) is H^{2}(Ω/F, T)_{tors}. Since Gal(Ω/F) is (topologically) finitely generated, H^{2}(Ω/F, T)_{tors} is finite. Now the corollary is immediate from Theorem C.2.1.

APPENDIX D p-adic Calculations in Cyclotomic Fields

In this appendix we carry out some p-adic calculations in cyclotomic fields which are used in examples in Chapters 3 and 8. Everything here is basically well-known, due originally to Iwasawa and Coleman.

For every n ∈ Z^{+} fix a primitive n-th root of unity ζ_{n} such that ζ_{mn}^{n} = ζ_{m} for every m and n. By slight abuse of notation, for every n we will write Z_{p}[μ_{n}] = Z[μ_{n}] ⊗ Z_{p}, the p-adic completion of Z[μ_{n}], and similarly Q_{p}(μ_{n}) = Q(μ_{n}) ⊗ Q_{p}.

Define

log : Z_{p}[μ_{n}][[X]]^{×} = Z_{p}[μ_{n}]^{×} × (1 + XZ_{p}[μ_{n}][[X]]) ⟶ Q_{p}(μ_{n})[[X]]

to be the usual p-adic logarithm on Z_{p}[μ_{n}]^{×} and the power series expansion of log(1 + Xf(X)) on 1 + XZ_{p}[μ_{n}][[X]]. If α ∈ Z_{p} define

[α](X) = (1 + X)^{α} − 1 ∈ XZ_{p}[[X]].

Let D be the derivation (1 + X)\frac{d}{dX} of \overline{Q_{p}}[[X]]. Then for every α ∈ Z_{p} and g ∈ \overline{Q_{p}}[[X]], we have

D[α] = α · ([α](X) + 1) and D(g ∘ [α]) = α · (Dg) ∘ [α].

If m is prime to p let Fr_{p} be the Frobenius of p in Gal(Q(μ_{m})/Q), i.e., the automorphism which sends ζ_{m} to ζ_{m}^{p}. We let Fr_{p} act on Q_{p}(μ_{m})[[X]] by acting on the power series coefficients.

D.1. Local Units in Cyclotomic Fields

In this section we will construct, for every positive integer n, a homomorphism λ_{n} : Z_{p}[μ_{n}]^{×} → Z_{p}. These maps are used in §3.4 to construct an Euler system for the trivial representation Z_{p}.

Fix an integer m prime to p. Define

f_{m}(X) = mζ_{m}[m^{−1}](X) − \frac{mζ_{m}^{p}}{|(Z_{p}^{×})_{tors}|} ∑_{β∈(Z_{p}^{×})_{tors}} \frac{[m^{−1}β](X)}{β}

∈ |(Z_{p}^{×})_{tors}|^{−1}Z_{p}[μ_{m}][[X]]

212 D. p-ADIC CALCULATIONS IN CYCLOTOMIC FIELDS

and

G_{m}(X) = ζ_{m} log(1 + X) − m ∑_{i=1}^{∞} p^{i}ζ_{m}^{p^{−i}}

+ ∑_{i=0}^{∞}(\frac{f_{m}^{Fr_{p}^{i}}([p^{i}](X))}{p^{i}} − (ζ_{m}^{p^{i}} − ζ_{m}^{p^{i+1}}) log(1 + X)).

Lemma D.1.1(i) below shows that the latter sum converges to an element of Q_{p}(μ_{m})[[X]], and a direct computation shows that

DG_{m}(X) = ζ_{m} + ∑_{i=0}^{∞}(ζ_{m}^{p^{i}}[m^{−1}p^{i}](X) − \frac{ζ_{m}^{p^{i+1}}}{|(Z_{p}^{×})_{tors}|} ∑_{β∈(Z_{p}^{×})_{tors}}[m^{−1}βp^{i}](X)).

(D.1)

Lemma D.1.1. (i) G_{m}(X) ∈ Q_{p}(μ_{m})[[X]], i.e., the sum in the defini-

tion of G_{m}(X) converges. (ii) There is a unique g_{m}(X) ∈ 1 + (p, X)Z_{p}[μ_{m}][[X]] such that

log(g_{m}(X)) = G_{m}(X).

(iii) If ℓ is a prime different from p, then

Tr_{Q_{p}(μ_{mℓ})/Q_{p}(μ_{m})}DG_{ℓm}(X) = \begin{cases}−ℓDG_{m}^{Fr_{ℓ}^{−1}}([ℓ^{−1}](X)) & if ℓ ∤ m, \\ 0 & if ℓ | m. \end{cases}

(iv) ∑_{ζ∈μ_{p}} G_{m}(ζ(1 + X) − 1) = G_{m}^{Fr_{p}}([p](X)).

(v) If g_{m} is as in (ii), then ∏_{ζ∈μ_{p}} g_{m}(ζ(1 + X) − 1) = g_{m}^{Fr_{p}}([p](X)).

Proof. The first two assertions follow from Theorem 24 of [Co] applied with a = −m ∑_{i=1}^{∞} p^{i}ζ_{m}^{p^{−i}}, with b = ζ_{m}, and with f(X) = f_{m}(X) − (ζ_{m} − ζ_{m}^{p})X. Assertion (iii) follows directly from (D.1) and the fact that

Tr_{Q(μ_{mℓ})/Q(μ_{m})}ζ_{mℓ} = \begin{cases}−ζ_{m}^{Fr_{ℓ}^{−1}} & if ℓ ∤ m, \\ 0 & if ℓ | m. \end{cases}

The fourth assertion is similarly a direct computation, and then (v) follows from (iv), since log is injective on 1 + (p, X)Z_{p}[μ_{m}][[X]].

Definition D.1.2. Suppose m ≥ 1 is prime to p, and let N(m) = ∏_{ℓ|m} ℓ, product over primes ℓ dividing m. Let g_{m}(X) ∈ Z_{p}[μ_{m}][[X]]^{×} be as in Lemma D.1.1(ii). For n ≥ 0 define

α_{mp^{n}} = ∏_{d|m,N(m)|d}(g_{d}^{Fr_{p}^{−n}}(ζ_{p^{n}} − 1)) ∈ Z_{p}[μ_{mp^{n}}]^{×}.

By Lemma D.1.1(v),

N_{Q(μ_{mp^{n+1}} )/Q(μ_{mp^{n}})}α_{mp^{n+1}} = \begin{cases}α_{mp^{n}} & if n ≥ 1, \\ α_{m}^{1−Fr_{p}^{−1}} & if n = 0. \end{cases}

Suppose 𝔓 is a prime of Q(μ_{m}) above p. We will also write 𝔓 for the unique prime of Q(μ_{mp^{n}}) above 𝔓, for every n. We let

α_{m,𝔓} ∈ Gal(Q(μ_{mp^{∞}})_{𝔓}^{ab}/Q(μ_{mp^{∞}})_{𝔓})

be the image of {α_{mp^{n}}}_{n≥1} under the Artin map of local class field theory. Using the Kummer pairing we define

λ_{mp^{n}} : Z_{p}[μ_{mp^{n}}]^{×} ⟶ Z_{p}

so that, writing u ∈ Z_{p}[μ_{mp^{n}}]^{×} as (u_{𝔓}) ∈ ⊕_{𝔓}Z[μ_{mp^{n}}]_{𝔓}^{×}, for every k ≥ 0 we have

∏_{𝔓|p}(u_{𝔓}^{p^{−k}})^{α_{m,𝔓}−1} = ζ_{p^{k}}^{mλ_{mp^{n}} (u)}.

The explicit reciprocity law gives the following description of the map λ_{mp^{n}}. Recall that D is the derivation (1 + X)\frac{d}{dX}.

Proposition D.1.3. If m is prime to p and n ≥ 0 then

λ_{mp^{n}}(u) = p^{−n}Tr_{Q_{p}(μ_{mp^{n}})/Q_{p}}(x_{mp^{n}} log_{p}(u))

where log_{p} is the usual p-adic logarithm and

x_{mp^{n}} = \begin{cases}m^{−1} ∑_{d|m,N(m)|d}(DG_{d}^{Fr_{p}^{−n}})(ζ_{p^{n}} − 1) & if n > 0, \\ m^{−1} ∑_{d|m,N(m)|d}(DG_{d})(0) − \frac{1}{p}(DG_{d}^{Fr_{p}^{−1}})(0) & if n = 0. \end{cases}

Proof. The formula of the proposition is the explicit reciprocity law of Wiles [Wi] (see also [dS] Theorem I.4.2) in the present situation.

Lemma D.1.4. For every m ≥ 1 (not necessarily prime to p) and prime ℓ, there is a commutative diagram

\xymatrix{Z_{p}[μ_{mℓ}]^{×} \ar@{-->}[dr]^{λ_{mℓ}} \\  & Z_{p} \\ Z_{p}[μ_{m}]^{×} \ar@{-->}[ur]_{λ_{m}} \ar@{^{(}->}[uu]^{1 or −Fr_{ℓ}}}

where the vertical map is

\begin{cases}the inclusion Z_{p}[μ_{m}]^{×} ⊂ Z_{p}[μ_{mℓ}]^{×} & if ℓ | m or ℓ = p, \\ −Fr_{ℓ} followed by that inclusion & if ℓ ∤ mp. \end{cases}

Proof. Let the x_{m} be as defined in Proposition D.1.3. Using Lemma D.1.1(iii) and (iv) we see that

Tr_{Q_{p}(μ_{mℓ})/Q_{p}(μ_{m})}x_{mℓ} = \begin{cases}x_{m} & if ℓ | m or ℓ = p, \\ −Fr_{ℓ}^{−1}x_{m} & if ℓ ∤ mp, \end{cases}

for every m and ℓ. Now the lemma follows from Proposition D.1.3.

Let ω denote the Teichmüller character giving the action of G_{Q} on μ_{p} (if p is odd) or on μ_{4} (if p = 2).

Lemma D.1.5. Suppose O is the ring of integers of a finite extension of Q_{p}, and χ : G_{Q} → O^{×} is a character of finite prime-to-p order. Let f be the conductor of χ, and suppose that p^{2} ∤ f and χ^{−1}ω(p) ≠ 1 (where we view χ^{−1}ω as a primitive Dirichlet character). Let Δ = Gal(Q(μ_{f})/Q). Then ∑_{δ∈Δ} χ(δ)λ_{f}^{δ} generates the O-module Hom(Z_{p}[μ_{f}]^{×}, O)^{χ^{−1}} (the submodule of Hom(Z_{p}[μ_{f}]^{×}, O) on which Δ acts via χ^{−1}).

Proof. Let λ_{f,χ} = ∑_{δ∈Δ} χ(δ)λ_{f}^{δ}. Write f = mp^{ϵ} with m prime to p and ϵ = 0 or 1. Note that if p = 2, then we cannot have ϵ = 1. Let x_{f} be as in Proposition D.1.3, and let y_{f} be the “conductor f” part of x_{f}, namely

y_{f} = \begin{cases}f^{−1}((DG_{f})(0) − \frac{1}{p}(DG_{f}^{Fr_{p}^{−1}})(0)) = f^{−1}(ζ_{f} − \frac{1}{p}ζ_{f}^{Fr_{p}^{−1}}) & if ϵ = 0, \\ m^{−1}(DG_{m}^{Fr_{p}^{−1}})(ζ_{p} − 1) = m^{−1}(ζ_{f} + \frac{p}{p−1}ζ_{m}) & if ϵ = 1. \end{cases}

By Proposition D.1.3,

λ_{f,χ}(u) = p^{−ϵ} ∑_{δ∈Δ} χ(δ)Tr_{Q_{p}(μ_{f})/Q_{p}}x_{f}^{δ} log_{p}(u)

= p^{−ϵ} ∑_{δ∈Δ} ∑_{γ∈Δ} χ(δ)x_{f}^{δγ} log_{p}(u^{γ})

= p^{−ϵ} ∑_{δ∈Δ}(χ(δ)x_{f}^{δ}) ∑_{γ∈Δ}(χ^{−1}(γ) log_{p}(u^{γ}))

= p^{−ϵ} ∑_{δ∈Δ}(χ(δ)y_{f}^{δ}) ∑_{γ∈Δ}(χ^{−1}(γ) log_{p}(u^{γ}))

= \frac{1}{f} (1 − p^{−1}χ(p)) ∑_{δ∈Δ}(χ(δ)ζ_{f}^{δ}) ∑_{γ∈Δ}(χ^{−1}(γ) log_{p}(u^{γ})).

First suppose p ∤ f, so χ(p) ∈ O^{×}. Let g_{m} be as in Lemma D.1.1(ii) and let u = g_{m}(0)^{1/m} ∈ Z_{p}[μ_{f}]^{×}. Then log_{p}(u) = m^{−1}G_{m}(0) = − ∑_{i=1}^{∞} p^{i}ζ_{m}^{p^{−i}},

D.1. LOCAL UNITS IN CYCLOTOMIC FIELDS 215

so

∑_{γ∈Δ}(χ^{−1}(γ) log_{p}(u^{γ})) = − ∑_{i=1}^{∞} p^{i} ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{m}^{Fr_{p}^{−i}γ})

= − ∑_{i=1}^{∞} p^{i}χ^{−i}(p) ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{m}^{γ}).

Thus

λ_{f,χ}(u) = \frac{1}{f} (χ(p) − p) ∑_{i=0}^{∞} p^{i}χ^{−i−1}(p) ∑_{δ∈Δ}(χ(δ)ζ_{f}^{δ}) ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{f}^{γ})

= χ(−1)(χ(p) − p) ∑_{i=0}^{∞} p^{i}χ^{−i−1}(p) ∈ O^{×},

the last equality since the product of the two Gauss sums is χ(−1)f.

Now suppose p | f but p^{2} ∤ f (so p ≠ 2), and set

u = (g_{m}^{Fr_{p}^{−1}}(ζ_{p} − 1))^{1/m} ∈ Z_{p}[μ_{f}]^{×}.

Then

log_{p}(u) = m^{−1}G_{m}^{Fr_{p}^{−1}}(ζ_{p} − 1)

= (1 − \frac{1}{p − 1} ∑_{{σ∈Δ \atop σ|_{Q(μ_{m})}=Fr_{p}}}ω(σ^{−1})σ)(ζ_{m}^{p^{−1}}(ζ_{p}^{m^{−1}} − 1)) − ∑_{i=1}^{∞} p^{i}ζ_{m}^{p^{−(i+1)}}

so with this choice, since ζ_{m}^{p^{−1}}ζ_{p}^{m^{−1}} = ζ_{f}, we have

∑_{γ∈Δ} χ^{−1}(γ) log_{p}(u^{γ}) = (1 − \frac{1}{p − 1} ∑_{{σ∈Δ \atop σ|_{Q(μ_{m})}=Fr_{p}}}ω(σ^{−1})χ(σ)) ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{f}^{γ})

= (1 − χω^{−1}(p)) ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{f}^{γ})

and

λ_{f,χ}(u) = \frac{1}{f} (1 − χω^{−1}(p)) ∑_{δ∈Δ}(χ(δ)ζ_{f}^{δ}) ∑_{γ∈Δ}(χ^{−1}(γ)ζ_{f}^{γ})
= χ(−1)(1 − χω(p)^{−1}) ∈ O^{×}.

In either case the p-adic logarithm shows that Hom(Z_{p}[μ_{f}]^{×}, O)^{χ^{−1}} is a rank-one O-module, which is clearly torsion-free and hence free. The formulas above show that

λ_{f,χ} ∉ 𝔭Hom(Z_{p}[μ_{f}]^{×}, O)^{χ^{−1}}

where 𝔭 is the maximal ideal of O, and the lemma follows.

D.2. Cyclotomic Units

For this section suppose that m > 1 and m is prime to p. Fix an embedding \overline{Q_{p}} ⊂ C and let ζ_{n} = e^{2πi/n} for every n ∈ Z^{+}. Define

u_{m}(X) = ζ_{m}(1 + X)^{m^{−1}} − 1 ∈ Z_{p}[μ_{m}][[X]].

Lemma D.2.1. Suppose m > 1 and m is prime to p. Suppose n ≥ 0 and γ ∈ Gal(Q(μ_{mp^{n}})/Q), and choose b ∈ Z such that ζ_{mp^{n}}^{γ} = ζ_{mp^{n}}^{b}. Then for every k ≥ 2,

(D^{k} log u_{m}^{Fr_{p}^{−n}γ})(ζ_{p^{n}}^{γ} − 1)

= (−1)^{k−1}Γ(k)(2πi)^{−k}p^{nk}(ζ(b, mp^{n}; k) + (−1)^{k}ζ(−b, mp^{n}; k))

where ζ(a, r; s) is the partial Riemann zeta function ∑_{0<j≡a (mod r)} j^{−s}.

Proof. Since m > 1 and m is prime to p, we see that u_{m}(0) ∈ Z_{p}[μ_{m}]^{×}. Therefore u_{m}^{Fr_{p}^{−n}γ} ∈ Z_{p}[μ_{m}][[X]]^{×} and log u_{m}^{Fr_{p}^{−n}γ} is defined. Thus

(D^{k} log u_{m}^{Fr_{p}^{−n}γ})(ζ_{p^{n}}^{γ} − 1) = D^{k−1} \frac{(1 + X)(u_{m}^{Fr_{p}^{−n}γ})′(X)}{u_{m}^{Fr_{p}^{−n}γ}(X)}|_{X=ζ_{p^{n}}^{b} −1}

= D^{k−1} \frac{m^{−1}ζ_{m}^{bp^{−n}}(1 + X)^{m^{−1}}}{ζ_{m}^{bp^{−n}}(1 + X)^{m^{−1}} − 1} |_{X=ζ_{p^{n}}^{b} −1} .

Substituting e^{Z} = (1 + X)^{m^{−1}} and m^{−1} \frac{d}{dZ} = (1 + X)\frac{d}{dX}, this becomes

(D^{k} log u_{m}^{Fr_{p}^{−n}γ})(ζ_{p^{n}}^{γ} − 1) = m^{−k} \frac{d^{k−1}}{dZ^{k−1}} \frac{ζ_{m}^{bp^{−n}}e^{Z}}{ζ_{m}^{bp^{−n}}e^{Z} − 1}|_{e^{Z}=ζ_{p^{n}}^{bm^{−1}}}

= m^{−k} \frac{d^{k−1}}{dZ^{k−1}} \frac{e^{Z}}{e^{Z} − 1}|_{Z= \frac{2πib}{mp^{n}}} .

By [Al] equation (10), p. 187 (or just observe that the difference is a bounded entire function which vanishes at 0)

\frac{e^{Z}}{e^{Z} − 1} = \frac{1}{2} + \frac{1}{Z} + ∑_{n=1}^{∞} (\frac{1}{Z − 2πin} + \frac{1}{Z + 2πin}) .

Thus for k ≥ 2, r > 1, and c ∈ Z − rZ, we have

\frac{d^{k−1}}{dZ^{k−1}} \frac{e^{Z}}{e^{Z} − 1}|_{Z= \frac{2πic}{r}} = (−1)^{k−1}(k − 1)!(2πi)^{−k}r^{k} ∑_{n∈Z} \frac{1}{(c + nr)^{k}}

= (−1)^{k−1}Γ(k)(2πi)^{−k}r^{k}(ζ(c, r; k) + (−1)^{k}ζ(−c, r; k)).

Combining these formulas proves the lemma.

Define

h_{m}(X) = ∏_{β∈(Z_{p}^{×})_{tors}}u_{m}((1 + X)^{β} − 1)ū_{m}((1 + X)^{β} − 1)

where ū_{m}(X) = 1 − ζ_{m}^{−1}(1 + X)^{m^{−1}}, and define

H_{m}(X) = log h_{m}(X) − \frac{1}{p} log h_{m}^{Fr_{p}}((1 + X)^{p} − 1).

For every n > 1 write Δ_{n} = Gal(Q(μ_{n})/Q) and Δ_{n}^{+} = Gal(Q(μ_{n})^{+}/Q).

Lemma D.2.2. Suppose p > 2, and let ω be the Teichmüller character giving the action of G_{Q} on μ_{p}. Suppose O is the ring of integers of a finite extension of Q_{p}, and χ : G_{Q} → O^{×} is a nontrivial even character of finite order, unramified at p. If m is the conductor of χ then

∑_{γ∈Δ_{m}^{+}} χ^{−1}(γ)D^{k}H_{m}^{γ}(ζ_{p} − 1)

= 2Γ(k)(−2πi)^{−k}L(χ^{−1}ω^{k}, k) × \begin{cases}−χ(p)p^{k} & if p − 1 ∤ k, \\ 1 − p^{k−1}χ(p) & if p − 1 | k. \end{cases}

Proof. We have

D^{k}H_{m}^{γ}(ζ_{p} − 1) = D^{k} log h_{m}^{γ}(ζ_{p} − 1) − p^{k−1}D^{k} log h_{m}^{Fr_{p}γ}(0).

If ζ = ζ_{p} or ζ = 1, then

D^{k} log h_{m}^{γ}(ζ − 1) = ∑_{β∈(Z_{p}^{×})_{tors}}β^{k}D^{k} log u_{m}^{γ}(ζ^{β} − 1) + β^{k}D^{k} log ū_{m}^{γ}(ζ^{β} − 1)

= ∑_{σ∈Gal(Q(μ_{mp})/Q(μ_{m})^{+})} ω^{k}(σ)D^{k} log u_{m}^{γσ}(ζ^{σ} − 1).

Thus by Lemma D.2.1, writing L_{r}(χ^{−1}ω^{k}, s) for the Dirichlet L-function with Euler factors for primes dividing r removed,

∑_{γ∈Δ_{m}^{+}} χ^{−1}(γ)D^{k}H_{m}^{γ}(ζ_{p} − 1)

= ∑_{γ∈Δ_{mp}} χ^{−1}ω^{k}(γ)(D^{k} log u_{m}^{γ}(ζ_{p}^{γ} − 1) − p^{k−1}D^{k} log u_{m}^{Fr_{p}γ}(0))

= (−1)^{k−1}Γ(k)(2πi)^{−k}p^{k}(1 + (−1)^{k}χ^{−1}ω^{k}(−1))χ(p)L_{mp}(χ^{−1}ω^{k}, k)

− p^{k−1}(−1)^{k−1}Γ(k)(2πi)^{−k}(1 + (−1)^{k})χ(p)L_{m}(χ^{−1}, k) ∑_{γ∈Δ_{p}} ω^{k}(γ).

218 D. p-ADIC CALCULATIONS IN CYCLOTOMIC FIELDS

Note that χ^{−1}ω^{k}(−1) = (−1)^{k}. If (p − 1) | k then ω^{k} = 1, and the formula above simplifies to

2Γ(k)(−2πi)^{−k}χ(p)L(χ^{−1}, k)(−p^{k}(1 − χ^{−1}(p)p^{−k}) + (p − 1)p^{k−1})

= 2Γ(k)(−2πi)^{−k}L(χ^{−1}, k)(1 − p^{k−1}χ(p)).

If (p − 1) ∤ k then ∑_{γ∈Δ_{p}} ω^{k}(γ) = 0, so in that case

∑_{γ∈Δ_{m}^{+}} χ^{−1}(γ)D^{k}H_{m}^{γ}(ζ_{p} − 1) = −2Γ(k)(−2πi)^{−k}p^{k}χ(p)L(χ^{−1}ω^{k}, k).

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Index of Symbols

Chapter 1

K 9 p 9 O 9 Φ 9 G_{K} 9 T 9 D 9 V 9 W 9 W_{M} 9 O_{ρ} 10 ε_{cyc} 10 O(1) 10 T^{∗} 10 V^{∗} 10 W^{∗} 10 T_{p}(A) 10 I 12 K^{ur} 12 Fr 12 H_{ur}^{1}(K, ·) 12 H_{f}^{1}(K, ·) 14 H_{s}^{1}(K, ·) 14 ( )_{div} 14 ⟨ , ⟩_{K} 18 M̄ 20 K_{Σ} 21 S_{Σ}(K, ·) 21 S^{Σ}(K, ·) 21 S(K, ·) 21 I_{v} 22 ι_{M} 22 O_{χ} 24 D_{χ} 24 Φ_{χ} 24 D_{w} 24 Bˆ 25 B^{χ} 25 Σ_{p} 27 loc_{Σ} 28 loc_{Σ,Σ_{0}}^{s} 28 loc_{Σ,Σ_{0}}^{f} 28 loc_{Σ_{p}}^{s} 31

Chapter 2

O_{K} 33 K(𝔮) 33 Fr_{𝔮} 33 P (Fr_{𝔮}^{−1}|T^{∗}; x) 33 ⊂_{f} 33 K 34 N 34 K_{∞} 34 c 34 Σ(F′/F) 34 K(𝔯) 35 1 35 K(1) 35 F (𝔯) 35 K_{min} 35 𝔭 36 𝕜 36 Hyp(K, T) 37 Hyp(K, V) 37 ind_{O} 37 ℓ_{O}( ) 37 Ω 37 K(W) 37 𝔫_{W} 37 𝔫_{W}^{∗} 37 Λ_{F} 40 Γ 40 Λ 40 char( ) 40 Hyp(K_{∞}/K) 41 Hyp(K_{∞}, T) 41 Hyp(K_{∞}, V) 41 S_{Σ_{p}}(K_{∞}, W^{∗}) 41 H_{∞}^{1}(K, T) 41 X_{∞} 41 c_{K,∞} 41 ind_{Λ}(c) 41 S(K_{∞}, W^{∗}) 42 H_{∞,s}^{1}(K_{p}, T) 42 ξ_{χ} 44 c_{F}^{χ} 44

Chapter 3

Q(μ_{m})^{+} 48 A_{n} 50 E_{n} 50 C_{n,χ} 50 A_{∞} 51 E_{∞} 51 C_{∞,χ} 51 U_{∞} 51 J 53 ω 54 L_{χ} 54 θ_{m} 56 Z_{p}[μ_{m}] 56 θ̄_{m}^{(b)} 56 λ_{m} 57 B_{1,χ^{−1}} 59 χ_{Λ} 60 ⟨ε⟩ 60 Tw_{⟨ε⟩} 60 η^{•} 60 θ_{fp^{∞}} 61 U 61 Tan(E_{/Q_{n,p}}) 64 exp_{E} 64 ω_{E} 64 Cotan(E_{/Q_{n,p}}) 64 E_{1}(Q_{n,p}) 64 𝔭_{n} 64 Ê 64 λ_{E} 64 exp_{E}^{∗} 64 ℓ_{q}(q^{−s}) 65 L_{m}(E, s) 65 L_{m}(E, χ, s) 65 Ω_{E} 66 r_{E} 66 ρ_{E,p} 67 L_{E} 69 Col_{∞} 70

Chapter 4

Γ_{𝔮} 75 R 75 Γ_{𝔯} 76 F (𝔯) 76 Γ_{F (𝔯)} 76 N_{𝔮} 76 R_{F,M} 76 X_{F (𝔯)} 78 x_{F (𝔯)} 78 X 79 X_{∞,𝔯} 79 D_{𝔮} 84 D_{𝔯} 84 N_{F (1)/F} 84 D_{𝔯,F} 84 𝕎_{M} 85 δ_{L} 85 δ_{𝔯} 87 d_{F} 87 κ_{[F,𝔯,M]} 89 Q_{𝔮}(x) 90 α_{𝔮} 91 β_{𝔮} 91 ϕ_{𝔮}^{fs} 91 W_{M}^{f} 94 𝕎_{M}^{f} 94 Ind_{D}(W_{M}) 94 δ_{L_{𝔮},W_{M}^{f}} 94

Chapter 5

order( , ) 105 (η)_{L} 105

Chapter 6

Tw_{ρ} 119 c^{ρ} 123

Chapter 7

Λ_{F,M} 129 θ^{∗} 129 Ω_{∞} 129 Ω_{∞}^{⟨τ⟩} 129 Ev^{∗} 130 a_{τ} 130 Z_{∞} 132 M 132 σ 132 Ω_{M} 132 π 132 𝔯(π) 132 Π(k, F, M) 132 Ψ(k, F, M) 132 ⟨κ_{[F,𝔯,M]}⟩ 132 q_{τ} (x) 135 θ 135 θ̄ 135 Ev 136 Ev_{F,M}^{∗} 136 Ev_{F,M} 136 A^{•} 136 Ann_{Λ}( ) 136 R_{F,M,τ} 137 Ev_{𝔮,f}^{∗} 137 Ev_{𝔮,f} 137 Ev_{𝔮,s} 137 Ẽv 138 A_{glob} 145 A_{v} 145 A_{N} 145

Z_{σ} 154
B_{0} 156

Chapter 8

D(V) 164 D_{F} (V) 164 ϵ_{χ} 164 H 165 K 165 L_{F} 166 d_{−} 166 L_{r}^{(p)} 166 Δ_{r} 167 Λ_{r} 167 ξ_{r} 168

Chapter 9

𝕋 176 S_{Σ′}^{Σ}(K, ·) 183

Subject Index

anticyclotomic Euler system, 181

characteristic ideal, 40 congruence condition on an Euler sys-

tem, 102 continuous cohomology groups, 195 crystalline representation, 164 cyclotomic character, 10 cyclotomic units, 50, 177

derivative cohomology classes, 89 dual exponential map, 64 dual representation, 10 duality, global, 28 duality, local, 18

eigen-components, 25 elliptic units, 55 Euler system, 34 Euler system of finite depth, 179 Euler system, universal, 79 evaluation map, 130, 136, 138

finite cohomology classes, 14 finite-singular comparison map, 91

global duality, 28 global units and Selmer groups, 25

Heegner points, 63, 180 Herbrand quotient, 189

ideal class groups as Selmer groups, 23,

50, 58 index of divisibility, 37, 41 induced module, 85, 201 Iwasawa algebra, 40

Kato’s Euler system for a modular elliptic curve, 66 Kolyvagin sequence, 132 Kolyvagin, Victor, 3

L-function (of elliptic curve), 65 local duality, 18 localization maps, 28

p-adic completion, 25 p-adic L-function (cyclotomic), 54 p-adic L-function (elliptic curve), 69 p-adic L-function (Perrin-Riou), 166 p-adic representation, 9 Perrin-Riou’s “logarithme élargi”, 165 pseudo-isomorphism of Iwasawa mod-

ules, 40 pseudo-null Iwasawa module, 40

representation, p-adic, 9 rigidity, 176

Selmer group, 21 Selmer group of an abelian variety, 27 Selmer sequence, 132 semilocal Galois cohomology, 93, 202 singular cohomology classes, 14 Stickelberger element, 56 symmetric square of an elliptic curve,

73, 170

Tate pairing, 18 Teichmüller character, 54 Thaine, Francisco, 3 twist (by arbitrary characters), 123 twist (by characters of finite order), 44 twisting homomorphism, 119

universal Euler system, 79 unramified cohomology classes, 12 unramified Galois module, 12