KOLYVAGIN SYSTEMS (VERSION 6/30/2001)

BARRY MAZUR AND KARL RUBIN

Contents

Introduction 2

Part I. Local cohomology groups 4 1. Local conditions 4 2. The finite/singular homomorphism 6 3. Local duality 8

Part II. Global cohomology groups and Selmer structures 9 4. Selmer modules 9 5. Global duality 12

Part III. Kolyvagin systems 13 6. Kolyvagin systems 13 7. Euler systems and Kolyvagin systems 16 8. Sheaves on graphs 17 9. Hypotheses on T, F, and P 19 10. Choosing useful primes 22 11. Some remarks about hypothesis (H.6) 25

Part IV. Kolyvagin systems over principal artinian rings 26 12. The core Selmer module 27 13. Kolyvagin systems and the core rank 32 14. The sheaf of stub Selmer modules 33 15. Kolyvagin systems and the stub Selmer sheaf 37 16. Kolyvagin systems over principal artinian rings 39

Part V. Kolyvagin systems over integral domains 42 17. Kolyvagin systems over a field 42 18. Kolyvagin systems over a discrete valuation ring 42 19. Kolyvagin systems over Λ 46

Part VI. Examples 54 20. The multiplicative group 54 21. Elliptic curves 59 22. The multiplicative group, revisited 61 Part VII. Appendices 63

Appendix A. Proof of Theorem 7.4 63 Appendix B. Proof of Theorem 14.3, by Benjamin Howard 69 References 75

Introduction

Since their introduction by Kolyvagin in [Ko], Euler systems have been used in several important applications in arithmetic algebraic geometry ([Ko], [Ka2], [Ru3]). These applications proceed by taking an Euler system (in Kolyvagin’s case, the “pre-existing” Euler systems of cyclotomic units, elliptic units, and Heegner points, and in Kato’s case a new Euler system which he constructed for the purpose), and applying Kolyvagin’s machinery for bounding Selmer groups.

The problem of finding (or constructing) Euler systems seems to be very difficult. On the other hand, Kolyvagin’s machinery lends itself well to axiomatization, and this has been done recently by several authors ([Ka1], [PR4], [Ru6]) in developing a theory of Euler systems for p-adic Galois representations.

In this paper we take a different approach to the general theory. To describe our approach we need to recall briefly how Kolyvagin’s machinery works.

Suppose T is a p-adic Galois representation of G_{Q} (see below for precise definitions). An Euler system is a collection of cohomology classes c_{n} ∈ H^{1}(Q(μ_{n}), T) for a large set of n ∈ Z^{+}, satisfying certain compatibility relations. Following Kolyvagin, one uses this collection to construct a new collection of “derivative” classes κ_{n} ∈ H^{1}(Q, T/I_{n}T) where I_{n} is a certain power of p depending on n and T. After this one never again uses the original classes c_{n}, but instead one works only with the derivative classes.

In this paper we study the Kolyvagin machine by considering the collection of derivative classes as the fundamental tool. We call such a collection a Kolyvagin system. Kolyvagin systems are simpler to study than Euler systems, because all of the cohomology classes are defined over Q. In a precise sense which we will not elaborate on here, a (good) Euler system corresponds to (and contains all the information in) a p-adic L-function, while a (good) Kolyvagin system corresponds to a special value, or a “leading term”, of an L-function. Euler systems appear in this paper only in that they are still the best way to produce explicit Kolyvagin systems.

We began with several questions.

1. How much information does a Kolyvagin system contain about the Selmer groups attached to T and to the dual representation T^{∗} = Hom(T, μ_{p^{∞}})?

2. For which T do Kolyvagin systems exist, and how large is the space of Kolyvagin systems for T?

3. In the simplest examples, what are these cohomology classes κ_{n}, and how can we compute them?

All three questions will be answered to some degree in this paper. In particular question 2 has a simple answer: the existence or nonexistence of Kolyvagin systems depends only on a single cohomological invariant (see §12).

The new information which allows us to answer these questions (to the extent that we do) is an additional axiom in our definition of Kolyvagin system (see Remarks 6.6 and 6.8). This additional property is not needed for Kolyvagin’s machinery, if one simply wants to bound the Selmer group of T^{∗} as in the standard applications of Euler systems. However, without this axiom, the definition of Kolyvagin system is not “rigid” enough: there would be too many of them, and they would not contain as much information. This additional axiom is satisfied by the derivative classes attached to an Euler system, so it is satisfied by all the standard examples.

The layout of the paper is as follows. in Parts I and II we introduce the local and global cohomology groups we will use, and recall basic facts about them. In particular in §4 we introduce the concept of a Selmer structure on a Galois module and its corresponding Selmer group.

In Part III we introduce Kolyvagin systems. We define them in §6 as global sections of the “Selmer sheaf”, a sheaf (whose stalks are generalized Selmer groups attached to T) on the graph whose vertices are positive integers and whose edges join integers m, n if n = mℓ for some prime ℓ. This point of view turns out to be very useful for studying the module of Kolyvagin systems. In §7 we describe the connection between Euler systems and Kolyvagin systems, although the proof that the derivative classes of an Euler system satisfy our definition of a Kolyvagin system is postponed until Appendix A. Section 8 studies general properties of sheaves on graphs, and §9 discusses several simplifying assumptions which we will make in most of the remainder of the paper.

In Part IV we study Kolyvagin systems in their simplest setting, where the ring R of coefficients is a principal artinian local ring (for example, Z/p^{k}Z). We introduce our fundamental cohomological invariant, the “core rank” of T, in §12, and in the following sections we show that the rank of the R-module of Kolyvagin systems is zero, one, or infinity according as the core rank is zero, one, or greater than one, respectively.^{1} Along the way we prove in §15 that a Kolyvagin system, a priori a global section of the Selmer sheaf, is actually a global section of a certain subsheaf. This extra information allows us in §12 to recover the structure of the Selmer group of T^{∗} from a Kolyvagin system for T.

In Part V we prove analogous results first in the case where the ring of coefficients R is a discrete valuation ring (by reducing to quotients of R which are principal artinian local rings), and then in the case where R is an Iwasawa algebra (by reducing to quotients of R which are subrings of discrete valuation rings).

Finally, Part VI we illustrate our results with the basic examples: cyclotomic units and ideal class groups in §20, and then Kato’s Kolyvagin system and Selmer groups of elliptic curves in §21. The reader may find it useful to follow these examples while reading the paper, rather than wait until the end.

Notation. Fix a rational prime p. Throughout this paper, R will denote a complete, noetherian, local ring with finite residue field of characteristic p. Let 𝔪 denote the maximal ideal of R and 𝕜 = R/𝔪 the residue field. The basic cases to keep in mind are R = F_{p}, R = Z_{p}, and R = Λ, a suitable Iwasawa algebra.

If A is an R-module and x ∈ A, then Ann_{R}(x) ⊂ R will denote the annihilator of x. If I is an ideal of R, we will write A[I] for the submodule of A killed by I. If A is a G_{Q}-module, we write Q(A) for the fixed field in Q̄ of the kernel of the map G_{Q} → Aut(A).

1 An essential step (Theorem 14.3) in the proof of this was supplied by Benjamin Howard.

The authors thank him for including his proof of this result in Appendix B.

If a group H acts on a set X, then the subset of elements of X fixed by H is denoted X^{H}.

We write ν(n) for the number of distinct prime factors of a nonzero integer n.

Part I. Local cohomology groups

For §§1-3, K will be a local field (archimedean or nonarchimedean), K̄ a fixed separable algebraic closure of K, and G_{K} = Gal(K̄/K).

If K is nonarchimedean let O be the ring of integers in K, F its residue field, and K^{unr} ⊂ K̄ the maximal unramified subfield of K̄. Let I denote the inertia group Gal(K̄/K^{unr}), and G_{F} = Gal(K^{unr}/K). These groups fit into the exact sequence

{1} ⟶ I ⟶ G_{K} ⟶ G_{F} ⟶ {1}.  (1)

Note that if F̄ is an algebraic closure of F, then G_{F} ≅ Gal(F̄/F) ≅ Ẑ (the latter isomorphism sending the Frobenius automorphism in Gal(F̄/F), x ↦ x^{|F|}, to 1 ∈ Ẑ).

Let T be an R-module endowed with a continuous G_{K}-action. By H^{∗}(K, T) = H^{∗}(G_{K}, T) we mean cohomology computed with respect to continuous cochains. If K is nonarchimedean, the vanishing of H^{2}(G_{F}, T^{I}) yields the canonical exact sequence

0 ⟶ H^{1}(G_{F}, T^{I}) ⟶ H^{1}(K, T) ⟶ H^{1}(I, T)^{G_{F}} ⟶ 0.  (2)

1. Local conditions

Definition 1.1. A local condition on T (over K) is a choice of an R-submodule of H^{1}(K, T). If we refer to the local condition by a symbol, say F, we will denote the corresponding R-submodule

H_{F}^{1}(K, T) ⊂ H^{1}(K, T).

If T is a category whose objects are certain R-modules endowed with continuous R-linear G_{K}-action (and possibly equipped with further structure) and whose morphisms are certain R-module homomorphisms, a local condition functorial over T will mean a subfunctor (of one-dimensional G_{K}-cohomology)

T ↦ H_{F}^{1}(K, T) ⊂ H^{1}(K, T)

where T ranges through the objects of the category T .

Example 1.2. Suppose T is an R[[G_{K}]]-module as above, and F is a local condition on T. If T′ is a submodule of T and T′′ is a quotient module, then F induces local conditions (also denoted by F) on T′ and T′′, by taking H_{F}^{1}(K, T′) and H_{F}^{1}(K, T′′) to be the inverse image and image, respectively, of H_{F}^{1}(K, T) in H^{1}(K, T′) and H_{F}^{1}(K, T′′) under the maps induced by

T′ ↪ T, T ↠ T′′.

In other words the local condition F “propagates” to submodules and quotients of T.

In particular if I is an ideal of R, then a local condition on T induces local conditions on T/IT and T [I].

One can similarly propagate the local condition F to arbitrary subquotients of T. Namely, if T_{1} ⊂ T_{2} ⊂ T, then one can define a local condition on T_{2}/T_{1} either by viewing it as a quotient of the submodule T_{2} of T, or as a submodule of the quotient T/T_{1} of T. It is an exercise to show that these two choices define the same local condition on the subquotient T_{2}/T_{1}.

Similarly, if R → R′ is a homomorphism of complete noetherian local rings, then F induces a local condition on the R′-module T ⊗_{R} R′ by taking H_{F}^{1}(K, T ⊗ R′) to be the image in H^{1}(K, T ⊗ R′) of H_{F}^{1}(K, T) ⊗ R′.

Example 1.3. Let Quot_{R}(T) be the category whose objects are quotients T/IT for all ideals I of R, and where the morphisms from T/IT to T/JT consist of all scalar multiplications r such that rI ⊂ J. It is immediate that the local condition F, propagated to the category Quot_{R}(T), is functorial over Quot_{R}(T).

Similarly one can define categories of submodules and subquotients of T, and again the propagated local conditions are functorial.

Definition 1.4. A local condition F is cartesian on a category T of R[[G_{K}]]- modules (or on a category of such modules equipped with some extra structure) if F is functorial over T and for any injective R[[G_{K}]]-module homomorphism (in the category T ) α : T_{1} → T_{2} the square

H_{F}^{1}(K, T_{1}) ↪ H^{1}(K, T_{1})
α↓  ↓^{α}
H_{F}^{1}(K, T_{2}) ↪ H^{1}(K, T_{2})

is cartesian.

Equivalently, F is cartesian on T if whenever α : T_{1} → T_{2} is injective, the local condition F on T_{1} is the same as the local condition obtained by propagating F from T_{2} to T_{1}.

Definition 1.5. Most (but not all) of the local conditions we deal with in this paper will be of the following form. Suppose L is a (possibly infinite) extension of K in K̄, and define

H_{L}^{1}(K, T) = H^{1}(L/K, T^{G_{L}}) = ker(H^{1}(K, T) → H^{1}(L, T)) ⊂ H^{1}(K, T).

It is clear that for every L this defines a local condition functorial over any category of R[[G_{K}]]-modules.

We have the following important special cases.

(i) the unrestricted or relaxed condition (L = K̄)

H_{relaxed}^{1}(K, T) = H1 ; K̄(K, T) = H^{1}(K, T),

(ii) the strict condition (L = K)

H_{strict}^{1}(K, T) = H_{K}^{1}(K, T) = 0 ⊂ H^{1}(K, T),

(iii) the unramified condition (L = K^{unr})

H_{unr}^{1}(K, T) = H_{K^{unr}}^{1}(K, T) = H^{1}(G_{F}, T^{I}) ⊂ H^{1}(K, T).

If K is nonarchimedean and T is unramified (i.e., I acts trivially on T), we will also call this the finite condition and write H_{f}^{1}(K, T) = H_{unr}^{1}(K, T) = H^{1}(G_{F}, T).

(iv) the L-transverse condition (K nonarchimedean, and L/K totally ramified and abelian of degree |F^{×}|)

H_{L−tr}^{1}(K, T) = H_{L}^{1}(K, T) ⊂ H^{1}(K, T).

Suppose now that K is nonarchimedean and T is unramified. We will call the R-submodule H_{f}^{1}(K, T) the finite part of H^{1}(K, T), and we call the quotient H_{s}^{1}(K, T) = H^{1}(K, T)/H_{f}^{1}(K, T) ≅ H^{1}(I, T)^{G_{F}} the singular quotient of H^{1}(K, T). The exact sequence (2) thus can be written in this case as

0 ⟶ H_{f}^{1}(K, T) ⟶ H^{1}(K, T) ⟶ H_{s}^{1}(K, T) ⟶ 0.  (3)

If c ∈ H^{1}(K, T) its image under projection to H_{s}^{1}(K, T) will be denoted c_{s} ∈ H_{s}^{1}(K, T).

Remark 1.6. The unrestricted, strict, and unramified local conditions can be expressed as étale one-dimensional cohomology over Spec O of natural sheaf-theoretic extensions to Spec O of the G_{K}-module T viewed as an étale sheaf over Spec K. This does not seem to be the case with the transverse conditions.

Remark 1.7. In general the local condition defined by an extension L of K will not be cartesian, nor will the propagation of the L-condition on a module T to its subquotients (see Example 1.2) coincide with the L-condition on the subquotients. However, for the finite condition we have the following lemma.

Lemma 1.8. (i) Suppose T is an unramified R[[G_{K}]]-module and T′ is a subquotient of T. Then the local condition induced on T′ by the finite condition on T is the same as the finite condition on T′,

(ii) The finite condition is cartesian on any category of unramified R[[G_{K}]]- modules.

Proof. Suppose T_{1} → T_{2} is a map of unramified R[[G_{K}]]-modules. Since they are unramified, the definition of H_{f}^{1} gives a diagram

0 ⟶ H_{f}^{1}(K, T_{1}) ⟶ H^{1}(K, T_{1}) ⟶^{d_{1}} Hom(I, T_{1})
a↓  ^{b}↓  ^{c}↓
0 ⟶ H_{f}^{1}(K, T_{2}) ⟶ H^{1}(K, T_{2}) ⟶^{d_{2}} Hom(I, T_{2}).

If T_{1} ↠ T_{2} is surjective, then coker(a) = H^{2}(K^{unr}/K, ker(T_{1} → T_{2})) = 0, and so H_{f}^{1}(K, T_{2}) is the image of H_{f}^{1}(K, T_{1}) in H^{1}(K, T_{2}), as desired.

If T_{1} ↪ T_{2} is injective, then c is injective, so b^{−1}(ker(d_{2})) = ker(d_{1}), i.e., the inverse image of H_{f}^{1}(K, T_{2}) in H^{1}(K, T_{1}) is H_{f}^{1}(K, T_{1}).

This proves (i), and (ii) is just (i) applied to the case of submodules. □

2. The finite/singular homomorphism

Suppose for this section that the local field K is nonarchimedean, and has residue characteristic ℓ ≠ p. Suppose also that the R-module T is of finite type, the action of G_{K} on T is unramified, and

|F^{×}| · T = 0.

Lemma 2.1. There are canonical functorial isomorphisms (i) H_{f}^{1}(K, T) ≅ T/(Fr − 1)T,

(ii) H_{s}^{1}(K, T) ≅ Hom(I, T^{Fr=1}) and H_{s}^{1}(K, T) ⊗ F^{×} ≅ T^{Fr=1}.

Proof. This is well known; see for example [Ru6] Lemma 1.3.2. The isomorphism of (i) is induced by evaluating cocycle classes in H^{1}(G_{F}, T) on Frobenius. For (ii), the Hochschild-Serre spectral sequence gives an isomorphism

H_{s}^{1}(K, T) ≅ Hom(I, T)^{Fr=1}.

Since |F^{×}| · T = 0 and I/|F^{×}|I is canonically isomorphic to F^{×}, we also have

Hom(I, T)^{Fr=1} = Hom(I/|F^{×}|I, T)^{Fr=1} = Hom(F^{×}, T)^{Fr=1} = Hom(F^{×}, T^{Fr=1})

and (ii) follows. □

Definition 2.2. Suppose that T is free of finite rank as an R-module, and that det(1 − Fr | T) = 0. Define P (x) ∈ R[x] by

P (x) = det(1 − Fr x | T).

Since P (1) = 0, there is a unique polynomial Q(x) ∈ R[x] such that

(x − 1)Q(x) = P (x) in R[x].

By the Cayley-Hamilton theorem, P (Fr^{−1}) annihilates T, so Q(Fr^{−1})T ⊂ T^{Fr=1}. We define the finite-singular comparison map ϕ_{ℓ}^{fs} on T to be the composition, using the isomorphisms of Lemma 2.1,

H_{f}^{1}(K, T) ⟶^{∼} T/(Fr − 1)T ⟶^{Q(Fr^{−1})} T^{Fr=1} ⟶^{∼} H_{s}^{1}(K, T) ⊗ F^{×}.

Lemma 2.3. Suppose that R is artinian, |F^{×}| · R = 0, and T is free of finite rank over R. Suppose further that T/(Fr − 1)T is a free R-module of rank one. Then det(1 − Fr | T) = 0 and the maps

Q(Fr^{−1}) : T/(Fr − 1)T ⟶ T^{Fr=1}, ϕ^{fs} : H_{f}^{1}(K, T) ⟶ H_{s}^{1}(K, T) ⊗ F^{×}

of Definition 2.2 are isomorphisms. In particular both H_{f}^{1}(K, T) and H_{s}^{1}(K, T) are free of rank one over R.

Proof. Choose an R-basis {x_{1}, …, x_{d}} of T such that {x_{2}, …, x_{d}} is a basis of (Fr − 1)T. With this basis it is clear that det(1 − Fr | T) = 0.

That ϕ^{fs} is an isomorphism will follow immediately once we show that Q(Fr^{−1}) is an isomorphism, and then the fact that H_{f}^{1}(K, T) and H_{s}^{1}(K, T) are free of rank one will follow from Lemma 2.1(i) and (ii).

It remains to show that Q(Fr^{−1}) is an isomorphism. When R is a field this is Corollary A.2.6 of [Ru6]. Applying that case to the R/𝔪-module T/𝔪T and using Nakayama’s Lemma we see that Q(Fr^{−1}) is surjective. The exact sequence

0 ⟶ T^{Fr=1} ⟶ T ⟶^{Fr−1} T ⟶ T/(Fr − 1)T ⟶ 0

shows that T/(Fr − 1)T and T^{Fr=1} have the same length, so Q(Fr^{−1}) is an isomorphism. □

Now fix an abelian extension L/K which is totally and tamely ramified, and moreover is a maximal such (abelian, totally, tamely ramified) extension of K. There is a natural isomorphism Gal(L/K) ≅ F^{×}. (When K = Q_{ℓ} we will take L = K(μ_{ℓ}).)

Since L is fixed we will suppress it from the notation and write simply H_{tr}^{1} for the local condition H_{L−tr}^{1} of Definition 1.5(iv).

Lemma 2.4. The transverse subgroup H_{tr}^{1}(K, T) ⊂ H^{1}(K, T) projects isomorphically to H_{s}^{1}(K, T) in (3). In other words, (3) has a functorial splitting (depending on L)

H^{1}(K, T) = H_{f}^{1}(K, T) ⊕ H_{tr}^{1}(K, T).

Proof. Since L/K is totally ramified and T is unramified, T^{G_{L}} = T^{G_{K}} = T^{Fr=1}. Hence we have a commutative diagram

\xymatrix{H_{tr}^{1}(K, T) \ar@{^{(}->}[r] \ar[d]_{∼} & H^{1}(K, T) \ar@{->>}[rr] &  & H_{s}^{1}(K, T) \ar[r]^{∼} & Hom(I, T^{Fr=1}) \\ H^{1}(L/K, T^{G_{L}}) \ar[rr]^{∼} &  & Hom(Gal(L/K), T^{Fr=1}) \ar[rr]^{∼} &  & Hom(I/|F^{×}|I, T^{Fr=1}) \ar[u]^{∼}}

in which the all maps marked as such are isomorphisms (the upper right map by Lemma 2.1(ii)). This proves the lemma. □

3. Local duality

Definition 3.1. Define the dual of T to be the R[[G_{K}]]-module

T^{∗} = Hom(T, μ_{p^{∞}}).

We have the (perfect) local Tate cup product pairing

⟨ , ⟩ : H^{1}(K, T) × H^{1}(K, T^{∗}) ⟶ H^{2}(K, μ_{p^{∞}}) ⟶ Q_{p}/Z_{p}.

A local condition F for T determines a local condition F^{∗} for T^{∗}, by taking H_{F^{∗}}^{1}(K, T^{∗}) to be the orthogonal complement of H_{F}^{1}(K, T) under the Tate pairing ⟨ , ⟩.

Clearly the dual of the unrestricted local condition in the strict condition, and the dual of the strict condition is the unrestricted condition. The next proposition says that (on suitable modules T) the finite and transverse conditions are self-dual.

Proposition 3.2. Suppose K is nonarchimedean of residue characteristic different from p, and T is unramified.

(i) H_{f}^{1}(K, T) and H_{f}^{1}(K, T^{∗}) are orthogonal complements under ⟨ , ⟩.

(ii) If |F^{×}| · T = 0 and L/K is a totally ramified abelian extension of degree |F^{×}|, then H_{tr}^{1}(K, T) and H_{tr}^{1}(K, T^{∗}) are orthogonal complements under ⟨ , ⟩.

Proof. The first assertion is (for example) Theorem I.2.6 of [Mi]. Assertion (ii) will follow from (i) and Lemma 2.4 once we show that H_{tr}^{1}(K, T) and H_{tr}^{1}(K, T^{∗}) are orthogonal.

Suppose first that R = Z_{p} and T = Z/p^{k}Z (with trivial G_{K}-action) with p^{k} dividing |F^{×}|. Then μ_{p^{k}} ⊂ K^{×} and we can identify

H_{tr}^{1}(K, T) = Hom(Gal(L/K), Z/p^{k}Z) ≅ Hom(K^{×}/N_{L/K}L^{×}, Z/p^{k}Z),

H_{tr}^{1}(K, T^{∗}) = Hom(Gal(L/K), μ_{p^{k}}) ≅ Hom(ker(K^{×}/(K^{×})^{p^{k}} → L^{×}/(L^{×})^{p^{k}}), μ_{p^{k}})

by class field theory and Kummer theory, respectively, and the cup product pairing is induced by the natural pairing

K^{×} × Hom(K^{×}, Z/p^{k}Z) ⟶ Z/p^{k}Z.

Suppose α ∈ ker(K^{×} → L^{×}/(L^{×})^{p^{k}}), say α = β^{p^{k}} with β ∈ L^{×}. Then a simple computation shows that N_{L/K}β = α^{|F^{×}|/p^{k}}. Since K^{×}/N_{L/K}L^{×} is cyclic of order |F^{×}|, α is divisible by p^{k} in K^{×}/N_{L/K}L^{×} and so α is sent to zero by every element of Hom(K^{×}/N_{L/K}L^{×}, Z/p^{k}Z). This proves (ii) in this case.

In general, since T is unramified and L/K is totally ramified we have T^{G_{L}} = T^{G_{K}}, (T^{∗})^{G_{L}} = (T^{∗})^{G_{K}}, and hence

H_{tr}^{1}(K, T) = H^{1}(L/K, T^{G_{L}}) = H^{1}(L/K, T^{G_{K}}) = H_{tr}^{1}(K, T^{G_{K}}),
H_{tr}^{1}(K, T^{∗}) = H^{1}(L/K, (T^{∗})^{G_{L}}) = H^{1}(L/K, (T^{∗})^{G_{K}}) = H_{tr}^{1}(K, (T^{G_{K}})^{∗}).

Writing T^{G_{K}} ≅ ⊕Z/p^{k_{i}}Z, the general case of (ii) follows from the case T = Z/p^{k}Z above. □

Example 3.3. Suppose F is a local condition on T and I is an ideal of R. There are two ways to induce a local condition on T^{∗}[I]: we can induce F to the quotient T/IT and then to the dual (T/IT)^{∗} = T^{∗}[I], or we can induce F to the dual T^{∗} and then to the submodule T^{∗}[I]. It is an easy exercise to show that these give the same local condition on T^{∗}[I].

Part II. Global cohomology groups and Selmer structures

For the rest of this paper, T will be a finitely generated R-module with a continuous action of G_{Q}, which is unramified outside a finite set of rational primes.

Global notation. Let Q̄ ⊂ C be the algebraic closure of Q in C, and for each rational prime ℓ fix an algebraic closure \overline{Q_{ℓ}} of Q_{ℓ} containing Q̄. If ℓ = ∞, then Q_{ℓ} = R and we take \overline{Q_{ℓ}} = C. Let D_{ℓ} = Gal(\overline{Q_{ℓ}}/Q_{ℓ}), which we identify with a closed subgroup of G_{Q} = Gal(Q̄/Q). In other words D_{ℓ} is a particular decomposition group at ℓ in G_{Q}, and H^{1}(D_{ℓ}, T) = H^{1}(Q_{ℓ}, T). Let I_{ℓ} ⊂ D_{ℓ} be the inertia group, and Fr_{ℓ} ∈ D_{ℓ}/I_{ℓ} the Frobenius element. If T is unramified at ℓ, then D_{ℓ}/I_{ℓ} acts on T, and hence so does Fr_{ℓ}. If we choose a different decomposition group at ℓ, then the action of Fr_{ℓ} changes by conjugation in G_{Q}.

The transverse submodule of H^{1}(Q_{ℓ}, T) is the submodule

H_{tr}^{1}(Q_{ℓ}, T) = H^{1}(Q_{ℓ}(μ_{ℓ})/Q_{ℓ}, T^{D_{ℓ}})

of Definition 1.5(iv) with L = Q_{ℓ}(μ_{ℓ}).

If c ∈ H^{1}(Q, T) we will write c_{ℓ} for the image of c under the localization map H^{1}(Q, T) → H^{1}(Q_{ℓ}, T). If further ℓ ≠ p, T is unramified at ℓ, and (ℓ − 1)T = 0, then we write c_{ℓ} = c_{ℓ,f} + c_{ℓ,tr}, where c_{ℓ,f} and c_{ℓ,tr} are the projections of c_{ℓ} under the decomposition H^{1}(Q_{ℓ}, T) ≅ H_{f}^{1}(Q_{ℓ}, T) ⊕ H_{tr}^{1}(Q_{ℓ}, T) of Lemma 2.4.

4. Selmer modules

Definition 4.1. A Selmer structure F on T is a collection of the following data:

• a finite set Σ(F) of places of Q, including ∞, p, and all primes where T is ramified,

• for every ℓ ∈ Σ(F) (including ℓ = ∞), a local condition (in the sense of Definition 1.1) on T viewed as an R[[D_{ℓ}]]-module, i.e., a choice of Rsubmodule

H_{F}^{1}(Q_{ℓ}, T) ⊂ H^{1}(Q_{ℓ}, T).

If ℓ ∉ Σ(F) we will sometimes write H_{F}^{1}(Q_{ℓ}, T) = H_{f}^{1}(Q_{ℓ}, T).

Let SS(T) denote the set of Selmer structures on T.

If F is a Selmer structure, we define the Selmer module H_{F}^{1}(Q, T) ⊂ H^{1}(Q, T) to be the kernel of the sum of restriction maps

H^{1}(Q_{Σ(F)}/Q, T) ⟶ ⨁_{ℓ∈Σ(F)} (H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T))

where Q_{Σ(F)} denotes the maximal extension of Q which is unramified outside Σ(F). In other words, H_{F}^{1}(Q, T) consists of all classes which are unramified (or equivalently, finite) outside of Σ(F) and which locally at ℓ belong to H_{F}^{1}(Q_{ℓ}, T) for every ℓ ∈ Σ(F).

There is a natural partial ordering on the set SS(T). Namely, we can say that F ≤ F′ if and only if

• H_{F}^{1}(Q_{ℓ}, T) ⊂ H_{F′}^{1}(Q_{ℓ}, T) if ℓ ∈ Σ(F) ∩ Σ(F′),
• H_{F}^{1}(Q_{ℓ}, T) ⊂ H_{f}^{1}(Q_{ℓ}, T) if ℓ ∈ Σ(F) − Σ(F′),
• H_{F′}^{1}(Q_{ℓ}, T) ⊃ H_{f}^{1}(Q_{ℓ}, T) if ℓ ∈ Σ(F′) − Σ(F).

If F ≤ F′ then we clearly have H_{F}^{1}(Q, T) ⊂ H_{F′}^{1}(Q, T).

For examples of classically important Selmer structures and Selmer modules, see §20 and §21.

Example 4.2. Suppose that F is a Selmer structure and Σ′ is a finite set of primes containing Σ(F). We can extend F to F′ with Σ(F′) = Σ′ by taking H_{F′}^{1}(Q_{ℓ}, T) = H_{f}^{1}(Q_{ℓ}, T) for ℓ ∈ Σ′ − Σ(F). Then F ≤ F′, F′ ≤ F, and H_{F}^{1}(Q, T) = H_{F′}^{1}(Q, T), so we will identify F and F′ inside SS(T).

Example 4.3. A Selmer structure F on T induces a Selmer structure (also denoted by F) on every subquotient T′ of T as follows. Keep the same set Σ(F), and for ℓ ∈ Σ(F) take H_{F}^{1}(Q_{ℓ}, T′) be the local condition on T′ induced by the one on T (as in Example 1.2). This construction defines a map SS(T) → SS(T′).

In particular if I is an ideal of R, then a Selmer structure on T induces Selmer structures on T/IT and T [I].

Similarly, if R → R′ is a homomorphism of complete noetherian local rings, then there is a natural map SS(T) → SS(T ⊗_{R} R′): a Selmer structure F on the R-module T induces a Selmer structure (denoted F ⊗ R′, or simply F) on the R′- module T ⊗_{R} R′ with the same set Σ(F) by taking H_{F}^{1}(Q_{ℓ}, T ⊗ R′) to be the image in H^{1}(Q_{ℓ}, T ⊗ R′) of H_{F}^{1}(Q_{ℓ}, T) ⊗ R′, for ℓ ∈ Σ(F).

Example 4.4. Suppose F is a Selmer structure and ℓ is a prime not in Σ(F). There are several natural ways to extend F to a new Selmer structure F′ with Σ(F′) = Σ(F)∪{ℓ}, which will be important in what follows. We can take H_{F′}^{1}(Q_{ℓ}, T) to be 0, or all of H^{1}(Q_{ℓ}, T). If we take H_{F′}^{1}(Q_{ℓ}, T) = H_{f}^{1}(Q_{ℓ}, T) then we have identified F′ with F (see Example 4.2). Finally, if we can take H_{F′}^{1}(Q_{ℓ}, T) = H_{tr}^{1}(Q_{ℓ}, T).

We can repeat these constructions with several primes, and we will use the following notation. If a, b, c ∈ Z^{+} are relatively prime, and c is not divisible by any primes in Σ(F), we write F_{a}^{b}(c) for the Selmer structure defined by

• Σ(F_{a}^{b}(c)) = Σ(F) ∪ {ℓ : ℓ | abc},

• H_{F_{a}^{b}(c)}^{1}(Q_{ℓ}, T) = \begin{cases}H_{F}^{1}(Q_{ℓ}, T) & if ℓ ∈ Σ(F) and ℓ ∤ ab \\ 0 & if ℓ | a, \\ H^{1}(Q_{ℓ}, T) & if ℓ | b, \\ H_{tr}^{1}(Q_{ℓ}, T) & if ℓ | c. \end{cases}

In other words, F_{a}^{b}(c) consists of F together with the strict condition at primes dividing a, the unrestricted condition at primes dividing b, and the transverse condition at primes dividing c.

If any of a, b, c are equal to 1, we will suppress them from the notation. If a′ | a, b | b′, and c = c′ then F_{a}^{b}(c) ≤ F_{a′}^{b′}(c′), and otherwise there will be in general no order relation between these Selmer structures. In particular for every n we have F_{n} ≤ F ≤ F^{n} and F_{n} ≤ F (n) ≤ F^{n}.

Definition 4.5. Suppose now that T is free over R, ℓ ≠ p, ∞ is prime, and T is unramified at ℓ. Let P_{ℓ}(x) = det(1 − Fr_{ℓ} x | T) ∈ R[x], and let I_{ℓ} be the ideal of R generated by ℓ − 1 and P_{ℓ}(1).

Let P denote a set of rational primes, disjoint from Σ(F). Generally P will be infinite; for example it could be the set of all primes not in Σ(F). Let N = N (P) ⊂ Z^{+} denote the set of squarefree products of primes in P (with the convention that 1 ∈ N). If n ∈ N let

I_{n} = ∑_{ℓ|n} I_{ℓ} ⊂ R

and let

G_{n} = ⨂_{ℓ|n}F_{ℓ}^{×} = ⨂_{ℓ|n}Gal(Q(μ_{ℓ})/Q).

Since each Gal(Q(μ_{ℓ})/Q) is cyclic of order ℓ − 1, we see that G_{n} ⊗ (R/I_{n}) is free of rank one over R/I_{n}. By convention, we set G_{1} = Z.

If ℓ is a prime dividing n, then I_{ℓ} annihilates T/I_{n}T, so we can apply the results of §2 to the local cohomology group H^{1}(Q_{ℓ}, T/I_{n}T). In particular we will write

ϕ_{ℓ}^{fs} : H_{f}^{1}(Q_{ℓ}, T/I_{n}T) ⟶ H_{s}^{1}(Q_{ℓ}, T/I_{n}T) ⊗ G_{ℓ}

for the finite-singular map of Definition 2.2 with K = Q_{ℓ}.

Example 4.6. Suppose P is a set of primes as above and n ∈ N. Recall that the Selmer structure F (n) is the Selmer structure F modified by replacing the finite local condition at primes dividing n by the transverse local condition. In what follows we will be interested in the modules H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} for n ∈ N.

If ℓ is a prime and nℓ ∈ N, then we can compare H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} and H_{F (nℓ)}^{1}(Q, T/I_{nℓ}T) ⊗ G_{nℓ} by localizing at ℓ and using the finite-singular map ϕ_{ℓ}^{fs}:

H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n}
loc_{ℓ}↓
H_{f}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{n}
ϕ_{ℓ}^{fs}⊗1↓
H_{F (nℓ)}^{1}(Q, T/I_{nℓ}T) ⊗ G_{nℓ} ⟶^{loc_{ℓ}} H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ}.

(4)

5. Global duality

Definition 5.1. The dual of T is the R[G_{Q}]-module

T^{∗} = Hom(T, μ_{p^{∞}}).

For every prime ℓ ≤ ∞ we have the local Tate pairing

⟨ , ⟩_{ℓ} : H^{1}(Q_{ℓ}, T) × H^{1}(Q_{ℓ}, T^{∗}) ⟶ Q_{p}/Z_{p}

as in §3. If c ∈ H^{1}(Q, T) and d ∈ H^{1}(Q, T^{∗}) we will write ⟨c, d⟩_{ℓ} for ⟨c_{ℓ}, d_{ℓ}⟩_{ℓ}.

Just as every local condition on T determines a local condition on T^{∗} (Definition 3.1), a Selmer structure F for T determines a Selmer structure F^{∗} for T^{∗}. Namely, take Σ(F^{∗}) = Σ(F), and for ℓ ∈ Σ(F) take H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}) to be the local condition induced by F: the orthogonal complement of H_{F}^{1}(Q_{ℓ}, T) under ⟨ , ⟩_{ℓ}. If F ≤ F′ then (F′)^{∗} ≤ F^{∗}.

Example 5.2. By Proposition 3.2, the dual F_{a}^{b}(c)^{∗} of the Selmer structure F_{a}^{b}(c) is (F^{∗})_{a}^{b}(c).

Theorem 5.3. Suppose that G_{1}, G_{2} are Selmer structures and G_{1} ≤ G_{2}. (i) There are exact sequences

0 ⟶ H_{G_{1}}^{1}(Q, T) ⟶ H_{G_{2}}^{1}(Q, T) ⟶^{loc_{G_{1}}^{G_{2}}} ⨁_{ℓ}H_{G_{2}}^{1}(Q_{ℓ}, T)/H_{G_{1}}^{1}(Q_{ℓ}, T),

0 ⟶ H_{G_{2}^{∗}}^{1}(Q, T^{∗}) ⟶ H_{G_{1}^{∗}}^{1}(Q, T^{∗}) ⟶^{loc_{G^{∗}_{2}}^{G^{∗}_{1}}} ⨁_{ℓ}H_{G_{1}^{∗}}^{1}(Q_{ℓ}, T^{∗})/H_{G_{2}^{∗}}^{1}(Q_{ℓ}, T^{∗})

where the sums ore over primes ℓ such that H_{G_{2}}^{1}(Q_{ℓ}, T) ≠ H_{G_{1}}^{1}(Q_{ℓ}, T), and the maps loc_{G_{1}}^{G_{2}}, loc_{G_{2}^{∗}}^{G_{1}^{∗}} are the natural localization maps.

(ii) The images loc_{G_{1}}^{G_{2}} and loc_{G_{2}^{∗}}^{G_{1}^{∗}} are orthogonal complements with respect to the pairing ∑_{ℓ} ⟨ , ⟩_{ℓ}.

Proof. Assertion (i) is immediate from the definition of these Selmer groups. The second statement is Poitou-Tate global duality; see for example [T] Theorem 3.1 or [Mi] Theorem I.4.10 (see also [Ru6] Theorem 1.7.3). □

Corollary 5.4. Suppose that R is artinian and T is free of finite rank over R. Suppose n ∈ Z^{+} is not divisible by any primes in Σ(F), and further that every prime ℓ dividing n satisfies the hypotheses of Lemma 2.3: I_{ℓ} = 0 and T/(Fr_{ℓ} − 1)T is free of rank one over R. Then

length(H_{F}^{1}(Q, T)) − length(H_{F^{∗}}^{1}(Q, T^{∗}))

= length(H_{F (n)}^{1}(Q, T)) − length(H_{F (n)^{∗}}^{1}(Q, T^{∗})).

Proof. Recall that ν(n) is the number of primes dividing n. Applying Theorem 5.3 with G_{1} = F and G_{2} = F^{n} we see that

length(H_{F^{n}}^{1}(Q, T)) − length(H_{F}^{1}(Q, T))

+ length(H_{F^{∗}}^{1}(Q, T^{∗})) − length(H_{(F^{n})^{∗}}^{1}(Q, T^{∗}))

= ∑_{ℓ|n} length(H_{s}^{1}(Q_{ℓ}, T)) = ν(n)length(R) the final equality by Lemma 2.3. Similarly, applying Theorem 5.3 with G_{1} = F (n) and G_{2} = F^{n} we see that

length(H_{F^{n}}^{1}(Q, T)) − length(H_{F (n)}^{1}(Q, T))

+ length(H_{F (n)^{∗}}^{1}(Q, T^{∗})) − length(H_{(F^{n})^{∗}}^{1}(Q, T^{∗}))

= ∑_{ℓ|n} length(H_{f}^{1}(Q_{ℓ}, T)) = ν(n)length(R).

Subtracting this from the previous equality proves the corollary. □

Part III. Kolyvagin systems

We assume for the rest of this paper that T is free of finite rank over R, in addition to being a G_{Q}-module which is unramified outside finitely many primes.

A Selmer triple is a triple (T, F, P) where F is a Selmer structure on T and P is a set of rational primes, disjoint from Σ(F).

6. Kolyvagin systems

Definition 6.1. If X is a simplicial complex, and C a category, a simplicial sheaf S on X with values in C is a rule assigning:

• to each simplex σ in X, an object S(σ) of C, and

• to each pair (σ, τ) of simplices of X such that σ is a face of τ (of codimen-

sion one), a morphism S(σ) → S(τ) in the category C. Note that such a simplicial sheaf gives rise to a sheaf on the topological realization of X which is locally constant on the open stars of simplices of X, and whose set of sections on the open star of a simplex σ is just S(σ).

In this paper we will only be concerned with the case where X is a graph. In that case, if V is the set of vertices of X and E is the set of edges, a simplicial sheaf S on X with values in the category of R-modules is a collection of the following data:

• an R-module S(v) (the stalk of X at v) for every vertex v ∈ V,

• an R-module S(e) for every edge e ∈ E,

• an R-module map ψ_{v}^{e} : S(v) → S(e) whenever the vertex v is an endpoint

of the edge e. A global section of S is a collection {κ_{v} ∈ S(v) : v ∈ V } such that for every edge e ∈ E, if e has endpoints v, v′ then ψ_{v}^{e}(κ_{v}) = ψ_{v′}^{e}(κ_{v′}) in S(e). We write Γ(S) = S(X) for the R-module of global sections of S.

Definition 6.2. Suppose (T, F, P) is a Selmer triple. We define a graph X = X (P) by taking the set of vertices of X to be N = N (P), and whenever n, nℓ ∈ N (with ℓ prime) we join n and nℓ by an edge.

The Selmer sheaf associated to (T, F, P) is the simplicial sheaf H = H_{(T,F,P)} of R-modules on X defined as follows. Take

• H(n) = H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} for n ∈ N, and if e is the edge joining n and nℓ we take

• H(e) = H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ},

• ψ_{nl}^{e} : H_{F (nℓ)}^{1}(Q, T/I_{nℓ}T) ⊗ G_{nℓ} → H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ} is localization at ℓ followed by projection to H_{s}^{1}, • ψ_{n}^{e} : H_{F (n)}^{1}(Q, T/I_{n}T)⊗G_{n} ⟶ H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T)⊗G_{nℓ} is the composition of localization at ℓ with the map ϕ_{ℓ}^{fs} of Definition 2.2 (see Example 4.6)

H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} ⟶ H_{f}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{n} ⟶ H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ}.

We call H(n) = H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} the Selmer stalk at n.

Definition 6.3. A Kolyvagin system for (T, F, P) is a global section (over X (P)) of the Selmer sheaf H_{(T,F,P)}. We write KS(T, F, P), or simply KS(T) when there is no risk of confusion, for the R-module of Kolyvagin systems Γ(H).

Concretely, a Kolyvagin system for (T, F, P) is a collection of cohomology classes

{κ_{n} ∈ H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n} : n ∈ N }

such that if ℓ is prime and nℓ ∈ N,

(κ_{nℓ})_{ℓ,s} = ϕ_{ℓ}^{fs}(κ_{n}) in H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ}.  (5)

In other words, the images of κ_{n} and κ_{nℓ} in H_{s}^{1}(Q_{ℓ}, T/I_{nℓ}T) ⊗ G_{nℓ} coincide in the diagram (4)

For examples of Kolyvagin systems, see the next section.

Remark 6.4. Suppose κ is a Kolyvagin system and nℓ ∈ N. Then κ_{nℓ} ∈ H_{F (nℓ)}^{1}(Q, T/I_{nℓ}T), so in particular (κ_{nℓ})_{ℓ} ∈ H_{tr}^{1}(Q_{ℓ}, T/I_{n}T). Thus by Lemma 2.4 and (5), (κ_{nℓ})_{ℓ} is completely determined by (κ_{n})_{ℓ}.

Remark 6.5. The assignments(T, F, P) ↦ H_{(T,F,P)} and (T, F, P) ↦ KS(T, F, P) have the following functorial properties (where we suppress the indices which are not varying).

• Commutation with direct sums: There are natural isomorphisms

H_{(T_{1}⊕T_{2},F_{1}⊕F_{2})} ≅ H_{(T_{1},F_{1})} ⊕ H_{(T_{2},F_{2})},

KS(T_{1} ⊕ T_{2}, F_{1} ⊕ F_{2}) ≅ KS(T_{1}, F_{1}) ⊕ KS(T_{2}, F_{2}).

• Change of base ring: If R → R′ is a homomorphism of complete local noetherian rings we have a natural homomorphism of sheaves of R′-modules over X

H_{(T,F)} ⊗_{R} R′ ⟶ H_{(T ⊗_{R}R′,F⊗_{R}R′)},
KS(T, F) ⊗_{R} R′ ⟶ KS(T ⊗_{R} R′, F ⊗_{R} R′).

• Change of P: If P′ ⊂ P then H_{P′} is the restriction of the sheaf H_{P} to the subgraph X (P′) ⊂ X (P), and there is a natural map KS(P) → KS(P′).

• Change of F: If F′ ≤ F, and P is disjoint from Σ(F) ∪ Σ(F′), then H_{F′} is naturally a subsheaf of R-modules in H_{F}, and KS(F′) ⊂ KS(F).

Remark 6.6. The requirement that κ_{n} ∈ H_{F (nℓ)}^{1}(Q, T/I_{n}T) in the definition of a Kolyvagin system is stronger than what is needed for standard applications found in the literature (bounding the size of the Selmer groups of the dual G_{Q}-module T^{∗}). For those purposes (see for example [Ru6]) the following collections of classes will suffice.

Definition 6.7. Define a sheaf Ĥ of R-modules on X as follows. Take • Ĥ(n) = H_{F^{n}}^{1}(Q, T/I_{n}T) ⊗ G_{n} for n ∈ N, and define Ĥ(e) and the maps ψ_{nl}^{e}, ψ_{n}^{e} exactly as in Definition 6.2. Clearly H is a subsheaf of Ĥ.

A weak Kolyvagin system for (T, F, P) is a global section of the sheaf Ĥ. Concretely, a weak Kolyvagin system is a collection of cohomology classes

{κ_{n} ∈ H_{F^{n}}^{1}(Q, T/I_{n}T) ⊗ G_{n} : n ∈ N }

satisfying (5).

Remark 6.8. In a weak Kolyvagin system, (κ_{n})_{ℓ,s} is determined, for every prime ℓ dividing n, by the κ_{d} for d properly dividing n. Equivalently, κ_{n} is uniquely determined modulo H_{F}^{1}(Q, T/I_{n}T) by the κ_{d} for d properly dividing n.

In a Kolyvagin system (see Remark 6.6), κ_{n} is determined modulo H_{F}^{1}(Q, T) ∩ H_{F (n)}^{1}(Q, T) = H_{F_{n}}^{1}(Q, T/I_{n}T) by the κ_{d} for d properly dividing n, and for sufficiently divisible n this group will be zero.

The following example exhibits this lack of rigidity of weak Kolyvagin systems, as compared to Kolyvagin systems.

Example 6.9. Suppose p > 2, R = Z_{p} and T = Z_{p}(1). Kummer theory shows that H^{1}(Q, T) is the p-adic completion of Q^{×}. Define F by Σ(F) = {p, ∞}, H_{F}^{1}(Q_{p}, T) = H^{1}(Q_{p}, Z_{p}(1)), and H_{F}^{1}(R, T) = 0. Let P be the set of all primes different from p.

For every squarefree n prime to p, and every prime ℓ dividing n, we have a commutative diagram (the upper isomorphism is Kummer theory, and the lower isomorphism is Lemma 2.4)

H_{F^{n}}^{1}(Q, T/I_{n}T) ⟶^{∼} Z[1/(pn)]^{×} ⊗ (Z_{p}/I_{n})
↓  ↓^{ord}_{ℓ}
H_{s}^{1}(Q_{ℓ}, T/I_{n}T) ⟶^{∼}  Z_{p}/I_{n}.

We can build a weak Kolyvagin system inductively as follows.

Choose any κ_{1} ∈ H^{1}(Q, Z_{p}(1)). Next, suppose n is a positive squarefree integer prime to p, and we have chosen κ_{t} for every proper divisor t of n. Choose κ_{n} ∈ Z[1/(pn)]^{×}⊗(Z_{p}/I_{n})⊗G_{n} so that ord_{ℓ}(κ_{n}) = ϕ_{ℓ}^{fs}(κ_{n/ℓ}) in the diagram above. (This determines κ_{n} up to a power of p; take any such choice.) In the limit this process will construct a weak Kolyvagin system for T. Because of the choices involved at each step, this will produce infinitely many weak Kolyvagin systems κ with the same κ_{1}.

Most of the weak Kolyvagin systems constructed in this way will not be Kolyvagin systems. We will see in §18 that in this setting every choice of κ_{1} extends uniquely to a Kolyvagin system.

Example 6.10. Suppose H_{F}^{1}(Q, T/IT) = 0 for every ideal I ⊂ R. We will show that every weak Kolyvagin system for T must be identically zero. More precisely, if κ is a weak Kolyvagin system, we will show by induction on the number of primes dividing n that κ_{n} = 0 for every n ∈ N.

By definition, κ_{1} ∈ H_{F}^{1}(Q, T) = 0. Suppose that κ_{d} = 0 for every d properly dividing n. We need to show that κ_{n} = 0.

We have κ_{n} ∈ H_{F^{n}}^{1}(Q, T/I_{n}T) ⊗ G_{n}, but the coherence relations (5) show that (κ_{n})_{ℓ,s} = 0 for every prime ℓ dividing n. Hence κ_{n} ∈ H_{F}^{1}(Q, T/I_{n}T) ⊗ G_{n} = 0.

Example 6.11. Suppose κ ∈ KS(T, F, P) and n ∈ N. Let P(n) be the set of primes in P not dividing n. If ξ ∈ Hom(G_{n}, R/I_{n}) then the collection κ^{(n)} defined by

κ_{m}^{(n)} = κ_{nm} ⊗ ξ for m ∈ N prime to n

is a Kolyvagin system for (T/I_{n}T, F (n), P(n)). (We view ⊗ξ here as a map from G_{nm} ⊗(R/I_{n}) = G_{m} ⊗G_{n} ⊗(R/I_{n}) to G_{m} ⊗(R/I_{n}) for every m.) This construction defines a homomorphism

KS(T, F, P) ⊗ Hom(G_{n}, R/I_{n}) → KS(T/I_{n}T, F (n), P(n)).

(Recall that Hom(G_{n}, R/I_{n}) is a free, rank-one (R/I_{n})-module.)

7. Euler systems and Kolyvagin systems

Suppose for this section that R is the ring of integers of a finite extension of Q_{p}.

Definition 7.1. We define a canonical Selmer structure F_{can} on T by • Σ(F_{can}) = {ℓ : T is ramified at ℓ} ∪ {p, ∞},

• if ℓ ∈ Σ(F_{can}) and ℓ ≠ p, ∞ then
H_{F_{can}}^{1}(Q_{ℓ}, T) = ker(H^{1}(Q_{ℓ}, T) → H^{1}(Q_{ℓ}^{unr}, T ⊗ Q_{p})),
• H_{F_{can}}^{1}(Q_{p}, T) = H^{1}(Q_{p}, T),
• H_{F_{can}}^{1}(R, T) = H^{1}(R, T).

If I is an ideal of R we define the canonical Selmer structure on T/IT to be F_{can}⊗R/I, the Selmer structure induced by F_{can} on T/IT. (Note that this depends on T, not only on T/IT.) We will write simply F_{can} instead of F_{can} ⊗ R/I. It is not true in general that H_{F_{can}}^{1}(Q_{p}, T/IT) = H^{1}(Q_{p}, T/IT); see Lemma A.1.

Definition 7.2. Fix a set P of primes, different from p, where T is unramified (so (T, F_{can}, P) is a Selmer triple). Let Ξ denote the collection of all finite abelian extensions of Q which are unramified outside of P and p. An Euler system c for (T, P) is a collection

{c_{F} ∈ H^{1}(F, T) : F ∈ Ξ}

such that whenever F, F′ ∈ Ξ and F ⊂ F′,

N_{F′/F} c_{F′} = (∏ P_{ℓ}(Fr_{ℓ}^{−1})) c_{F}

where N_{F′/F} is the corestriction map from F′ to F, the product is over primes ℓ ≠ p which ramify in F′/Q but not in F/Q, and P_{ℓ}(x) = det(1 − Fr_{ℓ} x | T) as in Definition 4.5.

Let ES(T, P) (or simply ES(T), if there is no danger of confusion) denote the collection of Euler systems for (T, P). Then ES(T) is a Z_{p}[[G_{Q}]]-module.

Remark 7.3. The “Euler factors” P_{ℓ}(Fr_{ℓ}^{−1}) in Definition 7.2 are slightly different from the ones used in [Ru6] Definition 2.1.1. However, it is easy to switch back and forth between the two choices, and they give equivalent theories and isomorphic modules ES(T). See §9.6 of [Ru6]. We use the present choice here because it simplifies the formulas in the proof of Theorem 7.4.

Theorem 7.4. Suppose that (a) H^{0}(Q_{p}, T^{∗}) is a divisible R-module,

(b) T/(Fr_{ℓ} − 1)T is a cyclic R-module for every ℓ ∈ P,

(c) Fr_{ℓ}^{p^{k}} − 1 is injective on T for every ℓ ∈ P and every k ≥ 0.

Then there is a canonical homomorphism ES(T) → KS(T, F_{can}, P) which is G_{Q}- equivariant (with G_{Q} acting trivially on KS(T)) with the property that if c maps to κ, then κ_{1} = c_{Q}.

The proof of this theorem is given in Appendix A. If we let {κ_{n}} denote the collection of derivative classes of the Euler system c as defined for example in Chapter 4 of [Ru6], then the results of that chapter show that this collection gives a weak Kolyvagin system. A minor modification of these classes gives a Kolyvagin system. Since the proof is tedious and unrelated to the rest of this paper, we defer it to Appendix A.

Remark 7.5. Suppose ρ : G_{Q} → R^{×} is a character of finite order, unramified outside of P and p. Writing P_{ρ} = {ℓ ∈ P : ρ is unramified at ℓ}, and again writing F_{can} for the canonical Selmer structure on T ⊗ ρ, we get a new Selmer triple (T ⊗ ρ, F_{can}, P_{ρ}).

As in §2.4 of [Ru6], if we fix a generator of the free rank-one R-module ρ, we obtain a map

ES(T, P) → ES(T ⊗ ρ, F_{can}, P_{ρ})

with the property that if c ↦ c^{ρ} and L is the fixed field of the kernel of ρ, then the image of c_{Q}^{ρ} under the composition

H^{1}(Q, T ⊗ ρ) ⟶^{res} H^{1}(L, T ⊗ ρ) ≅ H^{1}(L, T)

is ∑_{δ∈Gal(L/Q)} ρ(δ)c_{L}^{δ}. Thus, using Theorem 7.4 (and increasing the ring R as necessary, to include the values of ρ), an Euler system for T gives rise to a Kolyvagin system for T ⊗ ρ for every character ρ of finite order unramified outside of P and p.

Remark 7.6. In our definition of Euler system above, we require that Ξ contains all finite extensions of Q contained in the cyclotomic Z_{p}-extension Q_{∞} of Q. In other words, the Euler system “extends in the p-direction”, and in particular each class c_{F} is a universal norm from F Q_{∞}.

As discussed in §9.1 of [Ru6], it is possible to remove this requirement if we replace it with some other appropriate condition. For example, define ES_{0}(T, P) as in Definition 7.2 but with Ξ replaced by the subcollection Ξ_{0}, of all finite abelian extensions of Q which are unramified outside of P, and with the extra assumption that c_{F} ∈ H_{F_{can}}^{1}(F, T) for F ∈ Ξ_{0}. The discussion of §9.1 of [Ru6], combined with the proof of Theorem 7.4 in Appendix A, gives the following.

Theorem 7.7. Suppose that H^{0}(Q_{p}, T^{∗}) is a divisible R-module, and that there is a γ ∈ G_{Q} such that γ − 1 kills μ_{p^{∞}} and γ − 1 is injective on T. Then there is a canonical homomorphism ES_{0}(T) → KS(T, F_{can}) with the property that if c maps to κ, then κ_{1} = c_{Q}.

For examples of Euler systems and the corresponding Kolyvagin systems, see §20 and §21.

8. Sheaves on graphs

Suppose for this section that S is a (simplicial) sheaf on a graph X, as in Definition 6.1.

Definition 8.1. If v and w are vertices of X, a path in X from v to w is a sequence of vertices (v = v_{1}, v_{2}, …, v_{k} = w) in X such that for each i, v_{i} and v_{i+1} are joined by an edge e_{i}. The graph X is connected if every pair of vertices v, w there is a path from v to w. A loop in X (at v) is a path from v to v.

Let r be a positive integer. We say that S is locally free of rank r if all the R-modules S(v), S(e) are free of rank r and all the maps ψ_{v}^{e} are isomorphisms. If S is locally free and P = (v_{1}, v_{2}, …, v_{k}) is a path in X, we can define an isomorphism

ψ_{P} : S(v_{1}) ⟶^{∼} S(v_{k}) by

ψ_{P} = (ψ_{v_{k}}^{e_{k−1}})^{−1} ∘ ψ_{v_{k−1}}^{e_{k−1}} ∘ (ψ_{v_{k−1}}^{e_{k−2}})^{−1} ∘ ⋯ ∘ (ψ_{v_{2}}^{e_{1}})^{−1} ∘ ψ_{v_{1}}^{e_{1}}.

We say that S has trivial monodromy if S is locally free and for every vertex v of X and every loop P at v, ψ_{P} is the identity map in Aut(S(v)).

Definition 8.2. We say that S is locally cyclic if all the R-modules S(v), S(e) are cyclic and all the maps ψ_{v}^{e} are surjective.

If S is locally cyclic then a surjective path (relative to S) from v to w is a path (v = v_{1}, v_{2}, …, v_{k} = w) in X such that for each i, if v_{i} and v_{i+1} are joined by the edge e_{i}, then ψ_{v_{i+1}}^{e_{i}} is an isomorphism. We say that the vertex v is a hub of S if for every vertex w there is an S-surjective path from v to w.

Suppose now that the sheaf S is locally cyclic. If P = (v_{1}, v_{2}, …, v_{k}) is a surjective path in X, we can define a surjective map ψ_{P} : S(v_{1}) → S(v_{k}) by

ψ_{P} = (ψ_{v_{k}}^{e_{k−1}})^{−1} ∘ ψ_{v_{k−1}}^{e_{k−1}} ∘ (ψ_{v_{k−1}}^{e_{k−2}})^{−1} ∘ ⋯ ∘ (ψ_{v_{2}}^{e_{1}})^{−1} ∘ ψ_{v_{1}}^{e_{1}}

since all the inverted maps are isomorphisms. We will say that S has trivial monodromy if whenever v, w, w′ are vertices, P, P′ are surjective paths (v, …, w) and (v, …, w′), and w, w′ are joined by an edge e, then ψ_{w}^{e} ∘ ψ_{P} = ψ_{w′}^{e} ∘ ψ_{P′} ∈ Hom(S(v), S(e)). In particular for every pair v, w of vertices and and every pair P, P′ of surjective paths from v to w, we require that ψ_{P} = ψ_{P′} ∈ Hom(S(v), S(w)).

Remark 8.3. If S is locally free of rank one, then it is also locally cyclic, every path is a surjective path, and the definitions of “trivial monodromy” in Definitions 8.1 and 8.2 are equivalent. If S is locally free of rank one and connected, then every vertex is a hub.

Recall that a global section of S is a collection of elements κ_{v} ∈ S(v) for every vertex v, which are compatible with respect to the maps ψ_{v}^{e}.

Proposition 8.4. Suppose S is locally cyclic and v is a hub of S.

(i) The map f_{v} : Γ(S) → S(v) defined by κ ↦ κ_{v} is injective, and is surjective if and only if S has trivial monodromy.

(ii) Γ(S) is (noncanonically) isomorphic to an ideal of R.

(iii) If κ ∈ Γ(S), and if u is a vertex such that κ_{u} ≠ 0 and κ_{u} generates 𝔪^{i}S(u) for some i ∈ Z^{+}, then κ_{w} generates 𝔪^{i}S(w) for every vertex w.

Proof. For every vertex w fix a surjective path P_{w} from v to w. If κ ∈ Γ(S) then κ_{w} = ψ_{P_{w}}(κ_{v}) for every w, so the map f_{v} of (i) is injective.

Now fix c ∈ S(v) and for every vertex w define κ_{w} = ψ_{P_{w}}(c). If S has trivial monodromy then this is independent of the choice of P_{w}, and defines a global section κ. Thus c is in the image of f_{v} and hence f_{v} is surjective as well.

Conversely, suppose f_{v} is surjective. Then for every c ∈ S(v) we can find κ ∈ Γ(S) with κ_{v} = c. If w, w′ are vertices connected by an edge e, then we must have

ψ_{w}^{e} ∘ ψ_{P_{w}}(c) = ψ_{w}^{e}(κ_{w}) = ψ_{w′}^{e}(κ_{w′}) = ψ_{w′}^{e} ∘ ψ_{P_{w′}} (c) ∈ S(e).

Thus S has trivial monodromy.

This proves (i), (ii) is immediate from (i), and (iii) follows from the surjectivity of the maps ψ_{P_{w}}. □

Definition 8.5. A global section κ ∈ Γ(S) will be called primitive if for every vertex v, κ(v) ∈ S(v) is a generator of the R-module S(v).

It follows from Proposition 8.4 that a locally cyclic sheaf S has a primitive global section if and only if S has trivial monodromy.

9. Hypotheses on T, F, and P

In this section we record and discuss several hypotheses which will play a role in the following sections. Fix a Selmer triple (T, F, P).

Consider the following properties:

(H.0) T is a free R-module of finite rank.

(H.1) T/𝔪T is an absolutely irreducible 𝕜[G_{Q}]-representation, and the image of

R → End_{Z_{p}}(T) is contained in the image of Z_{p}[[G_{Q}]] → End_{Z_{p}}(T). (H.2) There is a τ ∈ G_{Q} such that τ = 1 on μ_{p^{∞}} and T/(τ − 1)T is free of rank

one over R. (H.3) H^{1}(Q(T, μ_{p^{∞}})/Q, T/𝔪T) = H^{1}(Q(T, μ_{p^{∞}})/Q, T^{∗}[𝔪]) = 0. (H.4) Either

(H.4a) T/𝔪T is not isomorphic to T^{∗}[𝔪] as F_{p}[[G_{Q}]]-modules, or (H.4b) |𝕜| > 4.

(H.5) P_{t} ⊂ P ⊂ P_{1} for some t ∈ Z^{+}, where for k ∈ Z^{+}

P_{k} = {ℓ ∉ Σ(F) : dim_{𝕜}(T/(𝔪T + (Fr_{ℓ} − 1)T)) = 1 and I_{ℓ} ⊂ 𝔪^{k}}.

(H.6) For every ℓ ∈ Σ(F), the local condition F at ℓ is cartesian (in the sense of Definition 1.4) on the category Quot_{R}(T) of quotients of T of Example 1.3.

These hypotheses hold in many, but not all, cases of interest. See §20 and §21, especially Lemmas 20.4 and 21.2. We have already assumed that (H.0) holds.

Remark 9.1. Note that (H.2) holds trivially when rank(T) = 1, with τ = 1. The condition P ⊂ P_{1} in (H.5) is reasonably harmless because we can always replace P by P ∩ P_{1}. Condition (H.6) holds trivially when R is a field, because in that case category Quot_{R}(T) has only two objects, 0 and T.

Also, if R ↠ R′ is a surjective homomorphism of (complete) local rings and T satisfies (H.i), then so does T ⊗_{R} R′ viewed as an R′[[G_{Q}]]-module, for any i = 1, …, 4.

Condition (H.1) implies that T/𝔪T and T^{∗}[𝔪] are irreducible F_{p}[[G_{Q}]]-repre-sentations.

Lemma 9.2. Suppose that (H.1) and (H.3) hold. Then for every subquotient S of T or of T^{∗}, S^{G_{Q}} = 0.

Proof. It will suffice to prove the lemma for subquotients annihilated by 𝔪. Suppose S is a subquotient of T and 𝔪S = 0. Then every Jordan-Holder factor of the 𝕜[[G_{Q}]]-module S is also a Jordan-Holder factor of T/𝔪T. By (H.1), T/𝔪T is an irreducible 𝕜[[G_{Q}]]-module, so in fact S ≅ (T/𝔪T)^{d} for some d. It follows from (H.3) that (T/𝔪T)^{G_{Q}} = 0, and so S^{G_{Q}} = 0. The proof for subquotients of T^{∗} is the same. □

Lemma 9.3. Suppose that (H.1) and (H.3) hold, and I is an ideal of R. Then the inclusion T^{∗}[I] ↪ T^{∗} induces an isomorphism

H_{F^{∗}}^{1}(Q, T^{∗}[I]) ⟶^{∼} H_{F^{∗}}^{1}(Q, T^{∗})[I].

Proof. Suppose first that I is principal with a generator β. Cohomology of the exact sequences

0 ⟶ T^{∗}[I] ⟶ T^{∗} ⟶^{β} IT^{∗} ⟶ 0
0 ⟶ IT^{∗} ⟶ T^{∗} ⟶ T^{∗}/IT^{∗} ⟶ 0

gives (writing G = Gal(Q_{Σ(F)}/Q), and using Lemma 9.2 to obtain the zeros on the left)

0 ⟶ H^{1}(G, T^{∗}[I]) ⟶ H^{1}(G, T^{∗}) ⟶^{β} H^{1}(G, IT^{∗})
0 ⟶ H^{1}(G, IT^{∗}) ⟶ H^{1}(G, T^{∗}).

Thus we have an isomorphism

H^{1}(Q_{Σ(F)}/Q, T^{∗}[I]) ⟶^{∼} H^{1}(Q_{Σ(F)}/Q, T^{∗})[I]  (6)

in this case.

By induction the isomorphism (6) extends to all ideals I: if I = (β_{1}, …, β_{i}) just apply (6) with R replaced by R′ = R/(β_{1}, …, β_{i−1}), T replaced by T ⊗ R′, and I = β_{i}R′.

The Selmer structure F^{∗} on T^{∗}[I] is the one induced on it (as a submodule) by the Selmer structure F^{∗} on T^{∗} (see Example 3.3). Consider the commutative diagram

0 → H_{F^{∗}}^{1}(Q, T^{∗}[I]) → H^{1}(Q_{Σ(F)}/Q, T^{∗}[I]) → ⨁_{ℓ∈Σ(F)}H^{1}(Q_{ℓ}, T^{∗}[I])/H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}[I])
↓  ↓  ↓
0 → H_{F^{∗}}^{1}(Q, T^{∗})[I] → H^{1}(Q_{Σ(F)}/Q, T^{∗})[I] → ⨁_{ℓ∈Σ(F)}H^{1}(Q_{ℓ}, T^{∗})/H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}).

The rows are exact by definition of H_{F^{∗}}^{1}, (6) shows that the center vertical map is an isomorphism, and by definition of the induced Selmer structure the right-hand vertical map is injective. Therefore the left-hand vertical map is an isomorphism, which proves the lemma. □

Lemma 9.4. Suppose R is artinian and principal of length k, T satisfies (H.0), (H.1), (H.3), and (H.6), 0 ≤ i ≤ k, and π is a generator of 𝔪. Then the injection π^{k−i} : T/𝔪^{i}T ↪ T induces isomorphisms

[π^{k−i}] : H_{F}^{1}(Q, T/𝔪^{i}T) ⟶ H_{F}^{1}(Q, T)[𝔪^{i}],
[π^{k−i}] : H^{1}(Q, T/𝔪^{i}T) ⟶ H^{1}(Q, T)[𝔪^{i}].

Proof. As in the proof of Lemma 9.3 (using Lemma 9.2), cohomology of the exact sequences

0 ⟶ T/𝔪^{i}T ⟶^{π^{k−i}} T ⟶ T/𝔪^{k−i}T ⟶ 0
0 ⟶ T/𝔪^{k−i}T ⟶^{π^{i}} T

shows that [π^{k−i}] : H^{1}(Q, T/𝔪^{i}T) → H^{1}(Q, T)[𝔪^{i}] is an isomorphism.

It is easy to check that [π^{k−i}] maps H_{F}^{1}(Q, T/𝔪^{i}T) into H_{F}^{1}(Q, T)[𝔪^{i}]. To prove the lemma we need to show that [π^{k−i}]^{−1}H_{F}^{1}(Q, T) satisfies the local conditions to lie in H_{F}^{1}(Q, T/𝔪^{i}T). For primes ℓ ∈ Σ(F) this holds by (H.6), so suppose ℓ ∉ Σ(F). We need to show that [π^{k−i}]^{−1}H_{f}^{1}(Q_{ℓ}, T) ⊂ H_{f}^{1}(Q_{ℓ}, T/𝔪^{i}T).

Writing I_{ℓ} for the inertia group in G_{Q_{ℓ}}, we have a diagram with exact rows

0 ↪ H_{f}^{1}(Q_{ℓ}, T/𝔪^{i}T) ⟶ H^{1}(Q_{ℓ}, T/𝔪^{i}T) ⟶ Hom(I_{ℓ}, T/𝔪^{i}T)
[π^{k−i}]↓  ^{[π^{k−i}]}↓  ^{[π^{k−i}]}↓
0 ↪ H_{f}^{1}(Q_{ℓ}, T) ⟶ H^{1}(Q_{ℓ}, T) ⟶ Hom(I_{ℓ}, T).

The right-hand vertical map is injective, so if c ∈ H^{1}(Q_{ℓ}, T/𝔪^{i}T) and [π^{k−i}]c is unramified, then c is unramified. □

Definition 9.5. Recall the set P_{k} defined in condition (H.5), and let N_{k} = N (P_{k}) be the corresponding set of positive integers as in Definition 4.5: all squarefree products of primes in P_{k}.

Lemma 9.6. Suppose k ∈ Z^{+} and T satisfies (H.0) and (H.2).

(i) Suppose τ ∈ G_{Q} satisfies (H.2). If ℓ ∉ Σ(F) and the Frobenius class of ℓ in Gal(Q(T/𝔪^{k}T, μ_{p^{k}})/Q) is the conjugacy class of τ, then ℓ ∈ P_{k}. (ii) Suppose R is principal and artinian of length k, and ℓ ∈ P_{k}. Then H_{f}^{1}(Q_{ℓ}, T), H_{s}^{1}(Q_{ℓ}, T), H_{f}^{1}(Q_{ℓ}, T^{∗}), and H_{s}^{1}(Q_{ℓ}, T^{∗}) are free of rank one over R, and the map ϕ_{ℓ}^{fs} of Definition 2.2 is an isomorphism.

Proof. If the Frobenius of ℓ in Gal(Q(T/𝔪^{k}T, μ_{p^{k}})/Q) belongs to the conjugacy class of τ, then

T/(𝔪^{k}T + (Fr_{ℓ} − 1)T) = T/(𝔪^{k}T + (τ − 1)T)

which is free of rank one over R/𝔪^{k} by (H.2). Further

P_{ℓ}(1) = det(1 − Fr_{ℓ}|T) ≡ det(1 − τ | T) = 0 (mod 𝔪^{k})

and, writing ε_{cycl} for the cyclotomic character,

ℓ − 1 = ε_{cycl}(Fr_{ℓ}) − 1 ≡ ε_{cycl}(τ) − 1 = 0 (mod p^{k})

so I_{ℓ} ⊂ 𝔪^{k}. Thus ℓ ∈ P_{k}, which proves (i).

For (ii), suppose ℓ ∈ P_{k}. By Nakayama’s Lemma, T/(Fr_{ℓ} − 1)T is a cyclic R-module. If d is the length of this cyclic module then P_{ℓ}(1) = det(Fr_{ℓ} − 1|T) generates 𝔪^{d}, so d = k and T/(Fr_{ℓ} −1)T is free of rank one. Now Lemma 2.3 shows that H_{f}^{1}(Q_{ℓ}, T) and H_{s}^{1}(Q_{ℓ}, T) are free of rank one and ϕ_{ℓ}^{fs} is an isomorphism.

To complete the proof of (ii) we want to apply the same argument to T^{∗} in place of T. The exact sequence

0 ⟶ (T/𝔪T)^{Fr_{ℓ}=1} ⟶ T/𝔪T ⟶^{Fr_{ℓ}−1} T/𝔪T ⟶ T/(𝔪T + (Fr_{ℓ} − 1)T) ⟶ 0

shows that dim_{𝕜}(T/𝔪T)^{Fr_{ℓ}=1} = 1. Since ℓ ∈ P_{k}, Fr_{ℓ} acts trivially on μ_{p^{k}}, so

det(1 − Fr_{ℓ}|T^{∗}) = det(1 − Fr_{ℓ}^{−1}|T) = det(−Fr_{ℓ}^{−1}|T) det(1 − Fr_{ℓ}|T) = 0

and

T^{∗}/(𝔪T^{∗} + (Fr_{ℓ} − 1)T^{∗}) ≅ T^{∗}[𝔪]/(Fr_{ℓ} − 1)T^{∗}[𝔪] ≅ ((T/𝔪T)^{Fr_{ℓ}=1})^{∗}

has 𝕜-dimension one. Thus the argument above applies to T^{∗} and shows that H_{f}^{1}(Q_{ℓ}, T^{∗}) and H_{s}^{1}(Q_{ℓ}, T^{∗}) are free of rank one. □

10. Choosing useful primes

In this section we apply the Cebotarev theorem carefully to produce primes with properties that we will need in the following sections. For this section we assume that R is artinian and principal, and that T is an R[[G_{Q}]]-module satisfying the Hypotheses (H.0-5) of §9. In particular, it is (only) for the results of this section that we use (H.4).

Note that T^{∗} also satisfies (H.0-5), so the Lemmas below, which are stated only for T, apply equally to T^{∗}.

This section is devoted to the proofs of the following two propositions.

Proposition 10.1. Suppose c_{1}, c_{2} ∈ H^{1}(Q, T) and c_{3}, c_{4} ∈ H^{1}(Q, T^{∗}) are all nonzero. For every k ∈ Z^{+} there is a set S ⊂ P_{k} of positive density such that for every ℓ ∈ S, the localizations (c_{i})_{ℓ} are all nonzero.

Proposition 10.2. Suppose that C ⊂ H^{1}(Q, T) is a finite submodule, ϕ : C → R is a homomorphism, and k ∈ Z^{+}.

(i) There is a set S ⊂ P_{k} of positive density such that for every ℓ ∈ S,

ker(loc_{ℓ} : C → H^{1}(Q_{ℓ}, T)) = ker(ϕ)

where loc_{ℓ} is localization at ℓ. (ii) Suppose in addition that (H.4a) holds (i.e., T/𝔪T ≅ T̸^{∗}[𝔪] as F_{p}[[G_{Q}]]- modules), D is a finite submodule of H^{1}(Q, T^{∗}), and ψ : D → R is a homomorphism. Then there is a set S ⊂ P_{k} of positive density such that for every ℓ ∈ S,

ker(loc_{ℓ} : C → H^{1}(Q_{ℓ}, T)) = ker(ϕ), ker(loc_{ℓ} : D → H^{1}(Q_{ℓ}, T^{∗})) = ker(ψ).

Before proving Propositions 10.1 and 10.2 we need two lemmas.

Lemma 10.3. If C is a finite R-module with trivial G_{Q}-action, and A is a subgroup of Hom_{R}(C, T) which is stable under the action of G_{Q}, then A is an R-submodule of Hom_{R}(C, T).

Proof. Let r ∈ R. By (H.1), we have that the endomorphism of T given by scalar multiplication by r is also given by the action of an element of α_{r} ∈ Z_{p}[[G_{Q}]] on T. Then multiplication by r on Hom_{R}(C, T) also coincides with the action of α_{r}, and so any Z_{p}-submodule of Hom_{R}(C, T) which is stable under G_{Q} is also stable under scalar multiplication by R. □

Fix k ∈ Z^{+} is large enough so that 𝔪^{k} = 0. Let F = Q(T, μ_{p^{k}}) = Q(T^{∗}, μ_{p^{k}}), and note that F is a finite Galois extension of Q. Fix τ ∈ G_{Q} satisfying (H.2).

Suppose that C ⊂ H^{1}(Q, T) is a finite submodule, and consider the composition

C ⟶^{res_{F}} H^{1}(F, T)^{G_{Q}} = Hom(G_{F} , T)^{G_{Q}} ⟶ Hom(G_{F} , T/(τ − 1)T). (7)

The first map is injective by (H.3). If f belongs to the kernel of the last map then the image of f is a G_{Q}-stable subgroup of (τ − 1)T. Using (H.1) and (H.2) we conclude that f = 0, and hence the composition (7) is injective. Let F_{C} be the smallest extension of F such that the image of C in Hom(G_{F} , T) factors through Gal(F_{C}/F). Then F_{C}/Q is Galois, and Gal(F/Q) acts on Gal(F_{C}/F) by conjugation.

Lemma 10.4. (i) The map

Gal(F_{C}/F) ⟶ Hom_{R}(C, T)

induced by the first part of (7) is an isomorphism. (ii) The map

Gal(F_{C}/F) ⟶ Hom_{R}(C, T/(τ − 1)T)

induced by (7) is surjective.

Proof. We will prove (i), and then (ii) follows from assumption (H.2).

The map of (i) is Gal(F/Q)-equivariant, so its image is G_{Q}-stable and hence by Lemma 10.3 is an R-submodule of Hom_{R}(C, T). We give Gal(F_{C}/F) the structure of an R-module by identifying it with its image in Hom_{R}(C, T). In particular every Jordan-Holder factor of Gal(F_{C}/F) is a Jordan-Holder factor of T, and hence (since T is irreducible by (H.1)) is equal to T/𝔪T.

With this definition, the composition (7) factors through an injection of Rmodules

C ↪ Hom_{R[[G_{Q}]]}(Gal(F_{C}/F), T).

Now using the fact that R is artinian and principal we have

Hom_{R[[G_{Q}]]}(T/𝔪T, T) = Hom_{R[[G_{Q}]]}(T/𝔪T, T [𝔪]) ≅ Hom_{R[[G_{Q}]]}(T/𝔪T, T/𝔪T)

which is free of rank one over 𝕜 since T/𝔪T is irreducible. Hence we see by induction on the length of Gal(F_{C}/F) that

length(C) ≤ length(Hom_{R[[G_{Q}]]}(Gal(F_{C}/F), T)) ≤ \frac{length(Gal(F_{C}/F))}{rank_{R}(T)}.

By definition the map of (i) is injective, so it must be surjective as well. □

Using Lemma 10.4(i), we give Gal(F_{C}/F) the structure of an R-module, and hence it is an R[[G_{Q}]]-module.

Proof of Proposition 10.1. Increasing k if necessary, we may suppose that 𝔪^{k} = 0. Further, replacing the c_{i} by suitable multiples we may assume that c_{1}, c_{2}, c_{3} and c_{4} are all annihilated by 𝔪.

Let C_{i} be the R-module generated by c_{i}. Let F, F_{C_{1}}, and F_{C_{2}} be as defined before Lemma 10.4, and F_{C_{3}}, F_{C_{4}} the analogous fields defined with T^{∗} instead of T.

Define H_{1}, H_{2} ⊂ G_{F} by

H_{i} = {γ ∈ G_{F} : c_{i}(τγ) = 0 in T/(τ − 1)T }.

Note that c_{i}(τγ) is well-defined in T/(τ − 1)T since γ acts trivially on T. Define H_{3}, H_{4} similarly with T^{∗} in place of T.

If γ ∈ G_{F} − H_{i} and ℓ is a rational prime whose Frobenius conjugacy class in Gal(F_{C_{i}}/Q) is the class of τγ, then (c_{i})_{ℓ} ≠ 0 and Lemma 9.6(i) shows that ℓ ∈ P_{k}.

Let μ be Haar measure on G_{F} normalized so that μ(G_{F}) = 1. By Lemma 10.4(2), the map c_{i} : G_{F} → (T/(τ − 1)T)[𝔪] is surjective for i = 1, 2, and the analogous statement holds for i = 3, 4, so μ(H_{i}) = 1/|𝕜|

Suppose first that (H.4b) holds. Then μ(H_{i}) < 1/4 by (H.4b), so

μ(H_{1} ∪ H_{2} ∪ H_{3} ∪ H_{4}) < 1 = μ(G_{F}).

Choose a γ ∈ G_{F} − (H_{1} ∪ H_{2} ∪ H_{3} ∪ H_{4}), and let S be the set of rational primes whose Frobenius conjugacy class in Gal(F_{C_{1}}F_{C_{2}}F_{C_{3}}F_{C_{4}}/Q) is the class of τγ. Then S satisfies the conclusions of the proposition.

Now suppose that hypothesis (H.4a) holds. By Lemma 10.4(i), we have R[[G_{Q}]]- module isomorphisms

Gal(F_{C_{1}}/F) ≅ Gal(F_{C_{2}}/F) ≅ T/𝔪T,

and analogously Gal(F_{C_{3}}/F) ≅ Gal(F_{C_{4}}/F) ≅ T^{∗}[𝔪]. Hence by (H.4a) and (H.1), F_{C_{1}}F_{C_{2}} ∩ F_{C_{3}}F_{C_{4}} = F. Also, it follows from (H.1) that if c_{2} is not a multiple of c_{1} then F_{C_{1}} ∩ F_{C_{2}} = F, and similarly for c_{3} and c_{4}.

Thus for each i we can choose γ_{i} ∈ G_{F} − H_{i}, such that there is a γ ∈ G_{F} satisfying γ |_{F_{C_{i}}} = γ_{i} |_{F_{C_{i}}} for every i. Again, if S is the set of rational primes whose Frobenius conjugacy class in Gal(F_{C_{1}}F_{C_{2}}F_{C_{3}}F_{C_{4}}/Q) is the class of τγ, then S satisfies the conclusions of the proposition. □

Proof of Proposition 10.2. Suppose k is large enough so that 𝔪^{k} = 0. With F and F_{C} as above, let G be the subgroup of Gal(F_{C}/Q) generated by Gal(F_{C}/F) and τ. There is a well-defined evaluation homomorphism

ev : G → Hom(C, T/(τ − 1)T)

where ev(γ) is evaluation of cocycles at γ.

Fix an isomorphism η : T/(τ − 1)T ⟶^{∼} R. By Lemma 10.4(ii) we can find γ ∈ Gal(F_{C}/F) so that

ev(γ) = (η^{−1} ∘ ϕ) − ev(τ).

Fix such a γ. If ℓ is a rational prime whose Frobenius in Gal(F_{C}/F) is the conjugacy class of τγ, and such that every element of C is finite at ℓ, then the composition

C ⟶^{loc_{ℓ}} H_{f}^{1}(Q_{ℓ}, T) ⟶^{∼} T/(Fr_{ℓ} − 1)T = T/(τ − 1)T

is the map

ev(Fr_{ℓ}) = ev(τγ) = η^{−1} ∘ ϕ

so in particular ker(loc_{ℓ}) = ker(ϕ). Also by Lemma 9.6(i) ℓ ∈ P_{k}, so this proves (i).

Now repeat the argument above with T replaced by T^{∗} and C replaced by D. Then with the obvious notation, there is an element δ ∈ Gal(F_{D}/F) such that if ℓ is a rational prime whose Frobenius in Gal(F_{D}/F) is the conjugacy class of τδ, and such that d is finite at ℓ, then ker(loc_{ℓ}) = ker(ψ) in D.

We claim that F_{C} ∩ F_{D} = F. For by Lemma 10.4(i), every Jordan-Holder factor of the Z_{p}[[G_{Q}]]-module Gal(F_{C} ∩ F_{D}/F) is isomorphic both to a subquotient of T/𝔪T and to a subquotient of T^{∗}[𝔪]. By (H.1) T/𝔪T and T^{∗}[𝔪] are irreducible over F_{p}[[G_{Q}]], so by our assumption (H.4a) every such Jordan-Holder factor must be zero.

Now define S to be the set of rational primes whose Frobenius in Gal(F_{C}/F) (resp. Gal(F_{D}/F)) is the conjugacy class of τγ (resp. τδ), and such that d and every element of C are finite at ℓ. Then S satisfies the conclusions of (ii). □

11. Some remarks about hypothesis (H.6)

In practice, the conditions (H.0) through (H.6) of §9 are generally straightforward to verify with the possible exception of (H.6). In this section we discuss (H.6), and give sufficient conditions for it to hold.

Lemma 11.1. Suppose R is a discrete valuation ring and for every ℓ ∈ Σ(F) the R-module H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T) is torsion-free. Then for every k ∈ Z^{+},

(i) the induced Selmer structure on the R/𝔪^{k}-module T/𝔪^{k}T satisfies (H.6), (ii) the map T → T/𝔪^{k}T induces an injection

H_{F}^{1}(Q, T)/𝔪^{k}H_{F}^{1}(Q, T) ↪ H_{F}^{1}(Q, T/𝔪^{k}T).

whose cokernel has order bounded independently of k.

Proof. Fix a prime ℓ ∈ Σ(F), integers i, j, with 0 ≤ i ≤ j ≤ k, and a generator π of 𝔪. Consider the diagram with exact rows (all cohomology groups are over Q_{ℓ})

H^{1}(T) ⟶^{π^{i}} H^{1}(T) ⟶ H^{1}(T/𝔪^{i}T) ⟶ H^{2}(T)[𝔪^{i}] ⟶ 0
id↓  _{π^{j−i}}↓  ^{[π^{j−i}]}↓  ↓^{∩}
H^{1}(T) ⟶^{π^{j}} H^{1}(T) ⟶ H^{1}(T/𝔪^{j}T) ⟶ H^{2}(T)[𝔪^{j}] ⟶ 0.

Suppose c ∈ H^{1}(Q_{ℓ}, T/𝔪^{i}T) and [π^{j−i}]c ∈ H_{F}^{1}(Q_{ℓ}, T/𝔪^{j}T). To prove that (H.6) holds for T/𝔪^{k}T, we need to show that c ∈ H_{F}^{1}(Q_{ℓ}, T/𝔪^{i}T).

By definition (Example 4.3) of the Selmer structure on T/𝔪^{j}T, the fact that [π^{j−i}]c ∈ H_{F}^{1}(Q_{ℓ}, T/𝔪^{j}T) means that there is a d′ ∈ H_{F}^{1}(Q_{ℓ}, T) whose image in H^{1}(Q_{ℓ}, T/𝔪^{j}T) is [π^{j−i}]c. From the diagram it follows that there is a d ∈ H^{1}(Q_{ℓ}, T) whose image in H^{1}(Q_{ℓ}, T/𝔪^{i}T) is c. But then π^{j−i}d−d′ ∈ π^{j}H^{1}(Q_{ℓ}, T), so adjusting d if necessary we may assume that π^{j−i}d = d′. If H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T) is torsion-free then we conclude that d ∈ H_{F}^{1}(Q_{ℓ}, T), and hence c ∈ H_{F}^{1}(Q_{ℓ}, T/𝔪^{i}T). This proves (i).

Write G = Gal(Q_{Σ(F)}/Q). Let f denote the map of (ii) and C its cokernel. By definition we have an injection

H^{1}(G, T)/H_{F}^{1}(Q, T) ↪ ⨁_{ℓ∈Σ(F)}H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T),  (8)

and the R-modules on the right are all torsion-free by our assumption on the H_{F}^{1}(Q_{ℓ}, T). Therefore H^{1}(G, T)/H_{F}^{1}(Q, T) is torsion-free as well. This proves the injectivity of the map g_{1} in

0 ⟶ H_{F}^{1}(Q, T)/𝔪^{k} ⟶^{f} H_{F}^{1}(Q, T/𝔪^{k}) ⟶  C  ⟶ 0
g_{1}↓  ^{g_{2}}↓  ^{g_{3}}↓
0 ⟶ H^{1}(G, T)/𝔪^{k} ⟶ H^{1}(G, T/𝔪^{k}) ⟶ H^{2}(G, T)[𝔪^{k}] ⟶ 0.

It follows that f is injective, and hence both rows of the diagram are exact. Since g_{2} is injective we have a snake lemma isomorphism

ker(g_{3}) ⟶^{∼} ker(coker(g_{1}) → coker(g_{2})),

and the latter is the kernel of the composition

H^{1}(G, T)/(H_{F}^{1}(Q, T) + 𝔪^{k}H^{1}(G, T))

⟶ ⨁_{ℓ∈Σ(F)}H^{1}(Q_{ℓ}, T)/(H_{F}^{1}(Q_{ℓ}, T) + 𝔪^{k}H^{1}(Q_{ℓ}, T)

⟶ ⨁_{ℓ∈Σ(F)}H^{1}(Q_{ℓ}, T/𝔪^{k}T)/H_{F}^{1}(Q_{ℓ}, T/𝔪^{k}T).

Another straightforward diagram-chasing argument shows that the second map is injective. The kernel of the first map is D[𝔪^{k}] where D is the cokernel of (8). Thus we have

length(C) ≤ length(D[𝔪^{k}]) + length(H^{2}(G, T)[𝔪^{k}]).

Since D and H^{2}(G, T) are finitely-generated R-modules, this is bounded independently of k. This proves (ii). □

Remark 11.2. Suppose R is the ring of integers of a local field. If H_{F}^{1}(Q_{ℓ}, T) is the subgroup H_{f}^{1}(Q_{ℓ}, T) defined by Bloch and Kato [BK], then H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T) is torsion-free and Lemma 11.1(i) shows that (H.6) is satisfied for all quotients of R. Similarly if H_{F}^{1}(Q_{ℓ}, T) = H^{1}(Q_{ℓ}, T) then Lemma 11.1 applies.

Lemma 11.3. Suppose that R is artinian and principal, F satisfies (H.6), and j ∈ Z^{+}. Then the induced Selmer structure on the R/𝔪^{j}-module T/𝔪^{j}T satisfies (H.6).

Proof. This clear from the statement of (H.6), since Quot_{R}(T/𝔪^{j}T) is a subcategory of Quot_{R}(T). □

Lemma 11.4. Suppose R is artinian and principal of length k, (H.2) holds, F satisfies (H.6), and n ∈ N_{k}. Then F (n) satisfies (H.6).

Proof. Fix a prime ℓ dividing n; we need to check the condition of (H.6) for ℓ.

Fix a generator π of 𝔪, and 0 ≤ i ≤ k. We have a commutative diagram, where the left-hand horizontal maps come from the splitting of Lemma 2.4(iii), and the isomorphisms on the right are from Lemma 2.4(i)

H^{1}(Q_{ℓ}, T/𝔪^{i}T) ⟶ H_{f}^{1}(Q_{ℓ}, T/𝔪^{i}T) ⟶^{∼} T/(π^{i}, Fr_{ℓ} − 1)T
[π^{k−i}]↓  ^{[π^{k−i}]}↓  ^{[π^{k−i}]}↓
H^{1}(Q_{ℓ}, T) ⟶ H_{f}^{1}(Q_{ℓ}, T) ⟶^{∼} T/(Fr_{ℓ} − 1)T.

We have ℓ ∈ P_{k}, so T/(Fr_{ℓ} − 1)T is free of rank one over R by Lemmas 2.1(i) and 9.6(ii), and so T/(π^{i}, Fr_{ℓ} − 1)T is free of rank one over R/𝔪^{i}. Hence the right-hand vertical map is injective, so the center one is as well. Thus if c ∈ H^{1}(Q_{ℓ}, T/𝔪^{i}T) and [π^{k−i}]c projects to zero in H_{f}^{1}(Q_{ℓ}, T), then c projects to zero in H_{f}^{1}(Q_{ℓ}, T/𝔪^{i}T), i.e., c ∈ H_{F (n)}^{1}(Q_{ℓ}, T/𝔪^{i}T). This proves the lemma. □

Part IV. Kolyvagin systems over principal artinian rings

We now study Kolyvagin systems in the simplest setting, where R is artinian and principal (e.g., R = Z/p^{k}Z). In Part V we will study Kolyvagin systems over integral domains by reducing to the case of principal artinian rings.

We assume for all of Part IV that R is a principal local artinian ring, and we let k = length(R), i.e., 𝔪^{k} = 0 and 𝔪^{k−1} ≠ 0. Clearly the quotient of a discrete valuation ring by the k-th power of its maximal ideal is such a ring; conversely it is not difficult to show that every principal local artinian ring is a quotient of a discrete valuation ring (but we will not need this).

We fix for all of Part IV a Selmer triple (T, F, P) satisfying hypotheses (H.0) through (H.6) of §9. By propagating the Selmer structure F to quotients of T we get Selmer triples (T/𝔪^{i}T, F, P) (with R replaced by R/𝔪^{i}) for 0 ≤ i ≤ k. We will usually suppress F and P from the notation. By restricting to P ∩ P_{k}, we will also assume that P ⊂ P_{k}. Thus N ⊂ N_{k}, so for all n ∈ N the ideal I_{n} vanishes, and the stalk of the Selmer sheaf H at n is H(n) = H_{F (n)}^{1}(Q, T) ⊗ G_{n}. Also, for every ℓ in P, Lemma 9.6(ii) shows that H_{f}^{1}(Q_{ℓ}, T), H_{s}^{1}(Q_{ℓ}, T), H_{f}^{1}(Q_{ℓ}, T^{∗}), and H_{s}^{1}(Q_{ℓ}, T^{∗}) are free of rank one over R, and ϕ_{ℓ}^{fs} is an isomorphism.

Note that since R is principal artinian, T^{∗} is also a free R-module; this is not true for general R.

Since we will use it frequently, we let T̄ = T/𝔪T. Then T̄^{∗} = T^{∗}[𝔪].

12. The core Selmer module

Lemma 12.1. Suppose n ∈ N and 0 ≤ i ≤ k.

(i) The exact sequence 0 → T [𝔪^{i}] → T → T/𝔪^{k−i}T → 0 induces an isomorphism H_{F (n)}^{1}(Q, T/𝔪^{i}T) ≅ H_{F (n)}^{1}(Q, T)[𝔪^{i}] and an exact sequence

0 → H_{F (n)}^{1}(Q, T)[𝔪^{i}] → H_{F (n)}^{1}(Q, T) ⟶ H_{F (n)}^{1}(Q, T/𝔪^{k−i}T).

(ii) The inclusion T^{∗}[𝔪^{i}] ↪ T^{∗} induces an isomorphism

H_{F (n)^{∗}}^{1}(Q, T^{∗}[𝔪^{i}]) ⟶^{∼} H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪^{i}].

Proof. By Lemma 11.4, (T, F (n), P) satisfies hypotheses (H.0) through (H.6), and so (i) follows from Lemma 9.4. The second assertion is a special case of Lemma 9.3. □

Definition 12.2. For every n ∈ N define

λ(n, T) = length(H_{F (n)}^{1}(Q, T)) = lengthH(n),
λ(n, T^{∗}) = length(H_{F (n)^{∗}}^{1}(Q, T^{∗})).

Note that the second definition is equivalent to the first applied to T^{∗}, since F (n)^{∗} = F^{∗}(n) by Example 5.2. These definitions apply equally well when T is replaced by T/𝔪^{i} for i ∈ Z^{+}.

Proposition 12.3. If n ∈ N then

λ(n, T̄) = 0 ⟺ λ(n, T) = 0, λ(n, T̄^{∗}) = 0 ⟺ λ(n, T^{∗}) = 0.

Proof. By Lemma 12.1 we have

H_{F (n)}^{1}(Q, T̄) = H_{F (n)}^{1}(Q, T)[𝔪], H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪]

and the proposition follows. □

Proposition 12.4. The differences λ(n, T) − λ(n, T^{∗}) are independent of n ∈ N.

Proof. This follows immediately from Corollary 5.4 and Lemma 9.6(ii). □

Proposition 12.5. For every m, n ∈ N there is a noncanonical isomorphism

H_{F (n)}^{1}(Q, T) ⊕ H_{F (m)^{∗}}^{1}(Q, T^{∗}) ≅ H_{F (m)}^{1}(Q, T) ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}).

Proof. Since R is principal, every finitely generated R-module B is of the form ⊕_{i}R/𝔪^{j_{i}} with positive integers j_{i}. Thus the isomorphism class of the R-module B is determined by the numerical function on {1, 2, …, k}

i ↦ length(B[𝔪^{i}]) = length(B/𝔪^{i}B).

For i ≥ 0 we have, by Lemma 12.1,

length((H_{F (n)}^{1}(Q, T) ⊕ H_{F (m)^{∗}}^{1}(Q, T^{∗}))[𝔪^{i}]) = λ(n, T/𝔪^{i}T) + λ(m, T^{∗}[𝔪^{i}]),

length((H_{F (m)}^{1}(Q, T) ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}))[𝔪^{i}]) = λ(m, T/𝔪^{i}T) + λ(n, T^{∗}[𝔪^{i}]).

These are equal by Proposition 12.4 applied to T/𝔪^{i}T, so the desired noncanonical isomorphism exists. □

The next lemma is an application of the Global Duality Theorem 5.3, and is crucial in many of the calculations that follow.

Lemma 12.6. Suppose ℓ is prime and nℓ ∈ N. We have the following diagrams of inclusions, in which the labels on the arrows are the lengths of the corresponding cyclic cokernels.

\xymatrix{ & H_{F^{ℓ}(n)}^{1}(Q, T) &  &  & H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) \\ H_{F (n)}^{1}(Q, T) \ar@{^{(}->}[ur]^{a} &  & H_{F (nℓ)}^{1}(Q, T) \ar@{^{(}->}[ul]_{b} & H_{F (n)^{∗}}^{1}(Q, T^{∗}) \ar@{^{(}->}[ur]^{c^{∗}} &  & H_{F (nℓ)^{∗}}^{1}(Q, T^{∗}) \ar@{^{(}->}[ul]_{d^{∗}} \\  & H_{F_{ℓ}(n)}^{1}(Q, T) \ar@{^{(}->}[ur]_{d} \ar@{^{(}->}[ul]^{c} &  &  & H_{F^{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) \ar@{^{(}->}[ur]_{b^{∗}} \ar@{^{(}->}[ul]^{a^{∗}}}

These lengths satisfy (i) 0 ≤ a, b, c, d, a^{∗}, b^{∗}, c^{∗}, d^{∗} ≤ k,

(ii) a + c = b + d, a^{∗} + c^{∗} = b^{∗} + d^{∗},
(iii) a + a^{∗} = b + b^{∗} = c + c^{∗} = d + d^{∗} = k,
(iv) a ≥ d, b ≥ c, c^{∗} ≥ b^{∗}, d^{∗} ≥ a^{∗}.

Proof. By definition

H_{F (n)}^{1}(Q, T) = ker(H_{F^{ℓ}(n)}^{1}(Q, T) → H_{s}^{1}(Q_{ℓ}, T)),
H_{F_{ℓ}(n)}^{1}(Q, T) = ker(H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T)),

etc. The two diagrams come from the definitions in this way. Since H_{f}^{1}(Q_{ℓ}, T), H_{s}^{1}(Q_{ℓ}, T), H_{f}^{1}(Q_{ℓ}, T^{∗}), and H_{s}^{1}(Q_{ℓ}, T^{∗}) are all free of rank one over R by Lemma 9.6(ii), the inequalities (i) hold. The equalities (ii) are immediate from the diagrams.

The equality a + a^{∗} = k follows from the Global Duality Theorem 5.3 with G_{1} = F (n) and G_{2} = F^{ℓ}(n), and similarly for the other three equalities of (iii). Finally, by definition H_{F (n)}^{1}(Q, T) ∩ H_{F (nℓ)}^{1}(Q, T) = H_{F_{ℓ}(n)}^{1}(Q, T). The first two inequalities of (iv) follow from this, and the other two similarly with (T, F) replaced by (T^{∗}, F^{∗}). □

Lemma 12.7. Suppose nℓ ∈ N with ℓ prime.

(i) |λ(nℓ, T) − λ(n, T)| ≤ k and |λ(nℓ, T^{∗}) − λ(n, T^{∗})| ≤ k.

(ii) If the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective, then

H_{F (nℓ)^{∗}}^{1}(Q, T^{∗}) = H_{F^{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) ⊂ H_{F (n)^{∗}}^{1}(Q, T^{∗}),

so if in addition H_{F (n)^{∗}}^{1}(Q, T^{∗}) = 0, then H_{F (nℓ)^{∗}}^{1}(Q, T^{∗}) = 0. (iii) The image of the composition

𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) ⟶^{loc_{ℓ}} H_{f}^{1}(Q_{ℓ}, T) ⟶^{ϕ_{ℓ}^{fs}} H_{s}^{1}(Q_{ℓ}, T)

is equal to the image of 𝔪^{λ(nℓ,T^{∗})}H_{F (nℓ)}^{1}(Q, T) ⟶^{loc_{ℓ}} H_{s}^{1}(Q_{ℓ}, T). (iv) If both localization maps

H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are nonzero, then λ(nℓ, T̄) = λ(n, T̄) − 1 and λ(nℓ, T̄^{∗}) = λ(n, T̄^{∗}) − 1.

Proof. Consider the diagrams of Lemma 12.6. Assertion (i) is immediate. Recall (Lemma 9.6(ii)) that H_{f}^{1}(Q_{ℓ}, T), H_{s}^{1}(Q_{ℓ}, T), H_{f}^{1}(Q_{ℓ}, T^{∗}), and H_{s}^{1}(Q_{ℓ}, T^{∗}) are free of rank one over R, and ϕ_{ℓ}^{fs} is an isomorphism.

If the localization map in (ii) is surjective, then c = k in the left-hand diagram of Lemma 12.6. Therefore by (iii) and (iv) of that lemma, b^{∗} = 0, which proves (ii).

To prove (iii) it is enough to show that length(C_{ℓ}) = length(C_{nℓ}) where C_{ℓ} and C_{nℓ} are the images of 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) and 𝔪^{λ(nℓ,T^{∗})}H_{F (nℓ)}^{1}(Q, T), respectively, under localization at ℓ. The left-hand diagram of Lemma 12.6 shows that

length(C_{ℓ}) = max{0, c − λ(n, T^{∗})}, length(C_{nℓ}) = max{0, d − λ(nℓ, T^{∗})}.

The right-hand diagram shows that

λ(n, T^{∗}) − λ(nℓ, T^{∗}) = d^{∗} − c^{∗} = c − d,

so length(C_{ℓ}) = length(C_{nℓ}).

For (iv), if the localization maps in question are nonzero, then using Lemma 12.1 we see that the localization maps

H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄), H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) → H_{f}^{1}(Q_{ℓ}, T̄^{∗})

are surjective. Applying Lemma 12.6 with T̄ in place of T and k = 1, we have c = a^{∗} = 1, so b = d^{∗} = 1 and d = b^{∗} = 0. Thus λ(nℓ, T̄^{∗}) = λ(n, T̄^{∗}) − 1 and λ(nℓ, T̄) = λ(n, T̄) − 1. □

Definition 12.8. If m ∈ N and either λ(m, T) = 0 or λ(m, T^{∗}) = 0, we say that m is a core vertex (of the graph X of Definition 6.2). By Proposition 12.3, this definition is unchanged if we replace T by T̄.

Recall that ν(n) denote the number of prime divisors of n ∈ Z^{+}.

Corollary 12.9. (i) There are core vertices n ∈ N with

ν(n) = min{dim_{𝕜} H_{F}^{1}(Q, T̄), dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗})}.

(ii) For every m ∈ N there are core vertices n divisible by m.

(iii) For every j ≥ k there are core vertices m ∈ N_{j}.

Proof. Suppose m ∈ N is not a core vertex, so λ(m, T̄) > 0 and λ(m, T̄^{∗}) > 0. It follows easily from Proposition 10.1 and Lemma 12.7(iv) that there is an ℓ ∈ P_{j} such that λ(mℓ, T̄) = λ(m, T̄)−1 and λ(mℓ, T̄^{∗}) = λ(m, T̄^{∗})−1. Proceeding inductively, after r = min{λ(m, T̄), λ(m, T̄^{∗})} steps we reach an integer n ∈ N satisfying ν(n) = ν(m) + r and λ(n, T̄)λ(n, T̄^{∗}) = 0. By Lemma 12.3, λ(n, T)λ(n, T^{∗}) = 0 as well.

This argument proves (ii), and when m = 1 it proves (i). Assertion (iii) is immediate from (ii). □

Theorem 12.10. If n ∈ N is a core vertex then H_{F (n)}^{1}(Q, T) and H_{F (n)^{∗}}^{1}(Q, T^{∗}) are free R-modules. The ranks of these modules are independent of the choice of core vertex n, and one of them is zero.

Proof. Fix a core vertex n ∈ N and suppose H_{F (n)}^{1}(Q, T) is not free. Then we can fix a homomorphism ϕ : H_{F (n)}^{1}(Q, T) → R which is not surjective, but such that ϕ(H_{F (n)}^{1}(Q, T)[𝔪]) ≠ 0. By Proposition 10.2(i) we can find an ℓ ∈ P such that the kernel of localization at ℓ in H_{F (n)}^{1}(Q, T) is ker(ϕ).

Now consider the diagrams of Lemma 12.6 with T̄ in place of T and k = 1. Since localization at ℓ is nontrivial on H_{F (n)}^{1}(Q, T̄) = H_{F (n)}^{1}(Q, T)[𝔪], we have c = 1, and hence c^{∗} = 0. Since n is a core vertex and H_{F (n)}^{1}(Q, T) ≠ 0 (we assumed it not to be free), we have H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = 0, and hence H_{F_{ℓ}(n)^{∗}}^{1}(Q, T̄^{∗}) = 0.

By Lemma 9.3 we deduce that H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) = 0. Thus in the diagrams of Proposition 12.6 applied to T we have c^{∗} = 0, and so c = k. But c is the length of the image of localization at ℓ, i.e., c = length(image(ϕ)) which by construction is strictly less than k. This contradiction shows that H_{F (n)}^{1}(Q, T) is free, as desired.

In exactly the same way, H_{F (n)^{∗}}^{1}(Q, T^{∗}) is a free R-module as well.

Now suppose that m is another core vertex. By Propositions 12.4 and 12.5, either H_{F (n)}^{1}(Q, T) = H_{F (m)}^{1}(Q, T) = 0 or H_{F (n)^{∗}}^{1}(Q, T^{∗}) = H_{F (m)^{∗}}^{1}(Q, T^{∗}) = 0. Now applying Proposition 12.5 shows that H_{F (n)}^{1}(Q, T) ≅ H_{F (m)}^{1}(Q, T) and H_{F (n)^{∗}}^{1}(Q, T^{∗}) = H_{F (m)^{∗}}^{1}(Q, T^{∗}). □

Definition 12.11. The core Selmer rank of T is the rank of the free R-module H_{F (n)}^{1}(Q, T) for any core vertex n. We will denote the core rank of T by χ(T). Similarly we define χ(T^{∗}) = rank_{R}(H_{F (n)^{∗}}^{1}(Q, T^{∗})) for any core vertex n. By Theorem 12.10 these nonnegative integers are well-defined, independent of the choice of n, and one of them is zero.

Theorem 12.12. (i) If χ(T) > 0 then χ(T^{∗}) = 0 and for every i ≥ 0 and every n ∈ N,

H_{F (n)}^{1}(Q, T/𝔪^{i}T) ≅ (R/𝔪^{i})^{χ(T)} ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}[𝔪^{i}])

(ii) If χ(T) = 0 then for every i ≥ 0 and every n ∈ N,

H_{F (n)^{∗}}^{1}(Q, T^{∗}[𝔪^{i}]) ≅ (R/𝔪^{i})^{χ(T^{∗})} ⊕ H_{F (n)}^{1}(Q, T/𝔪^{i}T).

Proof. When i ≥ k this is immediate from Proposition 12.5 and Theorem 12.10. Now taking the 𝔪^{i}-torsion and applying Lemma 12.1 proves the proposition for arbitrary i. □

Corollary 12.13. If χ(T) > 0 then for every n ∈ N

λ(n, T) − λ(n, T^{∗}) = kχ(T).

If χ(T) = 0 then for every n ∈ N

λ(n, T) − λ(n, T^{∗}) = −kχ(T^{∗}).

Proof. This follows immediately from Theorem 12.12. □

Definition 12.14. Suppose χ(T) > 0, so χ(T^{∗}) = 0. If ν(n) < dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}) then Lemma 12.7(i) (applied to T̄) shows that λ(n, T̄^{∗}) > 0, and hence n is not a core vertex. On the other hand, Corollary 12.9(i) shows that there exist core vertices n with ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}); we will call such an n a leading vertex.

Note that if 1 is a core vertex (i.e., if H_{F^{∗}}^{1}(Q, T^{∗}) = 0) then 1 is the only leading vertex.

Theorem 12.15. Suppose T satisfies hypothesis (H.4a) (in addition to our other standard hypotheses), χ(T) > 0, and 1 is not a core vertex.

Choose a direct sum decomposition H_{F}^{1}(Q, T) = L ⊕ M where dim_{𝕜}(L[𝔪]) = χ(T). Then there are infinitely many leading vertices n ∈ N such that L ⊂ H_{F (n)}^{1}(Q, T).

Proof. Let j = dim_{𝕜} H_{F}^{1}(Q, T̄)−χ(T). Then by Lemma 12.1 and Theorem 12.12(i),

j = dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗})[𝔪] ≥ 1

and

j = dim_{𝕜} H_{F}^{1}(Q, T)[𝔪] − χ(T) = dim_{𝕜}(M[𝔪]).

Choose homomorphisms ϕ_{1}, …, ϕ_{j} : H_{F}^{1}(Q, T) → R such that ∩_{i} ker(ϕ_{i}) = L and ψ_{1}, …, ψ_{j} : H_{F^{∗}}^{1}(Q, T^{∗}) → R such that ∩_{i} ker(ψ_{i}) = 0. Using Proposition 10.2(ii) choose primes ℓ_{1}, …, ℓ_{j} ∈ P so that for every i,

ker(loc_{ℓ_{i}}) : H_{F}^{1}(Q, T) → H^{1}(Q_{ℓ_{i}}, T) = ker(ϕ_{i}),
ker(loc_{ℓ_{i}}) : H_{F^{∗}}^{1}(Q, T^{∗}) → H^{1}(Q_{ℓ_{i}}, T^{∗}) = ker(ψ_{i}).

In particular if n = ∏_{i} ℓ_{i} then ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}) and L = H_{F_{n}}^{1}(Q, T) ⊂ H_{F (n)}^{1}(Q, T), so we only need to show that n is a core vertex. Since Proposition 10.2 provides infinitely many ℓ_{i} with the desired properties, this will give infinitely many suitable n.

By construction we have injections

H_{F}^{1}(Q, T̄)/L[𝔪] ↪ ⨁_{i}H_{f}^{1}(Q_{ℓ_{i}}, T̄), H_{F^{∗}}^{1}(Q, T̄^{∗}) ↪ ⨁_{i}H_{f}^{1}(Q_{ℓ_{i}}, T̄^{∗}).

Since all of these 𝕜-vector spaces have dimension j, both maps must be surjective as well.

Consider the diagram of the global duality Theorem 5.3 applied to T̄ with G_{1} = F_{n} and G_{2} = F^{n}. The remark of the previous paragraph shows that the image of H_{F^{n}}^{1}(Q, T̄) → ⊕_{i}H^{1}(Q_{ℓ_{i}}, T̄) contains ⊕_{i}H_{f}^{1}(Q_{ℓ_{i}}, T̄), and the image of H_{(F_{n})^{∗}}^{1}(Q, T̄^{∗}) → ⊕_{i}H^{1}(Q_{ℓ_{i}}, T̄^{∗}) contains ⊕_{i}H_{f}^{1}(Q_{ℓ_{i}}, T̄^{∗}). By Theorem 5.3(ii) these containments must be equalities, and it follows that

H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = H_{(F^{n})^{∗}}^{1}(Q, T̄^{∗}) = ∩_{i} ker(ψ_{i}) = 0.

Thus n is a core vertex as desired, and the proof is complete. □

Example 12.16. Suppose that p > 2, R is a quotient of a discrete valuation ring D, and T = T_{0} ⊗ R with a free D-module T_{0} of finite rank with a continuous action of G_{Q}, unramified outside a finite set of primes. Then the canonical Selmer structure F_{can} on T induced from T_{0} (Definition 7.1) satisfies (H.6) by Lemma 11.1. We will show in Theorem 18.13 below that if H^{0}(Q_{p}, T_{0}) is finite then

χ(T, F_{can}) = dim_{𝕜} T̄^{−} + dim_{𝕜} T̄^{G_{Qp}}

where T̄^{−} denotes the subspace of T̄ on which (some fixed) complex conjugation acts by −1.

See §20 and §21, especially Propositions 20.5 and 21.3, for important examples with χ(T) = 1.

13. Kolyvagin systems and the core rank

The existence or nonexistence of Kolyvagin systems is completely determined by the core Selmer rank. Namely,

χ(T) = 0 ⇒ KS(T) = 0,

χ(T) = 1 ⇒ KS(T) is free of rank one over R,

χ(T) > 1 ⇒ KS(T) contains a free R-module of rank r for every r.

The first assertion is Theorem 13.2 below. The other two will follow using Howard’s Theorem 14.3 and (for the second one) the other results of §14 and §15. See §16.

Lemma 13.1. Suppose κ ∈ KS(T) and n ∈ N. If ℓ ∈ P is such that

(a) localization at ℓ maps 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) to zero,
(b) κ_{nℓ} ∈ 𝔪^{λ(nℓ,T^{∗})}H_{F (nℓ)}^{1}(Q, T) ⊗ G_{nℓ},

then (κ_{n})_{ℓ} = 0.

Proof. Let c and d be as in the left-hand diagram of Lemma 12.6. Then

λ(nℓ, T) = λ(n, T) − c + d.

It follows from condition (a) that c ≤ λ(n, T^{∗}), and then by Proposition 12.4 d ≤ λ(nℓ, T^{∗}). Thus it follows from (b) that (κ_{nℓ})_{ℓ} = 0. Now by definition of a Kolyvagin system, we conclude that (κ_{n})_{ℓ} = 0. □

Theorem 13.2. If the core rank χ(T) = 0, then KS(T) = 0.

Proof. Since χ(T) = 0, Theorem 12.12 shows that λ(n, T) ≤ λ(n, T^{∗}) for every n ∈ N. In particular 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) = 0 for every n, so condition (a) of Lemma 13.1 is always satisfied.

Suppose κ ∈ KS(T). We will prove that κ_{n} = 0 for every n ∈ N by induction on λ(n, T̄). If λ(n, T̄) = 0 then H_{F (n)}^{1}(Q, T) = 0 by Proposition 12.3, and there is nothing to prove.

Choose n ∈ N with λ(n, T̄) > 0, and suppose that κ_{n} ≠ 0. Since χ(T) = 0, Theorem 12.12 shows that λ(n, T̄^{∗}) > 0 as well. Therefore we can use Proposition 10.1 to choose a prime ℓ satisfying

(a) (κ_{n})_{ℓ} ≠ 0, (b) the localization maps

H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are both nonzero. It follows from (b) and Lemma 12.7(iv) that λ(nℓ, T̄) < λ(n, T̄). Hence our induction hypothesis applies, and so κ_{nℓ} = 0. In particular we can apply Lemma 13.1 to conclude that (κ_{n})_{ℓ} = 0, which contradicts (a). Hence κ_{n} = 0. □

14. The sheaf of stub Selmer modules

We define a subsheaf of the Selmer sheaf H = H_{T} = H_{(T,F,P)} of Definition 6.2 as follows.

Definition 14.1. The sheaf of stub Selmer modules H′ = H′_{(T,F,P)} ⊂ H_{T} is the subsheaf of H defined by

• H′(n) = 𝔪^{λ(n,T^{∗})}H(n) = 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) ⊗ G_{n} ⊂ H(n). if n ∈ N,

• H′(e) is the image of H′(n) in H(e) = H_{s}^{1}(Q_{ℓ}, T) ⊗ G_{nℓ} under the vertex-

to-edge map of H, if e is an edge joining n and nℓ, and the vertex-to-edge maps are the restrictions of those of the sheaf H:

• H′(n) → H′(e) is localization at ℓ followed by ϕ_{ℓ}^{fs},

• H′(nℓ) → H′(e) is localization at ℓ.

The latter map is well-defined (its image lies in H′(e)), and both maps are surjective, by Lemma 12.7(iii).

Clearly we have Γ(H′) ⊂ Γ(H).

Lemma 14.2. Suppose n ∈ N. Then H′(n) = 0 if λ(n, T^{∗}) ≥ k, and otherwise H′(n) is free of rank χ(T) over R/𝔪^{k−λ(n,T^{∗})}.

Proof. This is immediate from Theorem 12.12. □

The following theorem is due to Benjamin Howard. The authors thank him for including his proof as Appendix B of this paper.

Theorem 14.3 (Howard).

(i) For every n the map Γ(H′) → H′(n) is surjective.

(ii) If χ(T) = 1, then Γ(H′) has a free R-submodule of rank one.

(iii) If χ(T) > 1, then for every n, Γ(H′) has a free R-submodule of rank n.

Theorem 14.4. Suppose χ(T) = 1. Then the sheaf H′ is locally cyclic and connected, and every n ∈ N with λ(n, T^{∗}) = 0 is a hub.

Before proving Theorem 14.4 we give the following corollary.

Corollary 14.5. Suppose that χ(T) = 1. Then the locally cyclic sheaf H′ has trivial monodromy, and Γ(H′) is free of rank one over R.

Proof. Fix n so that λ(n, T^{∗}) = 0. Using Theorem 12.12(i) we see that

H′(n) ≅ H_{F (n)}^{1}(Q, T) ≅ R ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}) = R

is free of rank one over R. Howard’s Theorem 14.3 shows that Γ(H′) contains a free, rank-one R-submodule, so the corollary follows by Theorem 14.4 and Proposition 8.4(i). □

The rest of this section is devoted to the proof of Theorem 14.4. The reader may prefer to skip to the applications in the following sections.

We assume for the rest of this section that χ(T) = 1. The heart of the proof is a study of the restriction of H′ to a subgraph X^{0} defined below.

Definition 14.6. Define a subgraph X^{0} = X^{0}(T, F, P) of the graph X = X (P) of Definition 6.2 as follows. The vertices of X^{0} are the core vertices of X, the n ∈ N with λ(n, T̄^{∗}) = 0. We join n and nℓ by an edge in X^{0} if and only if the localization map H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) is nonzero (equivalently, is an isomorphism).

Now define the sheaf H^{0} on X^{0} to be the restriction of the Selmer sheaf H of Definition 6.2 to X^{0}. Then H^{0} is also the restriction of the stub Selmer sheaf H′ to X^{0}, since the vertices of X^{0} are precisely those n for which H′(n) = H(n), and the edges are those e for which H′(e) = H(e).

Lemma 14.7. The sheaf H^{0} is locally free of rank one.

Proof. By Proposition12.3, if n is a vertex of X^{0} then λ(n, T^{∗}) = 0 so by Lemma 14.2, H^{0}(n) is free of rank one.

Suppose e is an edge joining n and nℓ in X^{0}. As usual we have H_{F (n)}^{1}(Q, T̄) = H_{F (n)}^{1}(Q, T)[𝔪] and H_{f}^{1}(Q_{ℓ}, T̄) = H_{f}^{1}(Q_{ℓ}, T)[𝔪], and H_{F (n)}^{1}(Q, T), H_{f}^{1}(Q_{ℓ}, T) are both free of rank one, so the nontriviality of H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) implies that H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is an isomorphism. Now by Lemmas 12.7(iii) and 9.6(ii) the maps from H^{0}(n) and H^{0}(nℓ) to H^{0}(e) are both isomorphisms. □

The key to the proof of Theorem 14.4 is the fact that the graph X^{0} is connected (Theorem 14.12 below). Since X^{0} was defined in terms of T̄, we can work with T̄ and 𝕜 instead of T and R,

Lemma 14.8. Suppose n and nℓ are vertices of X^{0}. The following are equivalent. (i) There is an edge of X^{0} joining n and nℓ. (ii) The localization map H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) is nonzero. (iii) The localization map H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) is an isomorphism. (iv) The localization map H_{F (nℓ)}^{1}(Q, T̄) → H_{s}^{1}(Q_{ℓ}, T̄) is nonzero.

(v) The localization map H_{F (nℓ)}^{1}(Q, T̄) → H_{s}^{1}(Q_{ℓ}, T̄) is an isomorphism.

(vi) H_{F (n)}^{1}(Q, T̄) ≠ H_{F (nℓ)}^{1}(Q, T̄).

Proof. By the various definitions (i) is equivalent to (ii), (ii) is equivalent to the assertion H_{F (n)}^{1}(Q, T̄) ⊄ H_{F (nℓ)}^{1}(Q, T̄), and (iv) is equivalent to the assertion H_{F (nℓ)}^{1}(Q, T̄) ⊄ H_{F (n)}^{1}(Q, T̄). Since (using Lemma 9.6(ii))

dim_{𝕜} H_{F (n)}^{1}(Q, T̄) = dim_{𝕜} H_{F (nℓ)}^{1}(Q, T̄) = dim_{𝕜} H_{f}^{1}(Q_{ℓ}, T̄) = dim_{𝕜} H_{f}^{1}(Q_{ℓ}, T̄) = 1,

the remaining equivalences follow. □

Lemma 14.9. Suppose n and nℓ are vertices of the graph X^{0}. Then there is a path in X^{0} from n to nℓ.

Proof. If n and nℓ are connected by an edge there is nothing to prove. So suppose not, i.e., the localization map H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) is zero. Then by Lemma 14.8 we have H_{F (n)}^{1}(Q, T̄) = H_{F (nℓ)}^{1}(Q, T̄).

Further, in the diagrams of Lemma 12.6 we have c = 0, so c^{∗} = 1. By definition of X^{0} we have H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = 0, so we deduce that dim_{𝕜} H_{F_{ℓ}(n)^{∗}}^{1}(Q, T̄^{∗}) = 1.

Now applying Proposition 10.1 we can choose (using (H.5)) a prime q ∈ P_{t} ⊂ P, prime to nℓ, such that the localization maps

(a) H_{F (n)}^{1}(Q, T̄) = H_{F (nℓ)}^{1}(Q, T̄) ⟶^{∼} H_{f}^{1}(Q_{q}, T̄),

(b) H_{F_{ℓ}(n)^{∗}}^{1}(Q, T̄^{∗}) ⟶^{∼} H_{f}^{1}(Q_{q}, T̄^{∗})

are both isomorphisms. By Lemma 12.7(ii) it follows from (a) that nq and nℓq are both vertices of X^{0}, and there is an edge from n to nq and from nℓ to nℓq.

By (b) we have H_{F_{ℓ}^{q}(n)^{∗}}^{1}(Q, T̄^{∗}) = 0, so applying Theorem 5.3 with G_{1} = F_{ℓ}^{q}(n) and G_{2} = F^{q}(n) shows that the localization map H_{F^{q}(n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄) is nonzero. From the left-hand diagram of Lemma 12.6 (with ℓ replaced by q), using that H_{F (n)}^{1}(Q, T̄) ≠ H_{F (nq)}^{1}(Q, T̄) by Lemma 14.8, we see that H_{F^{q}(n)}^{1}(Q, T̄) = H_{F (n)}^{1}(Q, T̄)⊕H_{F (nq)}^{1}(Q, T̄). Since the image of H_{F (n)}^{1}(Q, T̄) in H_{f}^{1}(Q_{ℓ}, T̄) is zero, we conclude that the image of H_{F (nq)}^{1}(Q, T̄) in H_{f}^{1}(Q_{ℓ}, T̄) is nonzero. Therefore there is an edge in X^{0} joining nq to nℓq, and so there is a path (n, nq, nℓq, nℓ) in X^{0} from n to nℓ. □

Proposition 14.10. Suppose that n is a vertex of X^{0} and ν(n) ≥ dim_{𝕜} H_{F}^{1}(Q, T̄). Then there is a vertex m of X^{0} with ν(m) < ν(n) such that there is a path in X^{0} from n to m.

Proof. We consider two cases. Case 1: for some ℓ dividing n, the localization map H_{F (n)}^{1}(Q, T̄) → H^{1}(Q_{ℓ}, T̄) is nonzero. In the left-hand diagram of Lemma 12.6 applied to n/ℓ and ℓ, we have d = 1. Since λ(n/ℓ, T̄) ≥ χ(T̄) = λ(n, T̄), we conclude that n/ℓ is a vertex of X^{0}. By Lemma 14.8 there is an edge of X^{0} joining n and n/ℓ. Case 2: for every ℓ dividing n, the localization map H_{F (n)}^{1}(Q, T̄) → H^{1}(Q_{ℓ}, T̄) is zero. Then

H_{F (n)}^{1}(Q, T̄) = H_{F_{n}}^{1}(Q, T̄) = ⋂_{ℓ|n} H_{F_{ℓ}}^{1}(Q, T̄).

Since this intersection of at least dim_{𝕜} H_{F}^{1}(Q, T̄) subspaces of H_{F}^{1}(Q, T̄) is nonempty, there is a proper divisor g of n, say g = n/q for some prime q, such that H_{F_{n}}^{1}(Q, T̄) = H_{F_{g}}^{1}(Q, T̄).

If λ(g, T̄) = 1 then by Lemma 14.9 there is a path from n to g, and again we are done. So we may assume that λ(g, T̄) > 1. By Lemma 12.7(i), λ(g, T̄) = 2, so by Corollary 12.13 we have H_{F (g)^{∗}}^{1}(Q, T̄^{∗}) ≠ 0.

Choose nonzero elements c ∈ H_{F (n)}^{1}(Q, T̄) ⊂ H_{F (g)}^{1}(Q, T̄), d ∈ H_{F (g)^{∗}}^{1}(Q, T̄^{∗}), and apply Proposition 10.1 to get a prime ℓ ∈ P, prime to n, such that c_{ℓ} ≠ 0 and d_{ℓ} ≠ 0.

By Lemma 12.7(iv) we have λ(gℓ, T̄) = 1, so gℓ is a vertex of X^{0}. From the left-hand diagram of Lemma 12.6 it follows that H_{F (gℓ)}^{1}(Q, T̄) ⊂ H_{F (g)}^{1}(Q, T̄), and hence

H_{F (n)}^{1}(Q, T̄) = ker(H_{F (g)}^{1}(Q, T̄) → H_{f}^{1}(Q_{q}, T̄)),
H_{F (gℓ)}^{1}(Q, T̄) = ker(H_{F (g)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄)).

Since the class c belongs to the first kernel but not the second, these two kernels are different; since both are one-dimensional, they are disjoint. It follows that the localization maps

H_{F (n)}^{1}(Q, T̄) → H_{f}^{1}(Q_{ℓ}, T̄), H_{F (gℓ)}^{1}(Q, T̄) → H_{f}^{1}(Q_{q}, T̄)

are both nonzero, so by Lemmas 12.7(ii) and 14.8, nℓ is a vertex of X^{0} and there are edges joining n to nℓ and gℓ to nℓ.

Finally, we claim that there is a prime r dividing gℓ such that the localization map H_{F (gℓ)}^{1}(Q, T̄) → H_{s}^{1}(Q_{r}, T̄) is nonzero. If not, then

H_{F (gℓ)}^{1}(Q, T̄) = H_{F_{gℓ}}^{1}(Q, T̄) ⊂ H_{F_{n}}^{1}(Q, T̄) = H_{F (n)}^{1}(Q, T̄)

which we have seen is not the case. Thus by Case 1 there is an edge connecting gℓ and gℓ/r, and so the path (n, nℓ, gℓ, gℓ/r) satisfies the proposition. □

Proposition 14.11. Suppose that n and m are vertices of X^{0} and ν(n) = ν(m) = dim_{𝕜} H_{F}^{1}(Q, T̄) − 1. Then there is a path in X^{0} from n to m.

Proof. We will prove this by induction on dim_{𝕜} H_{F}^{1}(Q, T̄) − ν(gcd(n, m)). This quantity is always at least one, and when it is equal to one we have n = m and there is nothing to prove.

Suppose n ≠ m, and fix distinct primes q | n and r | m. Applying Lemma 12.7(i) repeatedly we conclude that λ(n/q, T̄) = λ(m/r, T̄) = 2. As in the proof of Proposition 14.10 it follows that

H_{F (n)}^{1}(Q, T̄) = ker(H_{F (n/q)}^{1}(Q, T̄) → H_{f}^{1}(Q_{q}, T̄)),
H_{F (m)}^{1}(Q, T̄) = ker(H_{F (m/r)}^{1}(Q, T̄) → H_{f}^{1}(Q_{r}, T̄)),
dim_{𝕜} H_{F (n/q)^{∗}}^{1}(Q, T̄^{∗}) = dim_{𝕜} H_{F (m/r)^{∗}}^{1}(Q, T̄^{∗}) = 1.

Choose nonzero elements c ∈ H_{F (n)}^{1}(Q, T̄), c′ ∈ H_{F (m)}^{1}(Q, T̄), d ∈ H_{F (n/q)^{∗}}^{1}(Q, T̄^{∗}), and d′ ∈ H_{F (m/r)^{∗}}^{1}(Q, T̄^{∗}). By Proposition 10.1 we can find a prime ℓ ∈ P, prime to nm, such that c_{ℓ}, c′_{ℓ}, d_{ℓ}, d′_{ℓ} are all nonzero.

Exactly as in the proof of Proposition 14.10, it follows that

• λ(nℓ/q, T̄) = λ(nℓ, T̄) = 1 and there are edges in X^{0} connecting n to nℓ

and nℓ/q to nℓ,

• λ(mℓ/r, T̄) = λ(mℓ, T̄) = 1 and there are edges in X^{0} connecting m to mℓ

and mℓ/r to mℓ.

We have ν(gcd((nℓ/q)(mℓ/r))) = ν(gcd(n, m)) + 1, so our induction hypothesis shows that there is a path in X^{0} connecting nℓ/q and mℓ/r. Therefore there is a path (n, nℓ, nℓ/q, ⋯, mℓ/r, mℓ, m) connecting n and m. □

Theorem 14.12. The graph X^{0} is connected.

Proof. Suppose n and m are vertices of X^{0}. By Lemma 12.7(i), we must have ν(n), ν(m) ≥ dim_{𝕜} H_{F}^{1}(Q, T̄) − 1. Applying Proposition 14.10 inductively we can find paths connecting n to a vertex n′ and m to a vertex m′ with ν(n′) = ν(m′) = dim_{𝕜} H_{F}^{1}(Q, T̄) − 1. By Proposition 14.11 there is a path from n′ to m′, and the proof is complete. □

Proof of Theorem 14.4. Since χ(T) = 1, Lemma 14.2 shows that the stalks of H′ are cyclic. If e is an edge joining vertices n and nℓ, then the vertex-to-edge map H′(n) → H′(e) is surjective by definition, and then H′(nℓ) → H′(e) is surjective by Lemma 12.7(iii).

It remains to show that if n ∈ N and λ(n, T^{∗}) = 0, then n is a hub of H′. Fix such a vertex n, and let m be any other vertex. We will show by induction on λ(m, T̄^{∗}) that there is a surjective path from n to m.

Note that n is a vertex of X^{0}. If λ(m, T̄^{∗}) = 0 then m is also a vertex of X^{0}, and so there is a path in X^{0} from n to m. Every path in X^{0} is a surjective path (because all vertex-to-edge maps in X^{0} are isomorphisms by Lemma 14.7), so there is an H′-surjective path from n to m.

Now suppose λ(m, T̄^{∗}) > 0. Using Proposition 10.1, choose a prime ℓ ∈ P such that the localization maps

𝔪^{k−1}H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are both nonzero. (Note that 𝔪^{k−1}H_{F (n)}^{1}(Q, T) ≠ 0 by Theorem 12.12.) Then by Lemma 12.7(iv), λ(mℓ, T̄^{∗}) < λ(m, T̄^{∗}), so by our induction hypothesis there is an H′-surjective path from n to mℓ. Further we see that localization H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective, so 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) → 𝔪^{λ(n,T^{∗})}H_{f}^{1}(Q_{ℓ}, T) is surjective as well. These two modules are both cyclic of length max{0, k − λ(n, T^{∗})}, so if e is the edge joining m and mℓ, we conclude that the map H′(m) → H′(e) is an isomorphism. Therefore the path from mℓ to m is a surjective path, so by composition we obtain a surjective path from n to m. This concludes the proof. □

15. Kolyvagin systems and the stub Selmer sheaf

The main result of this section is that, under fairly general hypotheses, a Kolyvagin system, a priori a global section of the Selmer sheaf H, is actually a global section of the subsheaf H′.

Theorem 15.1. Suppose that (at least) one of the following three conditions is satisfied.

(a) χ(T) = 1,

(b) k = 1, i.e., R is a field, or

(c) (H.4a) holds, i.e., T̄ ≅ ¯T̸^{∗}. Then the inclusion Γ(H′) ⊂ Γ(H) is an isomorphism. In other words, for every κ ∈ KS(T) and n ∈ N we have κ_{n} ∈ H′(n).

Proof. Fix a κ ∈ KS(T). We treat the three hypotheses separately, but in each case we will show by induction on λ(n, T̄^{∗}) that κ_{n} ∈ H′(n) for every n ∈ N. If λ(n, T̄^{∗}) = 0 we have λ(n, T^{∗}) = 0 (Proposition 12.3) so H′(n) = H(n) and there is nothing to prove. Case 1: k = 1. Suppose λ(n, T̄^{∗}) > 0. In this case H′(n) = 𝔪^{λ(n,T^{∗})}H(n) = 0, so we need to show κ_{n} = 0. If κ_{n} ≠ 0, then using Proposition 10.1 we can fix a prime ℓ ∈ P not dividing n such that

(a) (κ_{n})_{ℓ} ≠ 0,

(b) the localization map H_{F (n)^{∗}}^{1}(Q, T^{∗}) → H_{f}^{1}(Q_{ℓ}, T^{∗}) is nonzero.

It follows from (a), (b) and Lemma 12.7(iv) that λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}), so our induction hypothesis shows that κ_{nℓ} ∈ H′(nℓ). Thus by Lemma 13.1 we conclude that (κ_{n})_{ℓ} = 0. But this contradicts (a), so we must have κ_{n} = 0.

Case 2: T̄ ≅ ¯T̸^{∗}. By Theorem 13.2 we may assume that χ(T) > 0. Suppose that λ(n, T̄^{∗}) > 0.

Since χ(T) > 0, it follows from Theorem 12.12 that

H(n)[𝔪] ⊄ H′(n).

If κ_{n} ∉ H′(n) we can find a homomorphism ϕ : H(n) → R such that

ϕ(H′(n)) = 0, ϕ(H(n)[𝔪]) ≠ 0, ϕ(κ_{n}) ≠ 0.

Hence by Proposition 10.2(ii) we can fix a prime ℓ ∈ P, prime to n, such that (a) the localization map 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is zero,

(b) (κ_{n})_{ℓ} ≠ 0,

(c) the localization maps

H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are both nonzero. By (c) and Lemma 12.7(iv), we have λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}), so by our induction hypothesis we have κ_{nℓ} ∈ H′(nℓ). Therefore by Lemma 13.1, (κ_{n})_{ℓ} = 0, which contradicts condition (b). This contradiction shows that κ_{n} ∈ H′(n). Case 3: χ(T) = 1. Choose a core vertex m. By Howard’s Theorem 14.3(i) there is a section κ′ ∈ Γ(H′) such that κ′(m) = κ(m). Let κ̄ = κ−κ′ ∈ KS(T), so κ̄_{m} = 0. We will show that κ̄ = 0.

Let X^{0} be the subgraph of X of Definition 14.6, whose vertices are the core vertices of X, and H^{0} the restriction of H to X^{0}. By Theorem 14.12 and Lemma 14.7, X^{0} is connected and H^{0} is locally free of rank one. Since κ̄_{m} = 0 at the vertex m of X^{0}, the restriction of κ̄ to X^{0} must be identically zero (Proposition 8.4(i)). In other words, κ̄_{n} = 0 for every core vertex n.

Now let n ∈ N be any vertex of X. We will show by induction on λ(n, T̄^{∗}) that κ̄_{n} = 0. We have already dealt with the case of core vertices, λ(n, T̄^{∗}) = 0.

Suppose now that λ(n, T̄^{∗}) > 0, and suppose κ̄_{n} ≠ 0. Using Proposition 10.1, choose a prime ℓ ∈ P, prime to n, such that

(a) (κ̄_{n})_{ℓ} ≠ 0,

(b) the localization maps

H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are both nonzero. By (b) and Lemma 12.7(iv), we have λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}), so by our induction hypothesis we have κ̄_{nℓ} = 0. But by (a) and the definition of a Kolyvagin system, we must have (κ̄_{nℓ})_{ℓ,s} ≠ 0. This contradiction shows that κ̄_{n} = 0, and so κ_{n} = κ′_{n} ∈ H′(n). □

The following corollary is the standard application of a Kolyvagin system (or Euler system): an upper bound on the size of the dual Selmer group H_{F^{∗}}^{1}(Q, T), in terms of the divisibility of κ_{1} (compare with, for example, Theorem 2.2.2 of [Ru6]). In Theorem 16.5 below, we will show that under some extra hypotheses this upper bound is sharp, so we get an exact formula for the size of H_{F^{∗}}^{1}(Q, T).

Corollary 15.2. Suppose κ ∈ KS(T) and one of the hypotheses (a), (b), or (c) of Theorem 15.1 holds. Then

length(H_{F^{∗}}^{1}(Q, T)) ≤ max{i : κ_{1} ∈ 𝔪^{i}H_{F}^{1}(Q, T)}.

Proof. By Theorem 15.1,

κ_{1} ∈ H′(1) = 𝔪^{λ(1,T^{∗})}H_{F}^{1}(Q, T) = 𝔪^{length(H_{F^{∗}}^{1} (Q,T^{∗}))}H_{F}^{1}(Q, T).  □

The next theorem gives a weaker conclusion than Theorem 15.1, but applies more generally.

Theorem 15.3. Let j = [k/2], the greatest integer less than or equal to k/2. If κ ∈ KS(T) then for every n ∈ N, κ_{n} ∈ 𝔪^{λ(n,T^{∗})}H(n) + H(n)[𝔪^{j}].

Proof. We will prove this by induction on both λ(n, T̄^{∗}) and k. If λ(n, T̄^{∗}) = 0 then λ(n, T^{∗}) = 0 and there is nothing to prove; similarly if k = 0 there is nothing to prove.

Suppose λ(n, T̄^{∗}) > 0 and k > 0. Let κ^{(i)} denote the image of κ in KS(T/𝔪^{i}T). Case 1: λ(n, T^{∗}) ≥ k−j. In this case we need to show that 𝔪^{j}κ_{n} = 0, or equivalently (by Lemma 12.1(i)) κ_{n}^{(k−j)} = 0.

If κ_{n}^{(k−j)} ≠ 0, then using Proposition 10.1 we can fix a prime ℓ ∈ P not dividing n such that

(a) (κ_{n}^{(k−j)})_{ℓ} ≠ 0,

(b) the localization maps

H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T), H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪] → H_{f}^{1}(Q_{ℓ}, T^{∗})

are both nonzero. It follows from (b) and Lemma 12.7(iv) that λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}), so our induction hypothesis shows that κ_{nℓ}^{(k−j)} ∈ 𝔪^{λ(nℓ,T^{∗}[𝔪^{k−j}])}H_{F (nℓ)}^{1}(Q, T/𝔪^{k−j}T) ⊗ G_{nℓ}. Thus

by Lemma 13.1 we conclude that (κ_{n}^{(k−j)})_{ℓ} = 0. But this contradicts (a), so we must have κ_{n}^{(k−j)} = 0. Case 2: λ(n, T^{∗}) < k−j. (Note that this implies k > 1.) Let i = k−λ(n, T^{∗}) ≥ j+1. We have λ(n, T^{∗}) ≤ j, so it follows from Theorem 12.12 that 𝔪^{λ(n,T^{∗})}H(n) + H(n)[𝔪^{j}] = H(n)[𝔪^{i}], so we need to show that 𝔪^{i}κ_{n} = 0.

Applying our induction hypothesis (on k) to κ^{(k−1)} we see that

κ_{n}^{(k−1)} ∈ 𝔪^{λ(n,T^{∗})}H_{F (n)}^{1}(Q, T/𝔪^{k−1}T) ⊗ G_{n} + H_{F (n)}^{1}(Q, T/𝔪^{k−1}T)[𝔪^{j}] ⊗ G_{n}

and so (since i − 1 ≥ j) 𝔪^{i−1}κ_{n}^{(k−1)} = 0. Now using Lemma 12.1(i) we conclude that 𝔪^{i}κ_{n} = 0, as desired. □

16. Kolyvagin systems over principal artinian rings

This section contains the main results and applications of Kolyvagin systems over principal artinian rings.

Corollary 16.1. (i) If χ(T) = 0 then KS(T) = 0.

(ii) If χ(T) ≥ 2 then for every d ∈ Z^{+}, KS(T) contains a free R-module of rank d.

Proof. Assertion (i) is Theorem 13.2, and (ii) is Theorem 14.3(iii). □

The rest of this section will deal with the case χ(T) = 1.

Corollary 16.2. Suppose χ(T) = 1.

(i) KS(T) is a free R-module of rank one.

(ii) Suppose κ ∈ KS(T). If there is a j ≥ 0 and an m ∈ N such that 𝔪^{j}H′(m) ≠ 0 and κ_{m} generates 𝔪^{j}H′(m), then κ_{n} generates 𝔪^{j}H′(n) for every n ∈ N.

(iii) Suppose κ ∈ KS(T). If n ∈ N is such that λ(n, T^{∗}) = 0, then the map KS(T) → H(n) which maps κ ↦ κ_{n} is an isomorphism.

Proof. Since KS(T) = Γ(H′) (Theorem 15.1), the Corollary is immediate from Corollary 14.5, Theorem 14.4, and Proposition 8.4. □

Corollary 16.3. Suppose that χ(T) = 1 and κ ∈ KS(T). Then the following are equivalent.

(i) There is an n ∈ N such that H′(n) ≠ 0 and κ_{n} generates H′(n).

(ii) κ_{n} generates H′(n) for every n ∈ N.

(iii) The global section of Γ(H′) corresponding to κ is primitive in the sense of

Definition 8.5. (iv) The image of κ in KS(T̄) is nonzero.

Proof. The equivalence (i) ⟺ (ii) is Corollary 16.2(ii), and (ii) ⟺ (iii) is Definition 8.5.

Suppose n ∈ N. By Lemma 12.1(i), κ_{n} maps to zero in H^{1}(Q, T̄) if and only if 𝔪^{k−1}κ_{n} = 0. Thus if the image of κ_{n} in H^{1}(Q, T̄) is nonzero, then κ_{n} must generate the free, rank-one R-module H′(n), so (iv) ⇒ (i). Conversely, suppose (ii) holds and n is a core vertex of X (these exist by Lemma 12.9). Then H′(n) is free of rank one over R, so 𝔪^{k−1}κ_{n} ≠ 0, and therefore (Lemma 12.1(i)) the image of κ_{n} in H^{1}(Q, T̄) is nonzero. Thus (ii) ⇒ (iv). □

Definition 16.4. We say that κ is primitive if it satisfies the equivalent conditions of Corollary 16.3.

The following theorem is a sharpening of the more general Corollary 15.2.

Theorem 16.5. Suppose χ(T) = 1 and κ ∈ KS(T) is primitive. Then

length(H_{F^{∗}}^{1}(Q, T^{∗})) = k − length(Rκ_{1}) = max{i : κ_{1} ∈ 𝔪^{i}H_{F}^{1}(Q, T)}.

Proof. By Corollary 16.3, κ_{1} generates H′(1) ⊂ 𝔪^{length(H_{F^{∗}}^{1} (Q,T^{∗}))}H_{F}^{1}(Q, T), and by Lemma 14.2, length(H′(1)) = k − length(H_{F^{∗}}^{1}(Q, T^{∗})). □

We can formulate a more precise version of Theorem 16.5, which determines the R-module structure of H_{F^{∗}}^{1}(Q, T^{∗}).

Definition 16.6. If κ ∈ KS(T) and r ≥ 0, define

∂^{(r)}(κ) = min{k − length(Rκ_{n}) : n ∈ N , ν(n) = r}.

Proposition 16.7. Suppose χ(T) = 1, κ ∈ KS(T), m ∈ N, and κ_{m} ≠ 0. Write H_{F^{∗}}^{1}(Q, T^{∗}) ≅ ⊕_{i}R/𝔪^{d_{i}} with nonnegative integers d_{1} ≥ d_{2} ≥ ⋯, and fix j ≥ 0 such that κ_{m} generates 𝔪^{j}H′(m).

Then for every r ≥ 0,

∂^{(r)}(κ) = min{k, j + ∑_{i>r} d_{i}}.

Proof. Note that since κ_{m} generates a nonzero submodule of the cyclic R-module H′(m), there is a unique j ≥ 0 such that κ_{m} generates 𝔪^{j}H′(m). By Corollary 16.2(ii), κ_{n} generates 𝔪^{j}H′(n) for every n. Therefore by Lemma 14.2 we have

∂^{(r)}(κ) = j + min{k − j, λ(n, T^{∗}) : n ∈ N , ν(n) = r}.

In particular when r = 0 we get the desired equality.

Suppose n ∈ N and ν(n) = r. Consider the map

H_{F^{∗}}^{1}(Q, T^{∗}) ⟶ ⨁_{ℓ|n}H_{f}^{1}(Q_{ℓ}, T^{∗}).

The right-hand side is free of rank r over R, so the image is a quotient of H_{F^{∗}}^{1}(Q, T^{∗}) generated by (at most) r elements. Hence the image has length at most ∑_{i≤r} d_{i}. Therefore the kernel has length at least ∑_{i>r} d_{i}. But by definition this kernel is contained in H_{F (n)^{∗}}^{1}(Q, T^{∗}), so we conclude that λ(n, T^{∗}) ≥ ∑_{i>r} d_{i}. Hence ∂^{(r)}(κ) ≥ min{k, j + ∑_{i>r} d_{i}}.

We will prove the opposite inequality by induction on r. The case r = 0 was proved above.

Since χ(T) = 1, Theorem 12.12(i) shows that 𝔪^{k−1}H_{F}^{1}(Q, T) ≠ 0. Fix a nonzero element c ∈ 𝔪^{k−1}H_{F}^{1}(Q, T) ⊂ H_{F}^{1}(Q, T)[𝔪]. If d_{1} > 0 then choose a nonzero element c′ ∈ 𝔪^{d_{1}−1}H_{F^{∗}}^{1}(Q, T^{∗}) ⊂ H_{F^{∗}}^{1}(Q, T^{∗})[𝔪]. Using Proposition 10.1, choose a prime ℓ ∈ P such that the localization c_{ℓ} is nonzero and, if d_{1} > 0, such that c′_{ℓ} is nonzero as well.

It follows that

• the localization map H_{F}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective, and
• H_{(F^{ℓ})^{∗}}^{1}(Q, T^{∗}) ≅ ⊕_{i≥1}R/𝔪^{d_{i+1}}.

By Theorem 12.7(ii) we have H_{F (ℓ)^{∗}}^{1}(Q, T^{∗}) = H_{(F^{ℓ})^{∗}}^{1}(Q, T^{∗}).

Let κ^{(ℓ)} ∈ KS(T, F (ℓ), P − {ℓ}) be the Kolyvagin system defined in Example 6.11, which is obtained by setting κ_{n}^{(ℓ)} = κ_{nℓ}⊗ξ for some generator ξ of Hom(G_{ℓ}, R). We have

∂^{(r)}(κ) ≥ ∂^{(r−1)}(κ^{(ℓ)}) = min{k, j + ∑_{i>r−1} d_{i+1}}.

where the inequality is clear from the definition and the equality follows from our induction hypothesis applied to κ^{(ℓ)}. This completes the proof. □

Theorem 16.8. Suppose χ(T) = 1, κ ∈ KS(T), and κ_{1} ≠ 0. Then

∂^{(0)}(κ) ≥ ∂^{(1)}(κ) ≥ ∂^{(2)}(κ) ≥ ⋯
∂^{(0)}(κ) − ∂^{(1)}(κ) ≥ ∂^{(1)} − ∂^{(2)}(κ) ≥ ⋯

and

H_{F^{∗}}^{1}(Q, T^{∗}) ≅ ⨁_{r≥0}R/𝔪^{∂^{(r)}(κ)−∂^{(r+1)}(κ)}.

Proof. This is immediate from Proposition 16.7. □

Part V. Kolyvagin systems over integral domains

17. Kolyvagin systems over a field

Suppose for this section that R = 𝕜 is a field, and that the Selmer structure (T, F, P) satisfies hypotheses (H.0) through (H.5) (hypothesis (H.6) is vacuous when R is a field).

In particular R is principal and artinian, so we can apply the results of Part IV with k = 1. The following theorem summarizes our results in this case.

Theorem 17.1. Suppose R = 𝕜 is a field.

(i) If χ(T) ≤ 1 then dim_{𝕜} KS(T) = χ(T).

(ii) If χ(T) = 1 and κ ∈ KS(T) is nonzero, then κ_{n} ≠ 0 if and only if

λ(n, T) = 1. (iii) If χ(T) = 1 and κ ∈ KS(T) is nonzero, then

dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}) = min{ν(n) : κ_{n} ≠ 0}.

Proof. Assertion (i) is Theorem 13.2 (when χ(T) = 0) and Corollary 16.2(i) (when χ(T) = 1). If χ(T) = 1 we have H′(n) ≠ 0 ⟺ λ(n, T^{∗}) = 0 ⟺ λ(n, T) = 1, so (ii) is Corollary 16.2(ii) in this setting.

Assertion (iii) is immediate from Proposition 16.7 applied to κ. □

Remark 17.2. Howard’s Theorem 14.3(iii) shows that if χ(T) > 1, then KS(T) is infinite dimensional over 𝕜.

Recall (Definition 12.14) that a leading vertex is an n ∈ N such that ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}) and H_{F (n)^{∗}}^{1}(Q, T^{∗}) = 0.

Theorem 17.3. Suppose χ(T) = 1, hypothesis (H.4a) is satisfied, and κ ∈ KS(T) is nonzero. If L is a line in H_{F}^{1}(Q, T), then there is a leading vertex n ∈ N such that κ_{n} generates L ⊗ G_{n}. If dim_{𝕜} H_{F}^{1}(Q, T) > 1 then there are infinitely many such n.

Proof. If dim_{𝕜} H_{F}^{1}(Q, T) > 1 then by Theorem 12.15 there are infinitely many leading vertices n such that L ⊂ H_{F (n)}^{1}(Q, T), and since a leading vertex has dim_{𝕜} H_{F (n)}^{1}(Q, T) = 1, we must have L = H_{F (n)}^{1}(Q, T). If dim_{𝕜} H_{F}^{1}(Q, T) = 1 then L = H_{F}^{1}(Q, T) and n = 1 is a leading vertex.

In either case, Theorem 17.1(ii) shows that κ_{n} ≠ 0, so κ_{n} is a generator of H_{F (n)}^{1}(Q, T) ⊗ G_{n} = L ⊗ G_{n}. □

18. Kolyvagin systems over a discrete valuation ring

For this section we assume that R is a discrete valuation ring. We assume that (T, F, P) satisfy Hypotheses (H.0) through (H.5). We assume also that the Selmer structure F is such that H^{1}(Q_{ℓ}, T)/H_{F}^{1}(Q_{ℓ}, T) is torsion free for every ℓ ∈ Σ(F). Then for every k, T/𝔪^{k}T (with the induced Selmer structure) satisfies (H.0) through (H.6) (this clear for (H.0) through (H.5), and for (H.6) it follows from Lemma 11.1(i)). Thus we can apply the results of Part IV to study the image of KS(T) in KS(T/𝔪^{k}T) for every k, and use this information to study KS(T).

Again we write T̄ = T/𝔪T. For simplicity we suppose that P = P_{1}.

Writing Frac(R) for the field of fractions of R, we define the rank of an Rmodule M to be rank_{R}M = dim_{Frac(R)} M ⊗ Frac(R) and the corank of M to be corank_{R}M = rank_{R}Hom(M, Q_{p}/Z_{p}).

Definition 18.1. If κ ∈ KS(T), define

∂^{(0)}(κ) = max{j : κ_{1} ∈ 𝔪^{j}H_{F}^{1}(Q, T)}

(we allow ∂^{(0)}(κ) = ∞). This is a special case of Definition 18.10 below.

Theorem 18.2. If κ ∈ KS(T) then

length_{R}H_{F^{∗}}^{1}(Q, T^{∗}) ≤ ∂^{(0)}(κ).

In particular if κ_{1} ≠ 0 then H_{F^{∗}}^{1}(Q, T^{∗}) is finite.

Proof. We may assume that κ_{1} ≠ 0 or else there is nothing to prove. Then ∂^{(0)}(κ) is finite, because H^{1}(Q, T) has no nonzero divisible submodules.

For every k ∈ Z^{+}, let κ^{(k)} be the image of κ in KS(T/𝔪^{k}T). Then κ_{1}^{(k)} is the image of κ_{1} under the injection H^{1}(Q, T)/𝔪^{k}H^{1}(Q, T) → H^{1}(Q, T/𝔪^{k}T). By Theorem 15.3, κ_{1}^{(2k)} ∈ 𝔪^{λ(1,T^{∗}[𝔪^{2k}])}H_{F}^{1}(Q, T/𝔪^{2k}T) + H_{F}^{1}(Q, T/𝔪^{2k}T)[𝔪^{k}]. Applying the natural map H_{F (n)}^{1}(Q, T/𝔪^{2k}T) → H_{F (n)}^{1}(Q, T/𝔪^{k}T) it follows (using

Lemma 12.1(i)) that κ_{1}^{(k)} ∈ 𝔪^{λ(1,T^{∗}[𝔪^{2k}])}H_{F}^{1}(Q, T/𝔪^{k}T). Since κ_{1} is the limit of the classes κ_{1}^{(k)} in H_{F}^{1}(Q, T) = \underleftarrow{lim} H_{F}^{1}(Q, T/𝔪^{k}T), and the sequence

λ(1, T^{∗}[𝔪^{k}]) = length_{R}H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{k}]) = length_{R}H_{F^{∗}}^{1}(Q, T^{∗})[𝔪^{k}]

converges to length_{R}H_{F^{∗}}^{1}(Q, T^{∗}), it follows that

κ_{1} ∈ 𝔪^{length_{R}H_{F^{∗}}^{1} (Q,T^{∗})}H_{F}^{1}(Q, T)

and the theorem follows from the definition of ∂^{(0)}(κ). □

Definition 18.3. For every k ∈ Z^{+} we have the Selmer triple (T/𝔪^{k}T, F, P_{k}) over R/𝔪^{k}, with the induced Selmer structure F. By Theorem 12.12, the core ranks χ(T/𝔪^{k}T) and χ(T^{∗}[𝔪^{k}]) are independent of k. We define χ(T), the core rank of T, to be this common value χ(T/𝔪^{k}T), and similarly χ(T^{∗}) = χ(T^{∗}[𝔪^{k}]).

In particular either χ(T) = 0 or χ(T^{∗}) = 0.

Theorem 18.4. For every k ∈ Z^{+} and every n ∈ N_{k} there is a noncanonical isomorphism

H_{F (n)}^{1}(Q, T/𝔪^{k}T) ⊕ (R/𝔪^{k})^{χ(T^{∗})} ≅ (R/𝔪^{k})^{χ(T)} ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}[𝔪^{k}]).

Proof. Since χ(T) = χ(T/𝔪^{k}T) and χ(T^{∗}) = χ(T^{∗}[𝔪^{k}]), this is just a restatement of Theorem 12.12. □

Corollary 18.5. rank_{R}(H_{F}^{1}(Q, T)) − corank_{R}(H_{F^{∗}}^{1}(Q, T^{∗})) = χ(T) − χ(T^{∗}).

Proof. We have

H_{F}^{1}(Q, T) = \underleftarrow{lim} H_{F}^{1}(Q, T/𝔪^{k}T), H_{F^{∗}}^{1}(Q, T^{∗}) = \underrightarrow{lim} H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{k}]),

so (using Lemma 12.1) this follows from Theorem 18.4. □

Lemma 18.6. For k ∈ Z^{+} consider the natural restriction map

ψ_{k} : KS(T, P) → KS(T/𝔪^{k}T, P_{k}).

Then ∩_{k>0} ker(ψ_{k}) = 0.

Proof. Suppose κ ∈ KS(T) is nonzero. Then we can find an n such that κ_{n} ≠ 0 in H_{F (n)}^{1}(Q, T/I_{n}T). If I_{n} ≠ 0 then let k be such that 𝔪^{k} = I_{n}. If I_{n} = 0 choose k so that κ_{n} ≠ 0 in H_{F (n)}^{1}(Q, T/(𝔪^{k})T). In either case I_{n} ⊂ 𝔪^{k}, so n ∈ N_{k} and ψ_{k}(κ) ≠ 0. □

Lemma 18.7. Suppose j ≤ k. The projection map T/𝔪^{k}T → T/𝔪^{j}T and restriction to P_{k} induce a surjection and an isomorphism, respectively

KS(T/𝔪^{k}T, P_{k}) ↠ KS(T/𝔪^{j}T, P_{k}) ⟵^{∼} KS(T/𝔪^{j}T, P_{j}).

Proof. Fix a core vertex n ∈ N_{k}. Then we have a commutative diagram

\xymatrix{KS(T/𝔪^{k}, P_{k}) \ar[r] \ar[d]^{≅} & KS(T/𝔪^{j}T, P_{k}) \ar[d]^{≅} & KS(T/𝔪^{j}T, P_{j}) \ar[l] \ar[d]^{≅} \\ H_{F (n)}^{1}(Q, T/𝔪^{k}) ⊗ G_{n} \ar@{->>}[r] & H_{F (n)}^{1}(Q, T/𝔪^{j}) ⊗ G_{n} & H_{F (n)}^{1}(Q, T/𝔪^{j}) ⊗ G_{n} \ar[l]_{id}}

In which the vertical maps are the specialization isomorphisms of Corollary 16.2(iii). Since n is a core vertex, Lemma 12.1(i) shows that bottom left-hand map is surjective, and the bottom right-hand map is the identity map. This proves the lemma. □

Proposition 18.8. The natural map KS(T) → \underleftarrow{lim} KS(T/𝔪^{k}, P_{k}), inverse limit with respect to the surjections of Lemma 18.7, is an isomorphism.

Proof. The injectivity is Lemma 18.6, so we need only show surjectivity. Suppose \{κ^{(k)}\} ∈ \underleftarrow{lim} KS(T/𝔪^{k}, P_{k}). If n ∈ N, let j be maximal such that n ∈ N_{j}. If

j < ∞ then I_{n} = 𝔪^{j}, and we define κ_{n} = κ_{n}^{(j)} ∈ H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n}. If j = ∞

(for example, when n = 1) then we define κ_{n} = lim_{k} κ_{n}^{(k)} ∈ H_{F (n)}^{1}(Q, T) ⊗ G_{n}. It is straightforward to verify that this defines an element κ ∈ KS(T) which maps to κ^{(k)} ∈ KS(T/𝔪^{k}T) for every k. □

Theorem 18.9. (i) If χ(T) = 0 then KS(T) = 0.

(ii) If χ(T) = 1 then KS(T) is a free R-module of rank one, and there is a κ ∈ KS(T) whose image in KS(T̄) is nonzero.

Proof. If χ(T) ≤ 1, then Theorem 13.2 and Corollary 16.2 show that for every k, KS(T/𝔪^{k}, P_{k}) is free of rank χ(T) over R/𝔪^{k}. The maps KS(T/𝔪^{k+1}, P_{k+1}) → KS(T/𝔪^{k}, P_{k}) of Lemma 18.7 are surjective, so the Theorem follows from Proposition 18.8. □

Definition 18.10. Suppose χ(T) = 1. We define the Kolyvagin rank of T to be

KolyvaginRk(T) = min{ν(n) : n ∈ N , κ_{n} ≠ 0}

where κ ∈ KS(T) is nonzero. That this is independent of the choice of κ follows from the next theorem.

Also, for κ ∈ KS(T) and r ∈ Z^{+} define (compare Definition 16.6)

∂^{(r)}(κ) = lim_{k→∞} max{j : κ_{n} ∈ 𝔪^{j}H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n}

for every n ∈ N_{k} with ν(n) = r} (we allow ∂^{(r)}(κ) = ∞). Then ∂^{(0)}(κ) = max{j : κ_{1} ∈ 𝔪^{j}H_{F}^{1}(Q, T)} as in Definition 18.1, and if r < KolyvaginRk(T) then ∂^{(r)}(κ) = ∞. Define

∂^{(∞)}(κ) = min{∂^{(r)}(κ) : r ≥ 0}.

Theorem 18.11. Suppose χ(T) = 1 and κ ∈ KS(T), κ ≠ 0. Then

corank_{R}(H_{F^{∗}}^{1}(Q, T^{∗})) = KolyvaginRk(T)

and

H_{F^{∗}}^{1}(Q, T^{∗})/(H_{F^{∗}}^{1}(Q, T^{∗}))_{div} ≅ ⨁_{i≥KolyvaginRk(T)}R/𝔪^{∂^{(i)}(κ)−∂^{(i+1)}(κ)}.

In particular • the sequence ∂^{(i)}(κ) is nonincreasing, and finite for i ≥ KolyvaginRk(T),

• KolyvaginRk(T) is independent of the choice of (nonzero) κ ∈ KS(T),

• length_{R}(H_{F^{∗}}^{1}(Q, T^{∗})/(H_{F^{∗}}^{1}(Q, T^{∗}))_{div}) = ∂^{(KolyvaginRk(T))}(κ) − ∂^{(∞)}(κ).

Proof. Let j ≥ 0 be minimal such that the image of κ in KS(T/𝔪^{j+1}T, P_{j+1}) is nonzero. (Such a j exists by Proposition 18.8.) By Proposition 18.7 it follows that for every k > j there is an n ∈ N_{k} such that κ_{n} generates 𝔪^{j}H_{F (n)}^{1}(Q, T/I_{n}T)⊗G_{n}.

Write r = corank_{R}(H_{F^{∗}}^{1}(Q, T^{∗})) and

H_{F^{∗}}^{1}(Q, T^{∗})/(H_{F^{∗}}^{1}(Q, T^{∗}))_{div} ≅ ⨁_{i>r}R/𝔪^{d_{i}}

with d_{r+1} ≥ d_{r+2} ≥ · · ·.

Fix k > h + ∑_{i>r} d_{i}. Then, writing d_{i} = k for i ≤ r,

H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{k}]) = H_{F^{∗}}^{1}(Q, T^{∗})[𝔪^{k}] ≅ ⨁_{i≥1}R/𝔪^{d_{i}}.

Therefore applying Proposition 16.7 to the image of κ in KS(T/𝔪^{k}, P_{k′}) for all k′ ≥ 2k − 1 shows that for every i ≥ r, ∂^{(i)}(κ) is finite and

∂^{(i)}(κ) − ∂^{(i+1)}(κ) = d_{i+1}.

This proves the theorem. □

As in Definition 16.4, we say that κ ∈ KS(T) is primitive if its image in KS(T̄) is nonzero. Note that (using Lemma 18.7) κ is primitive if and only if ∂^{(i)}(κ) = 0 for sufficiently large i.

Corollary 18.12. Suppose χ(T) = 1 and κ ∈ KS(T) is nonzero.

(i) length(H_{F^{∗}}^{1}(Q, T^{∗})) is finite ⟺ κ_{1} ≠ 0.

(ii) length(H_{F^{∗}}^{1}(Q, T^{∗})) ≤ ∂^{(0)}(κ) = max{j : κ_{1} ∈ 𝔪^{j}H_{F}^{1}(Q, T)}, with equality if and only if κ is primitive.

Proof. If κ_{1} = 0 then KolyvaginRk(T) ≥ 1, so length(H_{F^{∗}}^{1}(Q, T^{∗})) is infinite by Theorem 18.11. The other direction of (i) is Theorem 18.2.

Assertion (ii) is immediate from Theorem 18.11, since κ is primitive if and only if ∂^{(∞)}(κ) = 0. □

Let d^{−} = rank_{R}(T^{−}), where T^{−} is the minus-part of T for the action of some complex conjugation.

Theorem 18.13. Suppose R/2R is finite and F = F_{can}, the canonical Selmer structure on T given by Definition 7.1. Then χ(T^{∗}) = 0 and

χ(T) = d^{−} + corank_{R}(T^{∗}^{G_{Qp}}).

Proof. Suppose k > 0. By Proposition 1.6 of [Wi],

length(H_{F}^{1}(Q, T/𝔪^{k}T) − length(H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{k}])

= length(T^{∗}[𝔪^{k}]^{G_{R}}) + length((T/𝔪^{k}T)^{G_{Q}}) + length(T^{∗}[𝔪^{k}]^{G_{Q}})

+ ∑_{ℓ∈Σ(F),ℓ≠∞} (length(T^{∗}[𝔪^{k}]^{G_{Q_{ℓ}}}) − length(H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}[𝔪^{k}]))). (9)

By Theorem 18.4, the left-hand side of (9) is k(χ(T) − χ(T^{∗})). Writing d^{−} = rank_{R}T^{−}, where T^{−} is the minus-part of T for the action of complex conjugation, we have that |length(T^{∗}[𝔪^{k}]^{G_{R}}) − kd^{−}| is bounded independently of k (and is zero unless p = 2). By Lemma 9.2, (T/𝔪^{k}T)^{G_{Q}} = T^{∗}[𝔪^{k}]^{G_{Q}} = 0.

Suppose ℓ ∈ Σ(F), ℓ ≠ p. Recalling (Definition 1.5(iii))

H_{unr}^{1}(Q_{ℓ}, T^{∗}[𝔪^{k}]) = H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, T^{∗}[𝔪^{k}]^{I_{ℓ}}) = T^{∗}[𝔪^{k}]^{I_{ℓ}}/(Fr_{ℓ} − 1)T^{∗}[𝔪^{k}]^{I_{ℓ}}

we have an exact sequence

0 → T^{∗}[𝔪^{k}]^{G_{Q_{ℓ}}} → T^{∗}[𝔪^{k}]^{I_{ℓ}} ⟶^{Fr_{ℓ}−1} T^{∗}[𝔪^{k}]^{I_{ℓ}} → H_{unr}^{1}(Q_{ℓ}, T^{∗}[𝔪^{k}]) → 0

and so length(T^{∗}[𝔪^{k}]^{G_{Q_{ℓ}}}) = length(H_{unr}^{1}(Q_{ℓ}, T^{∗}[𝔪^{k}])). It now follows from Lemma 1.3.5 of [Ru6] that length(T^{∗}[𝔪^{k}]^{G_{Q_{ℓ}}}) − length(H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}[𝔪^{k}])) is bounded independently of k.

When ℓ = p, H_{F^{∗}}^{1}(Q_{p}, T^{∗}) = 0 by definition of the canonical Selmer structure, so H_{F^{∗}}^{1}(Q_{p}, T^{∗}[𝔪^{k}]) = ker(H^{1}(Q_{p}, T^{∗}[𝔪^{k}]) → H^{1}(Q_{p}, T^{∗})), which has order bounded by |H^{0}(Q_{p}, T^{∗})/H^{0}(Q_{p}, T^{∗})_{div}|, which is finite and independent of k.

Combining these observations we conclude that

k(χ(T) − χ(T^{∗})) − kd^{−} − k corank_{R}(T^{∗}^{G_{Qp}})

is bounded independently of k, which proves the theorem. □

For applications of these results see §20 and §21.

19. Kolyvagin systems over Λ

Let Q_{∞} denote the cyclotomic Z_{p}-extension of Q, and Q_{n} ⊂ Q_{∞} the (unique, cyclic) extension of degree p^{n} over Q. Let Λ denote the Iwasawa algebra

Λ = Z_{p}[[Gal(Q_{∞}/Q)]] = \underleftarrow{lim} Z_{p}[Gal(Q_{n}/Q)].

For this section suppose that T is a free Z_{p}-module of finite rank with a continuous action of G_{Q}, unramified outside a finite set of primes. We set T = T ⊗ Λ with G_{Q} acting on both factors, and we take R = Λ. Let A denote the augmentation ideal of Λ, so that T/AT = T, and let T̄ = T/pT = T/𝔪T.

We assume throughout this section that T satisfies hypotheses (H.0) through (H.4), and then it follows immediately that T satisfies (H.0) through (H.4) as well. For simplicity we fix P = P_{1}. Fix a finite set of primes Σ containing p, ∞, and the primes where T is ramified, and let Q_{Σ} denote the maximal extension of Q unramified outside of Σ.

Lemma 19.1. (i) For every i, H^{i}(Q_{Σ}/Q, T) = \underleftarrow{lim} H^{i}(Q_{Σ}/Q_{n}, T).

(ii) If ℓ ≠ p then H^{1}(Q_{ℓ}, T) = H_{unr}^{1}(Q_{ℓ}, T).
(iii) H^{1}(Q, T) = H^{1}(Q_{Σ}/Q, T).

Proof. (See [Co] Proposition II.1.1.) By Shapiro’s Lemma,

H^{1}(Q_{Σ}/Q, T ⊗ Z_{p}[Gal(Q_{n}/Q)]) = H^{1}(Q_{Σ}/Q_{n}, T),

and so there is an injection

H^{1}(Q, T) ↪ H^{1}(Q, \underleftarrow{lim} T ⊗ Z_{p}[Gal(Q_{n}/Q)]) = \underleftarrow{lim} H^{1}(Q_{n}, T).

Since the groups H^{1}(Q_{Σ}/Q_{n}, T) are compact for every n, this map is surjective as well, which gives (i). Similarly, if ℓ is a prime we have an isomorphism

H^{1}(Q_{ℓ}, T) ↪ \underleftarrow{lim} ⊕_{λ|ℓ}H^{1}(Q_{n,λ}, T).

By a standard argument (see [Ru6] Proposition B.3.4 for details),

\underleftarrow{lim} ⊕_{λ|ℓ}H^{1}(Q_{n,λ}, T) = \underleftarrow{lim} ⊕_{λ|ℓ}H_{unr}^{1}(Q_{n,λ}, T).

This proves (ii), and then (iii) follows from (i) and (ii). □

Definition 19.2. We fix the simplest possible Selmer structure F_{Λ} on T. Namely, we let Σ(F_{Λ}) = Σ, and we set H_{F}^{1}(Q_{v}, T) = H^{1}(Q_{v}, T) for v ∈ Σ. By Lemma 19.1(ii) we also have H_{f}^{1}(Q_{v}, T) = H^{1}(Q_{v}, T) for v ∉ Σ, so this Selmer structure is independent of the choice of Σ.

Note that the induced Selmer structure F_{Λ} on quotients T/IT (such as T and T̄) will not usually satisfy H_{F_{Λ}}^{1}(Q_{v}, T/IT) = H^{1}(Q_{v}, T/IT).

We have the following analogue of Theorem 7.4, which says that an Euler system for T gives rise to a Kolyvagin system for T. Let ES(T) be the Galois module of Euler systems for T as given in Definition 7.2.

Theorem 19.3. Suppose that H^{0}(Q_{p}, T^{∗}) is a divisible R-module, and P is a set of primes ℓ satisfying

• T/(Fr_{ℓ} − 1)T is a cyclic R-module,

• Fr_{ℓ}^{p^{k}} − 1 is injective on T for every k ≥ 0.

Then there is a canonical homomorphism ES(T, P) → KS(T, F_{Λ}, P) with the property that if c maps to κ, then

κ_{1} = \{c_{Q_{n}}\} ∈ \underleftarrow{lim} H^{1}(Q_{n}, T) = H^{1}(Q, T).

This theorem will be proved in Appendix A, along with Theorem 7.4.

Theorem 19.4. Suppose κ ∈ KS(T), and κ_{1} ≠ 0. Then H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}) is a cotorsion Λ-module.

Theorem 19.4 will be proved below. The assertion that H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}) is a cotorsion Λ-module is a form of the Weak Leopoldt conjecture. See for example [Gr2] Conjecture 2 or [PR3] §1.3.

Definition 19.5. If X is a finitely generated torsion Λ-module, then there is a pseudo-isomorphism (Λ-module map with map with finite kernel and cokernel) X → ⊕_{i}Λ/f_{i}Λ, where the f_{i} are nonzero elements of Λ. The characteristic ideal char(X) of X is the ideal (∏_{i} f_{i})Λ. If convenient we may assume that each f_{i} is a power of an irreducible element of Λ.

Definition 19.6. If c ∈ H^{1}(Q, T), we let Ind(c) denote the ideal of Λ

Ind(c) = {ψ(c) : ψ : H^{1}(Q, T) → Λ}.

Then Ind(c) is a principal ideal, and there is an ideal B of finite index in Λ such that Bc ∈ Ind(c)H^{1}(Q, T).

If κ ∈ KS(T) we write Ind(κ) = Ind(κ_{1}).

Let X_{∞} = Hom(H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}), Q_{p}/Z_{p}).

Theorem 19.7. Suppose κ ∈ KS(T).

(i) char(X_{∞}) divides Ind(κ).

(ii) If χ(T) = 1, κ_{1} ≠ 0, and 𝔓 is is a height-one prime of Λ not in the support

of κ, then ord_{𝔓}(char(X_{∞})) = ord_{𝔓}(Ind(κ)). (iii) If χ(T) = 1, κ_{1} ≠ 0, and κ is Λ-primitive then char(X_{∞}) = Ind(κ).

The rest of this section is devoted to the proofs of Theorems 19.4 and 19.7. We will apply the results of §18 to quotients T/𝔓T for appropriate primes 𝔓 of Λ, and deduce the desired results about T.

Suppose 𝔓 is a height-one prime ideal of Λ. Let S_{𝔓} denote the integral closure of Λ/𝔓. Then S_{𝔓} is a discrete valuation ring, [S_{𝔓} : Λ/𝔓] is finite, and T ⊗_{Λ} S_{𝔓} = T ⊗_{Z_{p}} S_{𝔓}. We will study Kolyvagin systems on T by studying their images in KS(T ⊗ S_{𝔓}), and applying the results of §18 to the latter. We will make frequent use of the exact sequence, obtained by fixing a generator ρ of 𝔓,

0 ⟶ T ⟶^{ρ} T ⟶ T/𝔓T ⟶ 0.  (10)

As in Definition 7.1, we have a canonical Selmer structure F_{can} on T ⊗ S_{𝔓} given by

• Σ(F_{can}) = {ℓ : T is ramified at ℓ} ∪ {p, ∞},
• if ℓ ∈ Σ(F_{can}) and ℓ ≠ p, ∞ then
H_{F_{can}}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) = ker(H^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) → H^{1}(Q_{ℓ}^{unr}, T ⊗ Frac(S_{𝔓}))),
• H_{F_{can}}^{1}(Q_{p}, T ⊗ S_{𝔓}) = H^{1}(Q_{p}, T ⊗ S_{𝔓}),
• H_{F_{can}}^{1}(R, T ⊗ S_{𝔓}) = H^{1}(R, T ⊗ S_{𝔓}),

where Frac(S_{𝔓}) is the field of fractions of S_{𝔓}.

Definition 19.8. Define an exceptional set of height-one primes of Λ by

Σ_{Λ} = {𝔓 : H^{2}(Q_{Σ}/Q, T)[𝔓] is infinite} ∪ {𝔓 : H^{2}(Q_{p}, T)[𝔓] is infinite} ∪ {pΛ}.

It follows from Lemma 19.9 below that Σ_{Λ} is finite.

Lemma 19.9. For every i ≥ 0, the Λ-modules H^{i}(Q_{Σ}/Q, T) and H^{i}(Q_{p}, T) are finitely generated, and H^{i}(Q_{Σ}/Q, T^{∗}) are co-finitely generated.

Further, H^{2}(Q_{p}, T) is a torsion Λ-module.

Proof. These are standard result. For the global cohomology groups see for example [Gr2] Proposition 3 or §3 of [PR1]. For the local cohomology groups see for example [Gr2] Proposition 1 or [PR2] Proposition 3.2.1. □

Lemma 19.10. Suppose 𝔓 is a height-one prime ideal of Λ. The inclusion

T/𝔓T = T ⊗ (Λ/𝔓) ↪ T ⊗ S_{𝔓}

induces maps

H_{F_{Λ}}^{1}(Q_{v}, T/𝔓T) → H_{F_{can}}^{1}(Q_{v}, T ⊗ S_{𝔓}),
H_{F_{can}^{∗}}^{1}(Q_{v}, (T ⊗ S_{𝔓})^{∗}) → H_{F_{Λ}^{∗}}^{1}(Q_{v}, (T/𝔓T)^{∗}),

for every place v. If 𝔓 ∉ Σ_{Λ}, then the kernels and cokernels of these maps are finite with order bounded by a constant depending only on T and [S_{𝔓} : Λ/𝔓].

Proof. First suppose that v is a prime ℓ ≠ p, and let I be the inertia group of ℓ. By definition, H_{F_{Λ}}^{1}(Q_{ℓ}, T/𝔓T) is the image of H^{1}(Q_{ℓ}, T) in H^{1}(Q_{ℓ}, T/𝔓T). Using Lemma 19.1(ii) for the first equality, the map H^{1}(Q_{ℓ}, T) → H^{1}(Q_{ℓ}, T/𝔓T) factors through

H^{1}(Q_{ℓ}, T) = H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, T^{I}) → H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, T^{I}/𝔓T^{I})

→ H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, (T/𝔓T)^{I}) = H_{unr}^{1}(Q_{ℓ}, T/𝔓T). (11)

Hence H_{F_{Λ}}^{1}(Q_{ℓ}, T/𝔓T) ⊂ H_{unr}^{1}(Q_{ℓ}, T/𝔓T), and so the image of H_{F_{Λ}}^{1}(Q_{ℓ}, T/𝔓T) in H^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) is contained in H_{unr}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}). By definition this is a submodule of H_{F_{can}}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}), so we obtain the desired map

H_{F_{Λ}}^{1}(Q_{ℓ}, T/𝔓T) ⟶ H_{F_{can}}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}).  (12)

The kernel of H^{1}(Q_{ℓ}, T/𝔓T) → H^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) is a quotient of H^{0}(Q_{ℓ}, T ⊗ (S_{𝔓}/(Λ/𝔓))), which has order bounded by [S_{𝔓} : Λ/𝔓]^{rank_{Zp} T}. Hence the same bound holds for the kernel of (12).

We now consider the cokernel of (12). In (11), the map

H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, T^{I}) → H^{1}(Q_{ℓ}^{unr}/Q_{ℓ}, T^{I}/𝔓T^{I})

is surjective because Gal(Q_{ℓ}^{unr}/Q_{ℓ}) has cohomological dimension one. There is an I-cohomology exact sequence (since 𝔓 is principal)

0 ⟶ T^{I}/𝔓T^{I} ⟶ (T/𝔓T)^{I} ⟶ H^{1}(I, T)[𝔓] ⟶ 0.

Since ℓ ≠ p the action of I on Λ is trivial, and since Λ is free over Z_{p} we get

H^{1}(I, T)[𝔓] = (H^{1}(I, T) ⊗ Λ)[𝔓].

But 𝔓 ≠ pΛ, so (recall that 𝔓 acts only on the second factor of this tensor product) H^{1}(I, T) ⊗ Λ has no 𝔓-torsion. Thus the composition (11) is surjective.

The cokernel of H_{unr}^{1}(Q_{ℓ}, T/𝔓T) → H_{unr}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) has order bounded by |T ⊗ (S_{𝔓}/(Λ/𝔓))|. A straightforward diagram chase (see the proof of Lemma 1.3.5 of [Ru6]) shows that

H_{F_{can}}^{1}(Q_{ℓ}, T ⊗ S_{𝔓})/H_{unr}^{1}(Q_{ℓ}, T ⊗ S_{𝔓}) ≅ H^{1}(I, T)_{tors}^{Fr_{ℓ}=1}

which has order bounded by |H^{1}(I, T)_{tors}|. Note that H^{1}(I, T)_{tors} is finite for every ℓ, and if T is unramified at ℓ then H^{1}(I, T)_{tors} = Hom(I, T)_{tors} = 0. Thus the cokernel of (12) is finite with a bound depending only on T.

Now consider the case v = p. Taking G_{Q_{p}}-cohomology of the sequence (10) shows that the cokernel of H^{1}(Q_{p}, T) → H^{1}(Q_{p}, T/𝔓T) is H^{2}(Q_{p}, T)[𝔓]. Since 𝔓 ∉ Σ_{Λ}, we have that H^{2}(Q_{p}, T)[𝔓] is finite. Hence this cokernel is no bigger than the maximal finite submodule of the finitely-generated Λ-module H^{2}(Q_{p}, T), so [H^{1}(Q_{p}, T/𝔓T) : H_{F_{Λ}}^{1}(Q_{p}, T/𝔓T) is finite and independent of 𝔓.

By definition H_{F_{can}}^{1}(Q_{p}, T ⊗ S_{𝔓}) = H^{1}(Q_{p}, T ⊗ S_{𝔓}), so the first map of the lemma is just the composition

H_{F_{Λ}}^{1}(Q_{p}, T/𝔓T) ↪ H^{1}(Q_{p}, T/𝔓T) → H^{1}(Q_{p}, T ⊗ S_{𝔓}).

The kernel and cokernel of the map H^{1}(Q_{p}, T/𝔓T) → H^{1}(Q_{p}, T ⊗ S_{𝔓}) are controlled by H^{i}(Q_{p}, T ⊗(S_{𝔓}/(Λ/𝔓))) with i = 0 and 1, respectively, and these groups have bounds of the desired sort.

When v = ∞, it is straightforward to check that H^{1}(R, T) → H^{1}(R, T/𝔓T) is surjective, so H_{F_{Λ}}^{1}(R, T/𝔓T) = H^{1}(R, T/𝔓T), and then the kernel and cokernel of H^{1}(R, T/𝔓T) → H^{1}(R, T ⊗ S_{𝔓}) are bounded exactly as for v = p.

This gives the desired properties of the first map of the lemma for every v. The proof for the second map is similar (the “dual” of the proof above). □

Proposition 19.11. For every height-one prime ideal 𝔓 of Λ, the composition T ↠ T ⊗ (Λ/𝔓) ↪ T ⊗ S_{𝔓} induces maps

π_{𝔓} : H_{F_{Λ}}^{1}(Q, T)/𝔓H_{F_{Λ}}^{1}(Q, T) ↪ H_{F_{can}}^{1}(Q, T ⊗ S_{𝔓}),
π_{𝔓}^{∗} : H_{F_{can}^{∗}}^{1}(Q, (T ⊗ S_{𝔓})^{∗}) → H_{F_{Λ}^{∗}}^{1}(Q, T^{∗})[𝔓].

For every 𝔓 the map π_{𝔓} is injective. If 𝔓 ∉ Σ_{Λ} then coker(π_{𝔓}), ker(π_{𝔓}^{∗}), and coker(π_{𝔓}^{∗}) are all finite with order bounded by a constant depending only on T and [S_{𝔓} : Λ/𝔓].

Proof. By Lemma 19.1(iii), H^{1}(Q_{Σ}/Q, T) = H^{1}(Q, T). Thus Gal(Q_{Σ}/Q)-coho-mology of (10) yields an injective map

H^{1}(Q, T)/𝔓H^{1}(Q, T) ↪ H^{1}(Q_{Σ}/Q, T/𝔓T)  (13)

with cokernel H^{2}(Q_{Σ}/Q, T)[𝔓]. Since 𝔓 ∉ Σ_{Λ}, H^{2}(Q_{Σ}/Q, T)[𝔓] is contained in the maximal finite submodule of the finitely-generated Λ-module H^{2}(Q_{Σ}/Q, T). Thus the cokernel of (13) is finite and bounded by a constant depending only on T.

The map

H^{1}(Q_{Σ}/Q, T/𝔓T) ⟶ H^{1}(Q_{Σ}/Q, T ⊗ S_{𝔓})  (14)

has kernel and cokernel controlled by H^{i}(Q_{Σ}/Q, T ⊗ (S_{𝔓}/(Λ/𝔓))) with i = 0 and 1, respectively.

The Λ-module S_{𝔓}/(Λ/𝔓) has a Jordan-Holder filtration in which all quotients are Z/pZ (with trivial Galois action). Hence by Lemma 9.2,

H^{0}(Q_{Σ}/Q, T ⊗ (S_{𝔓}/(Λ/𝔓))) = 0,

so (14) is injective. On the other hand, H^{1}(Q_{Σ}/Q, T ⊗ (S_{𝔓}/(Λ/𝔓))) is finite with order bounded by a function of rank_{Z_{p}}T, [S_{𝔓} : Λ/𝔓], and Σ. Thus the cokernel of (14) is bounded in terms of T and [S_{𝔓} : Λ/𝔓].

Lemma 19.10 shows that (14) restricts to a (still injective) map

H_{F_{Λ}}^{1}(Q, T/𝔓T) ↪ H_{F_{can}}^{1}(Q, T ⊗ S_{𝔓}).  (15)

It follows from the bounds on the kernels and cokernels in Lemma 19.10 that the cokernel of (14) is finite with order bounded by a constant depending only on T and [S_{𝔓} : Λ/𝔓]. The map π_{𝔓} is the composition of (13) and (14), so combining the observations above we get the desired properties of ker(π_{𝔓}) and coker(π_{𝔓}).

The map π_{𝔓}^{∗} and the bounds on its kernel and cokernel come from Lemma 19.10 in the same way, using Lemma 9.3 to identify H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}[𝔓]) with H_{F_{Λ}^{∗}}^{1}(Q, T^{∗})[𝔓].

□

Corollary 19.12. For every height-one prime 𝔓 of Λ, there is a natural map

KS(T, F_{Λ}) → KS(T ⊗ S_{𝔓}, F_{can}).

Proof. We have maps KS(T, F_{Λ}) → KS(T/𝔓T, F_{Λ}) → KS(T ⊗ S_{𝔓}, F_{can}), the first by functoriality (Remark 6.5) and the second by Lemma 19.10. □

Lemma 19.13. If 𝔓 is a height-one prime of Λ, and 𝔓 ∉ Σ_{Λ}, then

χ(T ⊗ S_{𝔓}, F_{can}) = rank_{Z_{p}}T^{−}

where T^{−} is the minus part of T for complex conjugation.

Proof. By Theorem 18.13,

χ(T ⊗ S_{𝔓}) = rank_{S_{𝔓}}(T ⊗ S_{𝔓})^{−} + corank_{S_{𝔓}}H^{0}(Q_{p}, (T ⊗ S_{𝔓})^{∗})

where (T ⊗ S_{𝔓})^{−} is the minus part of T ⊗ S_{𝔓} for complex conjugation. But (T ⊗ S_{𝔓})^{−} = T^{−} ⊗S_{𝔓}, so rank_{S_{𝔓}}(T ⊗S_{𝔓})^{−} = rank_{Z_{p}}T^{−}. Further, H^{0}(Q_{p}, (T/𝔓T)^{∗}) is dual to H^{2}(Q_{p}, T/𝔓T), and H^{2}(Q_{p}, T/𝔓T) ≅ H^{2}(Q_{p}, T)/𝔓H^{2}(Q_{p}, T) because G_{Q_{p}} has cohomological dimension 2. Since H^{2}(Q_{p}, T) is a finitely-generated torsion Λ-module (Lemma 19.9) and 𝔓 ∉ Σ_{Λ}, we see that H^{2}(Q_{p}, T)/𝔓H^{2}(Q_{p}, T) is finite. Hence H^{0}(Q_{p}, (T ⊗ S_{𝔓})^{∗}) is finite as well, and this proves the lemma. □

Definition 19.14. We define χ(T) to be the common value (by Lemma 19.13) of χ(T ⊗ S_{𝔓}, F_{can}) for 𝔓 ∉ Σ_{Λ}. Equivalently, χ(T) = rank_{Z_{p}}T^{−}.

Remark 19.15. By Proposition 1.3.2 of [PR3], if the weak Leopoldt conjecture holds for T then rank_{Λ}H_{F_{Λ}}^{1}(Q, T) = χ(T).

Definition 19.16. If κ ∈ KS(T) and 𝔓 is a height-one prime of Λ, let κ^{(𝔓)} denote the image of κ in KS(T ⊗S_{𝔓}, F_{can}). The support of κ is the set of height-one primes 𝔓 such that κ^{(𝔓)} = 0.

We will say that κ is Λ-primitive if the support of κ is empty, i.e., κ^{(𝔓)} ≠ 0 for every 𝔓. (This is weaker than being primitive (Definition 16.4), which requires that the image of κ be nonzero in KS(T̄).)

Lemma 19.17. The Λ-module H_{F_{Λ}}^{1}(Q, T) is finitely generated and torsion-free, and H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}) is co-finitely generated.

Proof. The fact that these modules are (co-) finitely generated follows from Lemma 19.9. By Lemma 9.2, T^{G_{Q}} = 0, so by the lemma of §1.3.3 of [PR3], H^{1}(Q, T) has no Λ-torsion. □

Corollary 19.18. Suppose κ ∈ KS(T) and κ_{1} ≠ 0. Then for all but finitely many height-one primes 𝔓 of Λ, the image of κ_{1} in H^{1}(Q, T ⊗ S_{𝔓}) is nonzero.

Proof. Since κ_{1} is a nonzero element of the finitely-generated torsion-free (Lemma 19.17) Λ-module H_{F_{Λ}}^{1}(Q, T), there are only finitely many height-one primes 𝔓 such that κ_{1} ∈ 𝔓H_{F_{Λ}}^{1}(Q, T). Thus the corollary follows from the injectivity of the map π_{𝔓} in Proposition 19.11. □

Proof of Theorem 19.4. By Corollary 19.18 we can choose a height-one prime 𝔓 of Λ such that 𝔓 ∉ Σ_{Λ} and κ_{1}^{(𝔓)} ≠ 0, where κ^{(𝔓)} denotes the image of κ in KS(T ⊗ S_{𝔓}, F_{can}).

Applying Theorem 18.2 to κ^{(𝔓)} shows that H_{F_{can}^{∗}}^{1}(Q, (T ⊗ S_{𝔓})^{∗}) is finite, and then Proposition 19.11 shows that H_{F_{Λ}^{∗}}^{1}(Q, T^{∗})[𝔓] is finite. But Λ^{∗}[𝔓] = (Λ/𝔓)^{∗} is infinite, and so H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}) must be co-torsion. □

Proof of Theorem 19.7. If κ_{1} ≠ 0 then Ind(κ) = 0 and there is nothing to prove. So we assume from now on that κ_{1} = 0.

Fix a height-one prime 𝔓 of Λ, and suppose first that 𝔓 ≠ pΛ. For convenience we identify Λ with the power series ring Z_{p}[[U]], and we fix a generator ρ(U) of 𝔓 which is a distinguished polynomial (a monic polynomial congruent to U^{deg(ρ)} modulo p). For every N let

𝔓_{N} = (ρ(U) + p^{N})Λ.

Since 𝔓 ≠ pΛ, the 𝔓_{N} are distinct ideals of Λ, and different from 𝔓.

Fix a pseudo-isomorphism

X_{∞} ⟶ ⨁_{i}(Λ/𝔓^{m_{i}})⨁_{j}(Λ/f_{j}Λ)  (16)

where each f_{j} is prime to 𝔓. In particular ord_{𝔓}(char(X_{∞})) = ∑_{i} m_{i}.

We first claim that if N is sufficiently large, then 𝔓_{N} satisfies the following properties:

(i) 𝔓_{N} is a prime ideal and Λ/𝔓 ≅ Λ/𝔓_{N}, (ii) the image of κ_{1} in H^{1}(Q, T ⊗ S_{𝔓_{N}}) is nonzero, (iii) the cokernel of the injection

H_{F_{Λ}}^{1}(Q, T)/𝔓_{N}H_{F_{Λ}}^{1}(Q, T) ↪ H_{F_{can}}^{1}(Q, T ⊗ S_{𝔓_{N}})

is finite with order bounded by a constant independent of N, (iv) 𝔓_{N} is prime to each f_{j} in (16), and 𝔓 ∉ Σ_{Λ}.

It follows without difficulty from Hensel’s Lemma that the first property holds for N sufficiently large. The second property holds for all but finitely many N by Corollary 19.18, and the third holds for all but finitely many N by Proposition 19.11 (and using property (i)). The fourth property clearly holds for all but finitely many N.

Fix an N large enough so that these properties hold, write 𝔔 = 𝔓_{N}, and let κ^{(𝔔)} denote the image of κ in KS(T ⊗ S_{𝔔}, F_{can}). we will apply the results of §18 to κ^{(𝔔)}. By Lemma 19.13 we have χ(T ⊗ S_{𝔔}, F_{can}) = 1.

Let d = ord_{𝔓}(Ind(κ)), and let e be the ramification degree of S_{𝔔}/Z_{p}. Since Λ/(𝔓^{d}, 𝔔) = Λ/(𝔔, p^{Nd}), it follows from (ii) and (iii) above that |∂^{(0)}(κ^{(𝔔)})−Nde| is bounded independently of N. Hence by Theorem 18.2 (writing O(1) for an integer bounded independently of N)

length_{S_{𝔔}}H_{F_{can}^{∗}}^{1}(Q, (T ⊗ S_{𝔔})^{∗}) ≤ Nde + O(1),  (17)

and so by Proposition 19.11

length_{Z_{p}}H_{F_{Λ}^{∗}}^{1}(Q, T^{∗})[𝔔] ≤ Ne ord_{𝔓}(Ind(κ)) + O(1).

On the other hand, using (16),

length_{Z_{p}}H_{F_{Λ}^{∗}}^{1}(Q, T^{∗})[𝔔] = length_{Z_{p}}(X_{∞}/𝔔X_{∞})

= ∑_{i} length_{Z_{p}}Λ/(𝔓^{m_{i}}, 𝔔) + O(1)

= ∑_{i} length_{Z_{p}}Λ/(𝔔, p^{Nm_{i}}) + O(1)

= Ne ∑_{i} m_{i} + O(1) = Ne ord_{𝔓}(char(X_{∞})) + O(1),

so taking N sufficiently large shows that ord_{𝔓}(char(X_{∞})) ≤ ord_{𝔓}(Ind(κ)). Since 𝔓 was arbitrary, this proves (i).

Now suppose further that χ(T) = 1 and 𝔓 is not in the support of κ. In other words, there is an n such that the image of κ_{n} in H^{1}(Q, (T ⊗ S_{𝔓})/I_{n}(T ⊗ S_{𝔓})) is nonzero. Fix k such that I_{n}S_{𝔓} ⊂ 𝔪^{k}S_{𝔓} and such that the image of κ_{n} in H^{1}(Q, (T ⊗ S_{𝔓})/𝔪^{k}(T ⊗ S_{𝔓})) is nonzero (we can suppose I_{n}S_{𝔓} = 𝔪^{k}S_{𝔓} unless I_{n}S_{𝔓} = 0).

Suppose N is large enough so that properties (i) through (iv) above hold, and in addition N > k. Then (T ⊗ S_{𝔔})/𝔪^{k}(T ⊗ S_{𝔔}) ≅ (T ⊗ S_{𝔓})/𝔪^{k}(T ⊗ S_{𝔓}) so the image of κ_{n} in H^{1}(Q, (T ⊗ S_{𝔔})/𝔪^{k}(T ⊗ S_{𝔔})) is nonzero. Also I_{n}S_{𝔔} ⊂ 𝔪^{k}S_{𝔔}, so it follows from Lemma 18.7 that ∂^{(∞)}(κ^{(𝔔)}) < k. Thus Theorem 18.11 shows that equality holds in (17), and with this equality the argument above shows that ord_{𝔓}(char(X_{∞})) = ord_{𝔓}(Ind(κ)).

This proves what we need for primes 𝔓 ≠ pΛ. When 𝔓 = pΛ, we can repeat the argument above with 𝔓_{N} = (U^{N} + p)Λ, to obtain the same result. Combining all of this information proves the theorem. □

We end this section with some questions and speculation.

Recall the module of Euler systems ES(T) = ES(T, P) given by Definition 7.2. The natural action of G_{Q} on H^{1}(F, T) for finite abelian extensions F of Q gives an action of G_{Q}^{ab} on ES(T). Further, H^{1}(Q, T) and all of the H^{1}(Q, T/I_{n}T) have a natural Λ-module structure induced by the Λ-module structure on T (alternatively this action can be defined using Lemma 19.1), so KS(T) and H_{F_{Λ}}^{1}(Q, T) have G_{Q}^{ab} actions which factor through Gal(Q_{∞}/Q).

Let A be the kernel of the restriction map Z_{p}[[G_{Q}^{ab}]] → Λ. Since A annihilates KS(T), AES(T) maps to zero under the Euler-system-to-Kolyvagin-system map of Theorem 19.3. Consider the composition

ψ : ES(T)/AES(T) ⟶^{ψ_{EK}} KS(T) ⟶^{ψ_{1}} H_{F_{Λ}}^{1}(Q, T)

where ψ_{EK} is the map induced by Theorem 19.3, and ψ_{1}(κ) = κ_{1}. Then ψ(c) = {c_{Q_{n}}}, and all of these maps are Λ-module homomorphisms.

Question 19.19. Is ψ injective?

Even in the basic examples we do do not know the answer to this question. Assuming the weak Leopoldt conjecture for T, we have (Proposition 1.3.2 of [PR3]) that rank_{Λ}H_{F_{Λ}}^{1}(Q, T) = χ(T). In many examples (see §20 and §21) we have that χ(T) = 1, and in some of those examples we know that the image of ψ is free of rank one, and we know an Euler system c such that ψ(c) generates the image of ψ. If ψ were injective, it would follow that ES(T)/AES(T) would be free of rank one over Λ, and therefore (Nakayama’s Lemma) ES(T) would be cyclic over Z_{p}[[G_{Q}^{ab}]], generated by c.

Assuming a conjecture of Greenberg [Gr1] and using a result of Seo (Theorem C of [Se]), it is possible to show that ψ is injective for the cyclotomic unit Euler system discussed in §20 below.

Question 19.20. Is ψ_{EK} surjective?

In other words, does every Kolyvagin system for T come from an Euler system? Again, this seems to be very difficult.

Question 19.21. Is ψ_{1} injective?

If we replace (T, Λ) by (T̄, F_{p}) then the analogous map is not injective if χ(T̄) > 1 (see Remark 17.2). However, the construction that leads to Remark 17.2 does not work over Λ.

Note that if the answers to Questions 19.19 and 19.20 are “yes”, then the answer to Question 19.21 is also “yes”.

Question 19.22. What is the image of ψ?

Let \overline{H_{F_{Λ}}^{1}(Q, T)} denote the double-dual Hom(Hom(H_{F_{Λ}}^{1}(Q, T), Λ), Λ). Then \overline{H_{F_{Λ}}^{1}(Q, T)} is a free Λ-module and there is a natural injection H_{F_{Λ}}^{1}(Q, T) ↪ \overline{H_{F_{Λ}}^{1}(Q, T)} (since H_{F_{Λ}}^{1}(Q, T) is torsion-free) with finite cokernel. Let \overline{ψ} denote the composition of ψ with this inclusion.

It follows from Theorem 19.7 that image(\overline{ψ}) ⊂ char(X_{∞})\overline{H_{F_{Λ}}^{1}(Q, T)}.

When χ(T) = 1, conjecturally we have rank_{Λ}H_{F_{Λ}}^{1}(Q, T) = 1 and it seems reasonable to hope that image(\overline{ψ}) = char(X_{∞})\overline{H_{F_{Λ}}^{1}(Q, T)}. If the weak Leopoldt conjecture holds for T and KS(T) contains a Λ-primitive Kolyvagin system, then this equality follows from Theorem 19.7. Both of these conditions do hold in many examples, see §20.

When χ(T) > 1, it is less clear what to expect. The simplest answer would be image(\overline{ψ}) = char(X_{∞})\overline{H_{F_{Λ}}^{1}(Q, T)}; some evidence for this can be found in §6 of [Ru4]. But at present there are no well-understood examples with χ(T) > 1.

Part VI. Examples

20. The multiplicative group

For this section fix a character ρ : G_{Q} → Z_{p}^{×} of finite order. Assume that p > 2, so that the order of ρ is prime to p. Let L be the cyclic extension of Q which is the fixed field of ρ, and Δ = Gal(L/Q).

First, suppose k ≥ 0, and let R = Z/p^{k}Z and T = μ_{p^{k}} ⊗ ρ^{−1}, a one-dimensional representation with G_{Q} acting via the product of ρ^{−1} and the cyclotomic character.

We have

H^{1}(Q, T) = H^{1}(Q, μ_{p^{k}} ⊗ ρ^{−1}) ≅ (H^{1}(L, μ_{p^{k}}) ⊗ ρ^{−1})^{Δ} ≅ (L^{×}/(L^{×})^{p^{k}})^{ρ}, (18)

the subgroup of L^{×}/(L^{×})^{p^{k}} on which G_{Q} acts via ρ. (The last isomorphism depends on a choice of generator of the free rank-one R-module ρ^{−1}.) Similarly for every prime ℓ we have

H^{1}(Q_{ℓ}, T) ≅ (L_{ℓ}^{×}/(L_{ℓ}^{×})^{p^{k}})^{ρ},  (19)

where L_{ℓ} = L ⊗ Q_{ℓ} is the sum of the completions of L at primes above ℓ.

The dual representation T^{∗} = Hom(T, μ_{p^{∞}}) is a one-dimensional representation with G_{Q} acting via ρ. We have

H^{1}(Q, T^{∗}) ≅ (H^{1}(L, Z/p^{k}Z) ⊗ ρ)^{Δ} = Hom(G_{L}, Z/p^{k}Z)^{ρ^{−1}},  (20)

the subgroup of Hom(G_{L}, Z/p^{k}Z) on which G_{Q} acts via ρ^{−1}. Similarly for every prime ℓ we have

H^{1}(Q_{ℓ}, T^{∗}) ≅ (⨁_{λ|ℓ}Hom(G_{L_{λ}}, Z/p^{k}Z))^{ρ^{−1}}.  (21)

Definition 20.1. Define a Selmer structure F by taking Σ(F) to be the set of primes where ρ is ramified, together with p and ∞. For every prime ℓ ∉ Σ(F) one can check that

H_{f}^{1}(Q_{ℓ}, T) ≅ (O_{L,ℓ}^{×}/(O_{L,ℓ}^{×})^{p^{k}})^{ρ}

under the identification (19), where O_{L,ℓ} = O_{L} ⊗ Z_{ℓ}. For primes ℓ ∈ Σ(F) we define H_{F}^{1}(Q_{ℓ}, T) = (O_{L,ℓ}^{×}/(O_{L,ℓ}^{×})^{p^{k}})^{ρ} as well. (Since p > 2, we have H^{1}(R, T) = 0 so we can safely ignore the infinite place.)

Let F_{can} be the “canonical” Selmer structure on T induced from Z_{p}(1) ⊗ ρ^{−1} as in Definition 7.1. One can check ([Ru6] §1.6.C) that F_{can} is obtained from F by relaxing the condition at p, i.e., H_{F_{can}}^{1}(Q_{p}, T) = H^{1}(Q_{p}, T) and H_{F_{can}}^{1}(Q_{ℓ}, T) = H_{F}^{1}(Q_{ℓ}, T) for ℓ ≠ p.

The dual Selmer structure F^{∗} is given as follows ([Ru6] §1.6.B). For every prime ℓ ∉ Σ(F^{∗}) = Σ(F) we have

H_{f}^{1}(Q_{ℓ}, T^{∗}) ≅ ⊕_{λ|ℓ}(Hom(G_{L_{λ}}/I_{λ}, Z/p^{k}Z))^{ρ^{−1}}

under the identification (21), where I_{λ} is the inertia group in G_{L_{λ}}. For primes ℓ ∈ Σ(F) we have H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}) = (⊕_{λ|ℓ}Hom(G_{L_{λ}}/I_{λ}, Z/p^{k}Z))^{ρ^{−1}} as well.

The Selmer structure F_{can}^{∗} is obtained from F^{∗} by strengthening the condition at p, i.e., H_{(F′)^{∗}}^{1}(Q_{p}, T^{∗}) = 0 and H_{(F′)^{∗}}^{1}(Q_{ℓ}, T^{∗}) = H_{F^{∗}}^{1}(Q_{ℓ}, T^{∗}) for ℓ ≠ p.

Proposition 20.2. We have natural exact sequences

0 ⟶ (O_{L}^{×}/(O_{L}^{×})^{p^{k}}))^{ρ} ⟶ H_{F}^{1}(Q, T) ⟶ Cl(L)[p^{k}]^{ρ} ⟶ 0
0 ⟶ (O_{L}[1/p]^{×}/(O_{L}[1/p]^{×})^{p^{k}}))^{ρ} ⟶ H_{F_{can}}^{1}(Q, T) ⟶ Cl′(L)[p^{k}]^{ρ} ⟶ 0

and isomorphisms

H_{F^{∗}}^{1}(Q, T^{∗}) ≅ Hom(Cl(L)^{ρ}, Z/p^{k}Z), H_{F_{can}^{∗}}^{1}(Q, T^{∗}) ≅ Hom(Cl′(L)^{ρ}, Z/p^{k}Z)

where Cl(L) is the ideal class group of L, and Cl′(L) is Pic(O_{L}[1/p]), the quotient of Cl(L) by the classes of primes above p.

Proof. From the definition of F, under the identification (18) we have

H_{F}^{1}(Q, T) = {x ∈ (L^{×}/(L^{×})^{p^{k}})^{ρ} : ord_{λ}(x) ≡ 0 (mod p^{k}) for every λ}.

This gives the first exact sequence of the proposition, where the map H_{F}^{1}(Q, T) → Cl(L)[p^{k}]^{ρ} sends x to the class of an ideal 𝔞 such that 𝔞^{p^{k}} is the ideal generated by (any representative of) x.

The second exact sequence is similar.

Under the identification (20), H_{F^{∗}}^{1}(Q, T^{∗}) consists of the unramified homomorphisms in H^{1}(Q, T^{∗}), and H_{F_{can}^{∗}}^{1}(Q, T^{∗}) consists of those unramified homomorphisms which are trivial on the decomposition group at p, i.e.,

H_{F^{∗}}^{1}(Q, T^{∗}) = Hom(Cl(L), Z/p^{k}Z)^{ρ^{−1}},
H_{F_{can}^{∗}}^{1}(Q, T^{∗}) = Hom(Cl′(L), Z/p^{k}Z)^{ρ^{−1}}.

This gives the rest of the proposition. □

Similarly if we take R = Z_{p} and T = Z_{p}(1) ⊗ ρ^{−1} we get isomorphisms

H_{F}^{1}(Q, T) ≅ (O_{L}^{×} ⊗ Z_{p})^{ρ}, H_{F_{can}}^{1}(Q, T) ≅ (O_{L}[1/p]^{×} ⊗ Z_{p})^{ρ}  (22)

since \underleftarrow{lim}_{k} Cl(L)[p^{k}] = 0, and

H_{F^{∗}}^{1}(Q, T^{∗}) ≅ Hom(Cl(L)^{ρ}, Q_{p}/Z_{p}), H_{F_{can}^{∗}}^{1}(Q, T^{∗}) ≅ Hom(Cl′(L)^{ρ}, Q_{p}/Z_{p}). (23)

Definition 20.3. Take P to be the set of all primes different from p. Fix a collection {ζ_{n} : n ∈ Z^{+}} such that ζ_{n} is a primitive n-th root of unity and ζ_{mn}^{m} = ζ_{n} for every m and n. If F is a finite abelian extension of Q of conductor f, we define c_{F} to be the image of N_{Q(μ_{fp})/F} (ζ_{fp} − 1) under the Kummer map

F^{×} ↪ H^{1}(F, Z_{p}(1)).

One can check without difficulty that this defines an Euler system in the sense of [Ru6] Chapter 2, the Euler system of cyclotomic units. To obtain an Euler system c ∈ ES(Z_{p}(1), P) satisfying Definition 7.2 we need to modify these units slightly (see Remark 7.3) as in [Ru6] §9.6.

Using Remark 7.5 the Euler system c gives an Euler system c^{ρ} ∈ ES(T, P_{ρ}) where P_{ρ} is the set of primes not dividing pm_{ρ} and m_{ρ} is the conductor of ρ. For r prime to m_{ρ}, using the identification (18), we have

c_{Q(μ_{r})}^{ρ} = ∏_{δ∈Gal(Q(μ_{rpmρ})/Q(μ_{r}))} (ζ_{rpm_{ρ}} − 1)^{ρ^{−1}(δ)δ}.  (24)

By Theorem 7.4, c^{ρ} gives rise to a Kolyvagin system

κ^{ρ} ∈ ES(T, F_{can}, P_{ρ}),

and

κ_{1}^{ρ} = c_{Q}^{ρ} = (  ∏_{δ∈Gal(Q(μ_{mρ})/Q)} (ζ_{m_{ρ}} − 1)^{ρ^{−1}(δ)δ})^{1−ρ^{−1}(p)}.  (25)

This will be trivial unless ρ is an even character and ρ(p) ≠ 1.

There is no way to get something nontrivial when ρ is odd, because cyclotomic units have no “odd components”. However, we can eliminate the assumption that ρ(p) ≠ 1. Namely, if we redefine the Euler system c^{ρ} by setting

c_{Q(μ_{r})}^{ρ} = ∏_{δ∈Gal(Q(μ_{rmρ})/Q(μ_{r}))} (ζ_{rm_{ρ}} − 1)^{ρ^{−1}(δ)δ}

in place of (24), then as long as ρ ≠ 1 we get an Euler system in the weaker sense of Remark 7.6, and then we get a Kolyvagin system κ^{ρ} ∈ ES(T, F_{can}, P_{ρ}) by Theorem 7.7. Thus for every even nontrivial ρ we get a Kolyvagin system κ^{ρ} satisfying

κ_{1}^{ρ} = ∏_{δ∈Gal(Q(μ_{mρ})/Q)}(ζ_{m_{ρ}} − 1)^{ρ^{−1}(δ)δ}.  (26)

If ρ ≠ 1, then using the fact that ζ_{f} is a unit when f is not a prime power one can show that in fact

κ^{ρ} ∈ ES(T, F, P_{ρ}).

Let ω : G_{Q} → Z_{p}^{×} denote the Teichmuller character giving the action of G_{Q} on μ_{p}.

Lemma 20.4. Suppose ρ ≠ 1 and ρ ≠ ω. Then T satisfies hypotheses (H.0), (H.1), (H.2), (H.3), and (H.4a) of §9, and F and F_{can} both satisfy (H.6).

Proof. Hypotheses (H.0) and (H.1) are trivially satisfied, and (H.2) holds with τ = 1. If ρ ≠ 1 and ρ ≠ ω, then (H.3) holds as well (note that since ρ has order prime to p, it is not congruent to either 1 or ω modulo p). Since ρ takes values in Z_{p}^{×}, ρ^{2} cannot equal ω and therefore (H.4a) holds. Lemma 11.1 shows that (H.6) holds for F and F_{can}. □

Proposition 20.5.

χ(T, F) = \begin{cases}1 & if ρ is even, ρ ≠ 1, \\ 0 & if ρ is odd, ρ ≠ ω \end{cases}

Proof. From Theorem 12.12 and Proposition 20.2 we have (when R = Z/p^{k}Z)

kχ(T, F) = length(H_{F}^{1}(Q, T)) − length(H_{F^{∗}}^{1}(Q, T^{∗})) = length((O_{L}^{×}/(O_{L}^{×})^{p^{k}}))^{ρ}

and the proposition follows. □

A similar result holds for χ(T, F_{can}), see Theorem 18.13.

The following result is the main application of the cyclotomic unit Kolyvagin system κ^{ρ}. Originally known as the Gras conjecture, it was proved by the first author and Wiles in [MW]. The proof sketched here is essentially due to Kolyvagin ([Ko], see also [Ru1]).

Theorem 20.6. If ρ is an even character, and C_{L} ⊂ O_{L}^{×} denotes the subgroup of cyclotomic units of L, then

|Cl(L)^{ρ}| = |O_{L}^{×}/C_{L}|^{ρ}.

Proof. We apply Corollary 18.12(ii) to the Kolyvagin system κ^{ρ} ∈ ES(T, F, P_{ρ}) satisfying (26). Equations (22) and (26) show that ∂^{(0)}(κ^{ρ}) = length((O_{L}^{×}/C_{L})^{ρ}), and together with (23) this yields

|Cl(L)^{ρ}| ≤ |O_{L}^{×}/C_{L}|^{ρ}.

By a standard method using the analytic class number formula, (see for example [Ru1] Theorem 4.2) this inequality for all ρ implies that equality must hold for all ρ. □

Remark 20.7. The proof of Theorem 20.6 shows, using Corollary 18.12(ii), that κ^{ρ} is primitive.

Now take R = Λ, T = Z_{p}(1) ⊗ ρ^{−1}, and T = T ⊗ Λ as in §19. By Theorem 19.3, the cyclotomic unit Euler system defined by (24) gives rise to a Kolyvagin system κ^{ρ,∞} ∈ KS(T, F_{Λ}) where F_{Λ} is the Selmer structure of Definition 19.2. We have

κ_{1}^{ρ,∞} = \{c_{Q_{n}}^{ρ}\} ∈ \underleftarrow{lim}_{n} (L_{n}^{×})^{ρ} ⊂ H^{1}(Q, T)

where L_{n} = LQ_{n} and c_{Q_{n}}^{ρ} ∈ (L_{n}^{×})^{ρ} is given by

c_{Q_{n}}^{ρ} = ∏_{δ∈Gal(Q(μ_{p^{n+1}mρ})/Q_{n})} (ζ_{p^{n+1}m_{ρ}} − 1)^{ρ^{−1}(δ)δ}.

Applying the results of §19 to κ^{ρ,∞} leads to the following theorem, Iwasawa’s “main conjecture”, proved by the first author and Wiles in [MW].

Theorem 20.8.

char(\underleftarrow{lim}_{n} (Cl(L_{n})[p^{∞}])^{ρ}) = L_{ρ}

where L_{ρ} ∈ Λ is the p-adic L-function attached to ρ.

Proof. One can show (see [Ru6] §3.2 for the details) that

Ind(κ^{ρ,∞}) = L_{ρ}

and

Hom(H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}), Q_{p}/Z_{p}) = \underleftarrow{lim}_{n} (Cl′(L_{n})[p^{∞}])^{ρ}

where we recall that Cl′(L_{n}) = Pic(O_{L_{n}}[1/p]).

Suppose first that ρ(p) ≠ 1. Let P_{n} denote the subgroup of Cl(L_{n}) generated by primes above p. Since the decomposition group of p acts trivially on P_{n}, the exact sequence

0 ⟶ P_{n} ⟶ Cl(L_{n}) ⟶ Cl′(L_{n}) ⟶ 0

shows that (Cl′(L_{n})[p^{∞}])^{ρ} = (Cl(L_{n})[p^{∞}])^{ρ}. Hence Hom(H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}), Q_{p}/Z_{p}) = \underleftarrow{lim}_{n} (Cl(L_{n})[p^{∞}])^{ρ} in this case. Further, it follows from Remark 20.7 that κ^{ρ,∞} is

Λ-primitive. Now the theorem follows in this case directly from Theorem 19.7. In the general case, one can show (again, see [Ru6] §3.2 for the details) that

• Gal(L_{∞}/L) acts trivially on \underleftarrow{lim}_{n} (Cl(L_{n})[p^{∞}])^{ρ}/\underleftarrow{lim}_{n} (Cl′(L_{n})[p^{∞}])^{ρ},
• char(\underleftarrow{lim}_{n} (Cl(L_{n})[p^{∞}])^{ρ}) is prime to A, the augmentation ideal of Λ.

Taking these two facts into account, Theorem 19.7 shows that

char(\underleftarrow{lim}_{n} (Cl(L_{n})[p^{∞}])^{ρ}) divides L_{ρ}.

Once again a standard argument using the analytic class number formula (see for example [MW] §1.6 or [Ru1] p. 414) allows us to turn this divisibility (for all ρ) into an equality. □

21. Elliptic curves

For this section fix an elliptic curve E defined over Q. We make the simplifying assumptions

p > 3,  (27)
the p-adic representation G_{Q} → Aut(E[p^{∞}]) ≅ GL_{2}(Z_{p}) is surjective, (28)
E(Q_{p})[p] = 0.  (29)

Suppose first that k ≥ 0, and let R = Z/p^{k}Z and T = E[p^{k}], the p^{k}-torsion in E(Q̄).

Definition 21.1. Define a Selmer structure F by taking Σ(F) to be the set of primes where E has bad reduction, together with p and ∞. For every prime ℓ ∉ Σ(F) one can check that H_{f}^{1}(Q_{ℓ}, E[p^{k}]) is the image of the Kummer map

E(Q_{ℓ})/p^{k}E(Q_{ℓ}) ↪ H^{1}(Q_{ℓ}, E[p^{k}]).  (30)

For primes ℓ ∈ Σ(F) we define H_{F}^{1}(Q_{ℓ}, T) to be the image of (30) as well.

With this Selmer structure H_{F}^{1}(Q, E[p^{k}]) is the classical p^{k}-Selmer group of E, which sits in an exact sequence

0 → E(Q)/p^{k}E(Q) ⟶ H_{F}^{1}(Q, E[p^{k}]) ⟶ � _{E}[p^{k}] ⟶ 0  (31)

where � _{E} is the Tate-Shafarevich group of E.

The Weil pairing identifies E[p^{k}]^{∗} = E[p^{k}], and one can show that F^{∗} = F. Similarly if R = Z_{p} and T is the p-adic Tate module T_{p}(E), we define F analo-

gously and get

0 → E(Q) ⊗ Z_{p} ⟶ H_{F}^{1}(Q, T_{p}(E)) ⟶ \underleftarrow{lim}_{k} � _{E}[p^{k}] ⟶ 0.

If � _{E} is finite then this becomes an isomorphism H_{F}^{1}(Q, T_{p}(E)) ≅ E(Q) ⊗ Z_{p}.

Now let F_{can} be the “canonical” Selmer structure on T as in Definition 7.1. Using (29) one can show that F_{can} is obtained from F by relaxing the condition at p, i.e.,

H_{F_{can}}^{1}(Q_{ℓ}, E[p^{k}]) = \begin{cases}H^{1}(Q_{p}, E[p^{k}])) & if ℓ = p, \\ H_{F}^{1}(Q_{ℓ}, E[p^{k}]) & if ℓ ≠ p. \end{cases}

Similarly F_{can}^{∗} is obtained from F by setting H_{F_{can}^{∗}}^{1}(Q_{p}, E[p^{k}]) = 0. Then F_{can}^{∗} ≤ F ≤ F_{can} and so

H_{F_{can}^{∗}}^{1}(Q, E[p^{k}]) ⊂ H_{F}^{1}(Q, E[p^{k}]) ⊂ H_{F_{can}}^{1}(Q, E[p^{k}]).

Let N be the conductor of E. Kato [Ka2] (see also [Sch] and §3.5 of [Ru6]) has constructed an Euler system for (T_{p}(E), P), where P is the set of primes not dividing NpDD′ with two auxiliary positive integers D, D′ used in Kato’s construction.

Using Theorem 7.4 (and (29)), Kato’s Euler system gives a Kolyvagin system κ^{Kato} for (T_{p}(E), F_{can}, P), and by reducing modulo p^{k} we get a Kolyvagin system for E[p^{k}] for every k.

Lemma 21.2. With assumptions as above, T satisfies hypotheses (H.0), (H.1), (H.2), (H.3), and (H.4) of §9, and F and F_{can} both satisfy (H.6).

Proof. Hypotheses (H.0) and (H.1) are trivially satisfied, (H.2) and (H.3) hold thanks to (28), and (H.4b) holds because of (27). Lemma 11.1 shows that (H.6) holds for F and F_{can}. □

Proposition 21.3. χ(T, F) = 0 and χ(T, F_{can}) = 1.

Proof. That χ(T, F) = 0 is immediate from Theorem 12.12, since H_{F}^{1}(Q, E[p]) = H_{F^{∗}}^{1}(Q, E[p]^{∗}). Also

χ(T, F_{can}) = χ(T_{p}(E), F_{can}) = rank_{Z_{p}}T_{p}(E)^{−} = 1

by Theorem 18.13. □

The following is an application of Kato’s Kolyvagin system κ^{Kato}. Let L(E, s) denote the Hasse-Weil L-function attached to E, and L_{N}(E, s) the L-function with the Euler factors at primes dividing the conductor N of E removed. Let Ω be the fundamental real period of E. We continue to suppose that p satisfies (27), (28), and (29).

Theorem 21.4. (i) If L(E, 1) ≠ 0 then E(Q) and � _{E}[p^{∞}] are finite. If L(E, 1) = 0 and κ^{Kato} ≠ 0, then either E(Q) is infinite or � _{E}[p^{∞}] is infinite (or both).

(ii) Suppose further that p satisfies (a) E has good reduction at p,

(b) p ∤ |E(F_{p})|,

(c) p does not divide the integer r_{E} of Theorem 7.1 of [Ru5].

If L(E, 1) ≠ 0 then

length(� _{E}[p^{∞}]) ≤ ord_{p}(\frac{L_{N}(E, 1)}{Ω}).

with equality if and only if κ^{Kato} is primitive.

Proof. Applying Corollary 18.12(ii) to κ^{Kato} shows that

length(H_{F_{can}^{∗}}^{1}(Q, E[p^{∞}])) ≤ ∂^{(0)}(κ^{Kato})

with equality if and only if κ^{Kato} is primitive. Combining this with Kato’s computation of κ_{1}^{Kato}, the exact sequence (31), and using global duality to compare the true Selmer group H_{F}^{1}(Q, E[p^{∞}]) with H_{F_{can}^{∗}}^{1}(Q, E[p^{∞}]), will prove the theorem. For the details see [Ru5] and Theorem 2.2.10 of [Ru6]. □

Remark 21.5. In general one does not expect equality in Theorem 21.4(ii). Besides the missing Euler factors in the L-value, the Birch and Swinnerton-Dyer conjecture for the order of � _{E} involves other terms as well, namely |E(Q)_{tors}| and c_{ℓ} = [E(Q_{ℓ}) : E_{0}(Q_{ℓ})] for primes ℓ of bad reduction.

The following corollary gives some evidence for the occurrence of the factors c_{ℓ} in the Birch and Swinnerton-Dyer formula for � _{E}.

Corollary 21.6. Suppose L(E, 1) ≠ 0 and p satisfies the hypotheses of Theorem 21.4(ii). If p | c_{ℓ} for some ℓ ≠ p, then

length(� _{E}[p^{∞}]) < ord_{p}(\frac{L_{N}(E, 1)}{Ω}).

Proof. By Theorem 21.4 we need to show that κ^{Kato} is not primitive, i.e., the image of κ^{Kato} in KS(E[p], F_{can}) is zero.

Define a Selmer structure F_{u} on T_{p}(E) by taking

H_{F_{u}}^{1}(Q_{ℓ}, T_{p}(E)) = \begin{cases}H_{unr}^{1}(Q_{ℓ}, T_{p}(E)) & if ℓ ≠ p, \\ H^{1}(Q_{p}, T_{p}(E)) & if ℓ = p. \end{cases}

We also write F_{u} for the Selmer structure on E[p] induced by this. Then F_{u} ≤ F_{can} (see for example [Ru6] Lemma 1.3.5), and we will take advantage of the fact that under the hypotheses of the corollary, F_{u} < F_{can}.

Suppose ℓ ≠ p and p | c_{ℓ}. Then (for example [Si], Corollary 15.2.1 of Appendix C) E has split multiplicative reduction at ℓ, so E(Q_{ℓ}) ≅ Q_{ℓ}^{×}/q^{Z} with q ∈ Q_{ℓ}^{×}, and p | ord_{ℓ}(q). It follows that the image of ℓ under

Q_{ℓ}^{×}/q^{Z} ↠ E(Q_{ℓ})/pE(Q_{ℓ}) ↪ H^{1}(Q_{ℓ}, E[p])

does not lie in H_{F_{u}}^{1}(Q_{ℓ}, E[p]). The image of this map is precisely H_{F_{can}}^{1}(Q_{ℓ}, E[p]), so we conclude that H_{F_{u}}^{1}(Q_{ℓ}, E[p]) is strictly smaller than H_{F_{can}}^{1}(Q_{ℓ}, E[p]).

It now follows from Theorem 5.3 and Proposition 21.3 that

dim_{F_{p}} H_{F_{u}}^{1}(Q, E[p]) − dim_{F_{p}} H_{F_{u}^{∗}}^{1}(Q, E[p])

< dim_{F_{p}} H_{F_{can}}^{1}(Q, E[p]) − dim_{F_{p}} H_{F_{can}^{∗}}^{1}(Q, E[p]) = 1,

and therefore χ(E[p], F_{u}) = 0.

Thus by Theorem 17.1(i), ES(E[p], F_{u}) = 0. But (see Remark A.5) the proof of Theorem 7.4 actually shows that the image of κ^{Kato} in KS(E[p], F_{can}) lies in KS(E[p], F_{u}). Hence this image must be zero, and κ^{Kato} is not primitive. As noted above, this proves the corollary. □

Now take R = Λ and T = T_{p}(E) ⊗ Λ as in §19. By Theorem 19.3, Kato’s Euler system gives rise to a Kolyvagin system κ^{Kato,∞} ∈ KS(T, F_{Λ}) where F_{Λ} is the Selmer structure of Definition 19.2. Applying the results of §19 to κ^{Kato,∞} leads to the following theorem.

Theorem 21.7. Suppose E has good ordinary reduction or nonsplit multiplicative reduction at p, and p does not divide the integer r_{E} of Theorem 7.1 of [Ru5]. Then

char(Hom(Sel_{p^{∞}}(E/Q_{∞}), Q_{p}/Z_{p})) divides L_{E,N}

where L_{E,N} ∈ Λ is the p-adic L-function attached to E with the Euler factors at primes dividing N removed, and Sel_{p^{∞}}(E/Q_{∞}) is the classical p-power Selmer group attached to E over Q_{∞}.

If E has split multiplicative reduction at p then the augmentation ideal A of Λ divides L_{E,N} and

char(Hom(Sel_{p^{∞}}(E/Q_{∞}), Q_{p}/Z_{p})) divides A^{−1}L_{E,N}.

Proof. We can relate Ind(κ^{Kato,∞}) with L_{E,N} and Sel_{p^{∞}}(E/Q_{∞}) with H_{F_{Λ}^{∗}}^{1}(Q, T^{∗}), and then the theorem follows from Theorem 19.7. See [Ru5] for the details. □

22. The multiplicative group, revisited

Return to the setting of §20: p > 2, ρ : G_{Q} → Z_{p}^{×} is a character of finite order, L is the cyclic extension of Q which is the fixed field of ρ, Δ = Gal(L/Q), R = Z_{p}, and T = Z_{p}(1) ⊗ ρ^{−1}. Let F be the Selmer structure on T given by Definition 20.1.

For this section we suppose that ρ is an odd character, different from the Teichmuller character ω. By Proposition 20.5 we have χ(T, F) = 0, so by Theorem 18.9(i), KS(T, F) = 0. However, we can modify F to obtain Kolyvagin systems as follows.

Recall (Example 4.4) that F^{ℓ} denotes the Selmer structure obtained from F by relaxing the local condition at ℓ, i.e., we set H_{F^{ℓ}}^{1}(Q_{ℓ}, T) = H^{1}(Q_{ℓ}, T).

Proposition 22.1. Suppose ℓ ≠ p is a prime such that ρ(ℓ) = 1. Then

χ(T, F^{ℓ}) = 1.

Proof. This follows from Theorem 5.3 (global duality) applied to F and F^{ℓ}, since (using (19) and Definition 20.1)

H_{F^{ℓ}}^{1}(Q_{ℓ}, T̄)/H_{F}^{1}(Q_{ℓ}, T̄) ≅ H^{1}(Q_{ℓ}, μ_{p})/H_{unr}^{1}(Q_{ℓ}, μ_{p}) ≅ (Q_{ℓ}^{×}/(Q_{ℓ}^{×})^{p})/(Z_{ℓ}^{×}/(Z_{ℓ}^{×})^{p})

is one-dimensional over F_{p}. □

Theorem 22.2. Suppose ρ : G_{Q} → Z_{p}^{×} is an odd character, ρ ≠ ω. For every prime ℓ ≠ p with ρ(ℓ) = 1, the Z_{p}-module KS(T, F^{ℓ}) is free of rank one and is generated by a primitive Kolyvagin system.

Proof. This follows from Theorem 18.9(ii) and Proposition 22.1 (using Lemma 20.4). □

Remark 22.3. Kolyvagin’s “Gauss sum Euler system” (see [Ko] or [Ru2]) gives an Euler system c^{(ℓ)} ∈ ES(T, F^{ℓ}, P^{(ℓ)}) for each ℓ with ρ(ℓ) = 1, where P^{(ℓ)} is the finite set of primes dividing ℓ − 1. These “finite” Euler systems give rise to Kolyvagin systems κ^{(ℓ)} ∈ KS(T, F^{ℓ}, P^{(ℓ)}). Theorem 22.2 says that κ^{(ℓ)} can be extended to KS(T, F^{ℓ}, P) where P is the set of all primes not dividing pℓm_{ρ} where m_{ρ} is the conductor of ρ. It is not clear whether the Euler systems themselves can be extended.

The family of Kolyvagin systems {κ^{(ℓ)}} satisfies a “compatibility in ℓ”, in that the classes {κ_{1}^{(ℓ)} ∈ (L^{×} ⊗Z_{p})^{ρ}}, which are given by Gauss sums, are the values ψ(ℓ) of a Hecke character ψ.

Remark 22.4. A similar phenomenon occurs in the work of Flach [Fl]. Let E be an elliptic curve defined over Q, R = Z_{p}, and T = Sym^{2}(T_{p}(E)), the symmetric square of the p-adic Tate module of E. Assume that the p-adic representation G_{Q} → Aut(T_{p}(E)) is surjective, so that the hypotheses of §9 will be satisfied.

Theorem 18.13 shows that χ(T, F_{can}) = 1. If F_{BK} denotes the Bloch-Kato Selmer structure on T used in [Fl], then F_{BK} differs from F_{can} only at p, F_{BK} ≤ F_{can}, and one can show that χ(F_{BK}, T) = 0. For every rational prime ℓ ≠ p where E has good reduction, the automorphism Fr_{ℓ} of T has the eigenvalue ℓ with multiplicity one, and it follows that with the relaxed-at-ℓ Selmer structure F_{BK}^{ℓ} we have

χ(T, F_{BK}^{ℓ}) = χ(T, F_{BK}) + 1 = 1.

Therefore by Theorem 18.9(ii) the Z_{p}-module KS(T, F_{BK}^{ℓ}) is free of rank one, generated by a primitive Kolyvagin system, for every prime ℓ ≠ p where E has good reduction.

In [Fl], Flach constructs what can be viewed as the classes κ_{1}^{(ℓ)} ∈ H_{F_{BK}^{ℓ}}^{1}(Q, T) for

these Kolyvagin systems. So far no further classes κ_{n}^{(ℓ)} have been constructed, but the classes κ_{1}^{(ℓ)} are compatible in an important way. See [Ma] or [We] for details.

Part VII. Appendices

Appendix A. Proof of Theorem 7.4

In this appendix we prove Theorems 7.4 and 19.3. The proof requires the setting, notation, and results from Chapter 4 of [Ru6]. We recall most of the essential information here and refer to [Ru6] for the details.

Let T, R, F = F_{can}, and P be as in §7. Thus R is the ring of integers of a finite extension of Q_{p}, P is a set of primes different from p where T is unramified, and such that for every ℓ ∈ P,

• T/(Fr_{ℓ} − 1)T is a cyclic R-module,

• Fr_{ℓ}^{p^{k}} − 1 is injective on T for every k ≥ 0.

As usual let N be the set of all squarefree products of primes in P. Also as in Theorem 7.4 we assume that H^{0}(Q_{p}, T^{∗}) is a divisible R-module.

Lemma A.1. If H^{0}(Q_{p}, T^{∗}) is a divisible R-module then for every ideal I of R, we have H_{F}^{1}(Q_{p}, T/IT) = H^{1}(Q_{p}, T/IT).

Proof. When I = 0 this is the definition of the Selmer structure F at p. For general I the Selmer structure on T/IT is the one induced from T, i.e., H_{F}^{1}(Q_{p}, T/IT) is the image of H_{F}^{1}(Q_{p}, T) = H^{1}(Q_{p}, T) in H^{1}(Q_{p}, T/IT). Fixing a generator α of I, the exact sequence 0 → T ⟶^{α} T → T/IT → 0 shows that

coker(H^{1}(Q_{p}, T) → H^{1}(Q_{p}, T/IT)) ≅ H^{2}(Q_{p}, T)[I],

the kernel of I in H^{2}(Q_{p}, T). By local duality

H^{2}(Q_{p}, T)[I] ≅ Hom(H^{0}(Q_{p}, T^{∗})/IH^{0}(Q_{p}, T^{∗}), Q_{p}/Z_{p})

which is zero since we are assuming that H^{0}(Q_{p}, T^{∗}) is divisible. □

Fix an Euler System c ∈ ES(T) = ES(T, F, P). Definition 4.4.10 of [Ru6] associates to c a collection

{κ_{n} ∈ H^{1}(Q, T/I_{n}T) ⊗ G_{n} : n ∈ N }

such that κ_{1} = c_{Q}. Here we write simply κ_{n} for the class denoted κ_{[Q,n,I_{n}]} of [Ru6]. (The arguments of [Ru6] use a fixed generator of each G_{ℓ}, and hence of each G_{n}, and produced κ_{[Q,n,I_{n}]} ∈ H^{1}(Q, T/I_{n}T). Without these choices the same construction gives κ_{[Q,n,I_{n}]} ∈ H^{1}(Q, T/I_{n}T) ⊗ G_{n}.)

Proposition A.2. The collection {κ_{n}} is a weak Kolyvagin system in the sense of Definition 6.7.

Proof. Theorem 4.5.1 of [Ru6], combined with Lemma A.1 above, shows that κ_{n} ∈ H_{F^{n}}^{1}(Q, T/I_{n}T) ⊗ G_{n}. Theorem 4.5.4 of [Ru6] shows that the κ_{n} satisfy (5) of §6. These properties say precisely that {κ_{n}} is a weak Kolyvagin system. □

The collection {κ_{n}} is not generally a Kolyvagin system, because we will not in general have κ_{n} ∈ H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n}. However, by computing the finite projections (κ_{n})_{ℓ,f} at primes ℓ dividing n, we will show that a slight modification of the κ_{n} gives a Kolyvagin system. Thus it remains to compute the (κ_{n})_{ℓ,f}.

Definition A.3. If I is an ideal of R and ℓ ∈ P, let A_{ℓ,I} denote the augmentation ideal of (R/I)[G_{ℓ} ⊗ R/I]. Then there is a canonical isomorphism

ρ_{ℓ} = ρ_{ℓ,I} : A_{ℓ,I}/A_{ℓ,I}^{2} ⟶ G_{ℓ} ⊗ R/I

which sends σ − 1 to σ.

If n ∈ N let Σ(n) denote the set of permutations of the primes dividing n, and let Σ_{1}(n) ⊂ Σ be the subset

{π ∈ Σ(n) : the ℓ not fixed by π form a single π-orbit}.

If π ∈ Σ(n) let d_{π} = ∏_{π(ℓ)=ℓ} ℓ.

Theorem A.4. If n ∈ N and ℓ | n then

(κ_{n})_{ℓ,f} = ∑_{{π∈Σ_{1}(n) \atop π(ℓ)≠ℓ}}(−1)^{ν(n/d_{π})}(κ_{d_{π}})_{ℓ,f} ⨂_{q|(n/d_{π})}ρ_{q}(P_{q}(Fr_{π(q)}^{−1}))

where as usual ν(d) is the number of prime factors of d.

We will prove Theorem A.4 below, after using it to prove Theorems 7.4 and 19.3.

Proof of Theorem 7.4. For every n ∈ N define

κ′_{n} = ∑_{π∈Σ(n)} sign(π)κ_{d_{π}} ⨂_{ℓ|(n/d_{π})}ρ_{ℓ}(P_{ℓ}(Fr_{π(ℓ)}^{−1})) ∈ H^{1}(Q, T/I_{n}T) ⊗ G_{n}

where sign(π) is the sign of the permutation π. By inspection we see that since the κ_{n} satisfy (5) of §6, so do the κ′_{n}. On further inspection we see that if ℓ | n then we can group the terms in the definition of κ′_{n} as

κ′_{n} = ∑_{{π∈Σ(n) \atop π(ℓ)=ℓ}} sign(π)κ_{d_{π}} ⨂_{q|(n/d_{π})}ρ_{q}(P_{q}(Fr_{π(q)}^{−1}))

+ ∑_{{π∈Σ(n) \atop π(ℓ)=ℓ}} ∑_{{π′∈Σ_{1}(d_{π}) \atop π′(ℓ)≠ℓ}} sign(ππ′)κ_{d_{π′}} ⨂_{q|(n/d_{π′})}ρ_{q}(P_{q}(Fr_{ππ′(q)}^{−1}))

= ∑_{{π∈Σ(n) \atop π(ℓ)=ℓ}} sign(π)s_{π} ⨂_{q|(n/d_{π})}ρ_{q}(P_{q}(Fr_{π(q)}^{−1}))

where

s_{π} = κ_{d_{π}} − ∑_{{π′∈Σ_{1}(d_{π}) \atop π′(ℓ)≠ℓ}}(−1)^{ν(d_{π}/d_{π′})}κ_{d_{π′}} ⨂_{q|(d_{π}/d_{π′})}ρ_{q}(P_{q}(Fr_{π′(q)}^{−1})).

Theorem A.4 shows that (s_{π})_{ℓ,f} = 0 for every π, so (κ′_{n})_{ℓ,f} = 0. Hence κ′_{n} ∈ H_{F (n)}^{1}(Q, T/I_{n}T) ⊗ G_{n}, so the collection {κ′_{n}} is a Kolyvagin system κ′. We have κ′_{1} = κ_{1} = c_{Q}, so the association c → κ′ gives the desired map ES(T) → KS(T) in Theorem 7.4. □

Remark A.5. The proof of Theorem 7.4 actually produces a Kolyvagin system for a possibly finer Selmer structure. Define F_{u} by

H_{F_{u}}^{1}(Q_{ℓ}, T) = \begin{cases}H^{1}(Q_{p}, T) & if ℓ = p \\ H_{unr}^{1}(Q_{ℓ}, T) & if ℓ ≠ p. \end{cases}

Then F_{u} ≤ F, and it can happen that F_{u} < F. More precisely, if T is unramified at ℓ then H_{F}^{1}(Q_{ℓ}, T) = H_{F_{u}}^{1}(Q_{ℓ}, T), but if T is ramified at ℓ they may be different (see [Ru6] Lemma 1.3.5(iv)).

The proof of Theorem 4.5.1 of [Ru6] shows that in fact the classes κ_{n} used in the proof of Theorem 7.4 lie in H_{(F_{u})^{n}}^{1}(Q, T/I_{n}T). Therefore the map of Theorem 7.4 factors through

ES(T) ⟶ KS(T, F_{u}) ⟶ KS(T, F).

See Corollary 21.6 for an application of this observation.

Proof of Theorem 19.3. The proof of Theorem 19.3 is the same as that of Theorem 7.4. If we replace I_{ℓ} by the (possibly larger) ideal I′_{ℓ} generated by ℓ − 1, P_{ℓ}(1), and Fr_{ℓ} − 1, then the results of [Ru6] Chapter 4 produce a weak Kolyvagin system {κ_{n} ∈ H_{F^{n}}^{1}(Q, T/I′_{n}T) ⊗ G_{n}}, which can be modified exactly as above to produce a Kolyvagin system {κ′_{n} ∈ H_{F (n)}^{1}(Q, T/I′_{n}T) ⊗ G_{n}}. To prove Theorem 19.3 it is necessary to repeat the construction of the classes κ_{n} (and the verification of their fundamental properties), to show that those methods do in fact give a weak Kolyvagin system {κ_{n} ∈ H_{F^{n}}^{1}(Q, T/I_{n}T) ⊗ G_{n}}. We omit the details. □

The rest of this appendix is devoted to the proof of Theorem A.4.

Fix n ∈ N and a prime ℓ | n. If F is a number field then we will write F_{ℓ} = F ⊗Q_{ℓ} = ⊕_{λ|ℓ}F_{λ} and H^{i}(F_{ℓ}, T) = ⊕_{λ|ℓ}H^{i}(F_{λ}, T), H_{f}^{1}(F_{ℓ}, T) = ⊕_{λ|ℓ}H_{f}^{1}(F_{λ}, T), etc. If F/Q is Galois then H^{i}(F_{ℓ}, T) and H_{f}^{1}(F_{ℓ}, T) have natural actions of Gal(F/Q).

For simplicity we will assume from now on that R = Z_{p}. The general case presents no additional difficulties. Let I be the power of p generating I_{n}. For every m dividing n let Q(m) denote the maximal extension of Q of exponent I inside Q(μ_{m}), and write G(m) = Gal(Q(m)/Q). We have a natural identification G(m) = Gal(Q(n)/Q(n/m)), and we will view G(m) either as a subgroup or as a quotient of G(n), as convenient.

Although in general I_{m} can be a proper divisor of I, we will modify our previous notation and replace I_{m} by I for every m dividing n. With this change we have identifications G(q) ≅ G_{q}/IG_{q} for every q ∈ P and G(m) ≅ ∏_{q|m} G(q).

Lemma A.6. (i) If L ⊂ Q(n) and H = Gal(Q(n)/L) then the natural restriction map H^{1}(L_{ℓ}, T) ⟶^{∼} H^{1}(Q(n)_{ℓ}, T)^{H} is an isomorphism, and it

induces an isomorphism H_{f}^{1}(L_{ℓ}, T) ⟶^{∼} H_{f}^{1}(Q(n)_{ℓ}, T)^{H}.
(ii) H^{1}(G(n/ℓ), H_{f}^{1}(Q(n)_{ℓ}, T)) = 0.

Proof. The first assertion follows from the inflation-restriction exact sequence, since we have assumed that T^{Fr_{ℓ}^{p^{k}} =1} = 0.

Let H be the decomposition group of ℓ in Gal(Q(n)/Q(ℓ)), and L = Q(n)^{H}. Then

H^{1}(L/Q(ℓ), H_{f}^{1}(Q(n)_{ℓ}, T)^{H}) = H^{1}(L/Q(ℓ), H_{f}^{1}(L_{ℓ}, T)) = 0

since ℓ splits completely in L/Q_{ℓ}(ℓ).

On the other hand, H_{f}^{1}(Q(n)_{ℓ}, T) = ⊕_{λ|ℓ}T/(Fr_{λ} − 1)T (Lemma 2.1(i)), where Fr_{λ} is the Frobenius in G_{Q(n)_{λ}}.

Write |H| = k. Since H is generated by Fr_{ℓ}, we can lift Fr_{ℓ} to G_{L_{λ}} so that Fr_{ℓ}^{k} = Fr_{λ}, and then

H^{1}(H, H_{f}^{1}(Q(n)_{λ}, T)) = H^{1}(⟨Fr_{ℓ}⟩, T/(Fr_{ℓ}^{k} − 1)T)

= {t ∈ T : (∑_{i=0}^{k−1} Fr_{ℓ}^{i})t ∈ (Fr_{ℓ}^{k} − 1)T }/(Fr_{ℓ} − 1)T.

By our assumptions on ℓ, multiplication by Fr_{ℓ}^{k} − 1 is injective on T, and so this final quotient is zero. Hence H^{1}(Q(n)/L, H_{f}^{1}(Q(n)_{ℓ}, T)) = 0, and (ii) follows from the inflation-restriction sequence. □

Let X_{n} be the “Universal Euler system” defined in §4.2 of [Ru6]. Then X_{n} is the free Z[G(n)]-module generated by symbols x_{m} for m dividing n, modulo the relations

• G(n/m) acts trivially on x_{m},

• N_{q}x_{qm} = P_{q}(Fr_{q}^{−1})x_{m} if q is a prime dividing n/m, where N_{q} = ∑_{σ∈G(q)} σ viewed as an element of Z[G(n)], and P_{q}(x) = det(1 − Fr_{ℓ} x | T) as in Definition 4.5.

Thus {x_{m}} “looks like” a piece of an Euler system. In particular our Euler system c induces a map c : X_{n} → H^{1}(Q(n), T) by taking c(x_{m}) to the restriction of c_{Q(m)}. Since (c_{Q(m)})_{ℓ} ∈ H_{f}^{1}(Q(m)_{ℓ}, T) ([Ru6] Proposition 4.6.1), we also get a map c_{f} : X_{n} → H_{f}^{1}(Q(n)_{ℓ}, T) by taking c_{f}(x_{m}) to be the restriction of (c_{Q(m)})_{ℓ}.

For each q dividing n fix a generator σ_{q} of G(q), so that we can identify G(q) with Z/IZ. Following Kolyvagin we define

D_{q} = ∑_{i=0}^{I−1} iσ_{q}^{i} ∈ Z[G(q)] ⊂ Z[G(n)]

and D_{m} = ∏_{q|m} D_{q}. We have the telescoping identity in Z[G(n)]

(σ_{q} − 1)D_{q} = I − N_{q},  (32)

which leads to the fundamental property ([Ru6] Lemma 4.4.2)

(σ − 1)D_{m}x_{m} ∈ IX_{n} for every σ ∈ G(n) and m dividing n.  (33)

Since X_{n} has no Z-torsion ([Ru6] Proposition 4.3.1), it follows from (33) that I^{−1}(σ − 1)D_{m}x_{m} is well-defined in X_{n} for every σ and m.

Definition A.7. Suppose m | n. The assignment σ ↦ c_{f}(I^{−1}(σ−1)D_{m}x_{m}) defines a 1-cocycle from G(n/ℓ) to H_{f}^{1}(Q(n)_{ℓ}, T). By Lemma A.6(ii) we conclude that there is a β_{m} ∈ H_{f}^{1}(Q(n)_{ℓ}, T) satisfying

(σ − 1)β_{m} = c_{f}(I^{−1}(σ − 1)D_{m}x_{m}) for every σ ∈ G(n/ℓ).  (34)

In particular

(D_{m}c_{Q(m)})_{ℓ} − Iβ_{m} ∈ H_{f}^{1}(Q(n)_{ℓ}, T)^{G(n/ℓ)},

so by Lemma A.6(i) there is a (unique) η_{m} ∈ H_{f}^{1}(Q(ℓ)_{ℓ}, T) whose restriction to Q(n)_{ℓ} is (D_{m}c_{Q(m)})_{ℓ} − Iβ_{m}.

Proposition A.8. The restriction of κ_{m} to H^{1}(Q(ℓ)_{ℓ}, T/IT) is the image of η_{m}.

Proof. We mimic the arguments of [Ru6] §4.6. Fix a prime λ of Q̄ above ℓ, and let D denote the decomposition group of λ in G_{Q}. Let

𝕋 = Maps(G_{Q}, T),

the group of continuous maps (not necessarily homomorphisms) from G_{Q} to T, with G_{Q} acting by

(γf)(g) = f(gγ) for f ∈ 𝕋 and γ, g ∈ G_{Q}.

Define 𝕋_{ℓ} ⊂ 𝕋 by

𝕋_{ℓ} = {f ∈ 𝕋 : f(hg) = h(f(g)) for all h ∈ D}.

Cohomology of the exact sequence 0 → 𝕋_{ℓ} → 𝕋 → 𝕋/𝕋_{ℓ} → 0 yields, for every subfield F of Q(n) (since H^{0}(F_{ℓ}, T) = 0 by our assumption on ℓ), a short exact sequence

0 ⟶ 𝕋^{G_{F}} ⟶ (𝕋/𝕋_{ℓ})^{G_{F}} ⟶^{δ_{F}} H^{1}(F_{ℓ}, T) ⟶ 0  (35)

and a map

(𝕋/(I𝕋 + 𝕋_{ℓ}))^{G_{F}} ⟶^{δ̄_{Q}} H^{1}(F_{ℓ}, T/IT)

(see [Ru6] Corollaries B.4.4 and B.5.2). The map c_{f} : X_{n} → H^{1}(Q(n)_{ℓ}, T) factors through a map d : X_{n} → (𝕋/𝕋_{ℓ})^{G_{Q(n)}} in (35) ([Ru6] Proposition 4.6.8). For every m dividing n we have d(D_{m}x_{m}) ∈ (𝕋/(I𝕋 + 𝕋_{ℓ}))^{G_{Q}}, and then

κ_{m} = δ̄_{Q}(d(D_{m}x_{m}))

([Ru6] Lemma 4.6.7 and Definition 4.4.10).

The assignment σ ↦ d(I^{−1}(σ − 1)D_{m}x_{m}) defines a 1-cocycle from G(n/ℓ) to (𝕋/𝕋_{ℓ})^{G_{Q(n)}}. The connecting map δ_{Q(n)} sends this cocycle to the coboundary σ ↦ (σ − 1)β_{m}, and hence to zero in H^{1}(G(n/ℓ), H^{1}(Q(n)_{ℓ}, T)). But 𝕋^{G_{Q}(n)} is a free Z_{p}[G(n)]-module (see [Ru6] Lemma 4.4.6), so H^{1}(G(n/ℓ), 𝕋^{G_{Q}(n)}) = 0 and hence there is a β̂_{m} ∈ (𝕋/𝕋_{ℓ})^{G_{Q(n)}} such that

(σ − 1)β̂_{m} = d(I^{−1}(σ − 1)D_{m}x_{m})

for every σ ∈ G(n/ℓ).

It follows that

δ_{Q(n)}(β̂_{m}) − β_{m} ∈ H^{1}(Q(n)_{ℓ}, T)^{G(n/ℓ)} = H^{1}(Q(ℓ)_{ℓ}, T)

the equality by Lemma A.6(i). Since δ_{Q(ℓ)} is surjective, after adjusting β̂_{m} by an element of (𝕋/𝕋_{ℓ})^{G_{Q(ℓ)}} we may assume that δ_{Q(n)}(β̂_{m}) = β_{m}.

Let η̂_{m} = d(D_{m}x_{m}) − Iβ̂_{m} ∈ (𝕋/𝕋_{ℓ})^{G_{Q(ℓ)}}. Then the restriction of κ_{m} to Q(ℓ) is

δ̄_{Q(ℓ)}(d(D_{m}x_{m})) = δ̄_{Q(ℓ)}(η̂_{m})

which is the image of δ_{Q(ℓ)}(η̂_{m}) = η_{m} in H^{1}(Q(ℓ)_{ℓ}, T/IT). This proves the lemma. □

Proposition A.8 will enable us to prove Theorem A.4, because for each m, the finite projection (κ_{m})_{ℓ,f} is determined by the restriction of κ_{m} to Q(ℓ)_{ℓ}.

Definition A.9. Define the ℓ-finite quotient \overline{X_{n}} of X_{n} to be the quotient of X_{n} by the elements \{x_{m} : ℓ | m, m | n\}. For x ∈ X_{n} let \overline{x} denote the image of x in \overline{X_{n}}.

Lemma A.10. (i) The map c_{f} : X_{n} → H_{f}^{1}(Q(n)_{ℓ}, T) factors through a map

X_{n} → \overline{X_{n}} ⟶^{c̄_{f}} H_{f}^{1}(Q(n)_{ℓ}, T).

(ii) P_{ℓ}(Fr_{ℓ}^{−1})\overline{X_{n}} = 0.

Proof. The “Euler system congruence relation”, Corollary 4.8.1 of [Ru6] adapted to account for the different choice of Euler factors in our definition of Euler system, says that (c_{Q(m)})_{ℓ,f} = 0 if ℓ | m. This proves (i).

For (ii), we need only observe that \overline{X_{n}} is generated by the \overline{x_{m}} for m dividing n/ℓ, and for these m, P_{ℓ}(Fr_{ℓ}^{−1})\overline{x_{m}} = N_{ℓ}\overline{x_{mℓ}} = 0. □

Definition A.11. Let A denote the augmentation ideal of Z[G(n)], and for each prime q dividing n define ρ̃_{q} : A → Z/IZ to be the composition

ρ̃_{q} : A ⟶ A_{q,I} ⟶^{ρ_{q,I}} G_{q} ⊗ Z/IZ ⟶ Z/IZ

where A_{q,I} and ρ_{q,I} are as in Definition A.3, the map A → A_{I,q} is induced by Z[G(n)] → Z[G(q)] = Z[G_{q} ⊗ Z/IZ], and the final map sends our chosen generator σ_{q} to 1. Concretely, A is generated by the σ_{q} − 1 for q dividing n, and

ρ̃_{q}(σ_{q′} − 1) = \begin{cases}1 & if q = q′, \\ 0 & if q ≠ q′. \end{cases}

Lemma A.12. Suppose mℓ | n.

(i) If f ∈ A^{2} + IZ[G(n)] then \overline{I^{−1}fD_{mℓ}x_{mℓ}} = 0 in \overline{X_{n}}.

(ii) If f ∈ A + IZ[G(n)], then

\overline{I^{−1}fD_{mℓ}x_{mℓ}} = − ∑_{q|m} ρ̃_{q}(f)\overline{I^{−1}P_{q}(Fr_{q}^{−1})D_{mℓ/q}x_{mℓ/q}} − ρ̃_{ℓ}(f)\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{m}x_{m}}.

Proof. Note that by (33), I^{−1}fD_{mℓ}x_{mℓ} is a well-defined element of X_{n}. If f ∈ IZ[G(n)] then \overline{I^{−1}fD_{mℓ}x_{mℓ}} = (I^{−1}f)D_{mℓ}\overline{x_{mℓ}} = 0.

Using (32) we see that

(σ_{q}−1)D_{mℓ}x_{mℓ} = ((q−1)−N_{q})D_{mℓ/q}x_{mℓ} = (q−1)D_{mℓ/q}x_{mℓ}−P_{q}(Fr_{q}^{−1})D_{mℓ/q}x_{mℓ/q}

in X_{n}. Dividing by I and projecting into \overline{X_{n}} proves (ii) when f = σ_{q} − 1.

Note that each P_{q}(Fr_{q}^{−1}) belongs to A + IZ[G(n)] by our assumption that I divides P_{q}(1). Proceeding inductively we can continue to expand those terms in the sum with index divisible by ℓ, and we conclude that if f ∈ A + IZ[G(n)] then \overline{I^{−1}fD_{mℓ}x_{mℓ}} is a linear combination of the elements \{\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{r}x_{r}} : r | (n/ℓ)\}.

Now if f, g ∈ A then \overline{I^{−1}fgD_{mℓ}x_{mℓ}} can be expressed as a linear combination of \{\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})gD_{r}x_{r}} : r | (n/ℓ)\}. But since gD_{r}x_{r} ∈ IX_{n},

\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})gD_{r}x_{r}} = P_{ℓ}(Fr_{ℓ}^{−1})\overline{I^{−1}gD_{r}x_{r}} = 0

by Lemma A.10(ii). This proves (i).

The right-hand side of (ii) is a linear function of

f ∈ (A + IZ[G(n)])/(A^{2} + IZ[G(n)]),

and thanks to (i) the left-hand side is as well. We have shown that (ii) holds for the generators σ_{q} − 1 of (A + IZ[G(n)])/(A^{2} + IZ[G(n)]), so (ii) holds for all f. □

Fix a representative Fr_{ℓ} ∈ G(n) so that Fr_{ℓ} = 1 on Q(ℓ).

Proposition A.13.

\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{n}x_{n}} = ∑_{{π∈Σ_{1}(n) \atop π(ℓ)≠ℓ}}(−1)^{ν(n/d_{π})} ∏_{q|(n/d_{π})} ρ̃_{q}(P_{q}(Fr_{π(q)}^{−1}))\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{d_{π}}x_{d_{π}}}

Proof. Apply Lemma A.12 repeatedly, beginning with m = n and f = P_{ℓ}(Fr_{ℓ}^{−1}). Expand all terms of the form \overline{I^{−1}gD_{m}x_{m}} with m divisible by ℓ, but not those with m prime to ℓ. (Note that at the first step we have ρ̃_{ℓ}(P_{ℓ}(Fr_{ℓ}^{−1})) = 0, since by convention Fr_{ℓ} restricts to 1 in G(ℓ).) The summand corresponding to π occurs as follows:

• expand \overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{n}x_{n}},

• take the resulting term \overline{I^{−1}P_{π(ℓ)}(Fr_{π(ℓ)}^{−1})D_{n/π(ℓ)}x_{n/π(ℓ)}} and expand that,

• take the resulting term \overline{I^{−1}P_{π^{2}(ℓ)}(Fr_{π^{2}(ℓ)}^{−1})D_{n/(π(ℓ)π^{2}(ℓ))}x_{n/(π(ℓ)π(ℓ))}} and ex-

pand that, and so forth until π^{i}(ℓ) = ℓ, which leaves us with the desired multiple of the term \overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{d_{π}}x_{d_{π}}}. □

Definition A.14. By Lemma 2.1(i),

H_{f}^{1}(Q(ℓ), T/IT) ≅ T/(I, Fr_{ℓ} − 1)T, H_{f}^{1}(Q(ℓ), T) ≅ T/(Fr_{ℓ} − 1)T.

By our assumptions on ℓ, both of these groups are cyclic and the latter is free of rank one over Z_{p}/P_{ℓ}(1). Hence we have a composition

H_{f}^{1}(Q_{ℓ}, T/IT) ⟶^{res} H_{f}^{1}(Q(ℓ)_{ℓ}, T/IT) ⟶^{P_{ℓ}(1)/I} H_{f}^{1}(Q(ℓ)_{ℓ}, T) ⟶^{res} H_{f}^{1}(Q(n)_{ℓ}, T)

(36)

in which all three maps are injective: the first by Lemma 2.4, the second by the observation above, and the third by Lemma A.6.

We denote the composition (36) by res_{I} : H_{f}^{1}(Q_{ℓ}, T/IT) → H_{f}^{1}(Q(n)_{ℓ}, T). We will test the identity of Theorem A.4 in H_{f}^{1}(Q(n)_{ℓ}, T), by applying res_{I}. Note that we could not simply test this identity in H_{f}^{1}(Q(n)_{ℓ}, T/IT) because the natural restriction map is not injective.

Proposition A.15. If m divides n then

res_{I}((κ_{m})_{ℓ,f}) = c̄_{f}(\overline{I^{−1}P_{ℓ}(Fr_{ℓ}^{−1})D_{m}x_{m}})

where c̄_{f} : \overline{X_{n}} → H_{f}^{1}(Q(n)_{ℓ}, T) is the map \overline{x_{m}} ↦ (c_{Q(m)})_{ℓ} of Lemma A.10(i).

Proof. The Cayley-Hamilton theorem shows that P_{ℓ}(Fr_{ℓ}^{−1})H_{f}^{1}(Q(n)_{ℓ}, T) = 0. Thus by Proposition A.8,

res_{I}((κ_{m})_{ℓ,f}) = \frac{P_{ℓ}(1)}{I}((D_{m}c_{Q(m)})_{ℓ} − Iβ_{m}) = \frac{P_{ℓ}(1)}{I}(D_{m}c_{Q(m)})_{ℓ} − P_{ℓ}(1)β_{m}

= \frac{P_{ℓ}(1)}{I}(D_{m}c_{Q(m)})_{ℓ} + (P_{ℓ}(Fr_{ℓ}^{−1}) − P_{ℓ}(1))β_{m}.

Since P_{ℓ}(Fr_{ℓ}^{−1}) − P_{ℓ}(1) is in the augmentation ideal of Z[G(n)], the definition of β_{m} shows that

(P_{ℓ}(Fr_{ℓ}^{−1}) − P_{ℓ}(1))β_{m} = c_{f}(I^{−1}(P_{ℓ}(Fr_{ℓ}^{−1}) − P_{ℓ}(1))D_{m}x_{m}).

Combining these identities proves the proposition. □

Proof of Theorem A.4. Theorem A.4 is now immediate from Propositions A.13 and A.15, and the injectivity of res_{I}. This concludes the proof of Theorem 7.4 as well. □

Appendix B. Proof of Theorem 14.3, by Benjamin Howard

In this appendix we give the proof of Theorem 14.3. It is a pleasure to thank Karl Rubin for his suggestions and encouragement.

Let R be a principal, Artinian, local ring with maximal ideal 𝔪 and finite residue field. If M is an R-module and ψ ∈ Hom(M, R) we define for any integer r a map, also denoted ψ

⋀^{r} M ⟶ ⋀^{r−1} M

by the rule

m_{1} ∧ ⋯ ∧ m_{r} ↦ ∑_{i=1}^{r}(−1)^{i+1}ψ(m_{i})m_{1} ∧ ⋯ ∧ m_{i−1} ∧ m_{i+1} ∧ ⋯ ∧ m_{r}.

We define a map

⋀^{s} Hom(M, R) ⟶ Hom(⋀^{r} M, ⋀^{r−s} M)

for s ≤ r by iteration of the above construction:

ψ_{1} ∧ ⋯ ∧ ψ_{s} = ψ_{s} ∘ ⋯ ∘ ψ_{1}.

Lemma B.1. Suppose M is a free R-module of rank ≥ r + 1 and we are given ψ_{1}, …, ψ_{r} ∈ Hom(M, R). Define

ψ = ψ_{1} ⊕ … ⊕ ψ_{r} : M ⟶ R^{r}
Ψ = ψ_{1} ∧ … ∧ ψ_{r} : ⋀^{r+1} M ⟶ M.

Then

Ψ( ⋀^{r+1} M) = 𝔪^{length(coker(ψ))} ker(ψ).

Proof. First suppose ψ is surjective with kernel A ⊂ M. The image is projective, and so there is a B ⊂ M such that M = A ⊕ B and ψ maps B isomorphically onto R^{r}. The map

⋀^{r+1} M = ⨁_{p+q=r+1}(⋀^{p} A ⊗ ⋀^{q} B) ⟶^{Ψ} A ⊕ B

takes the factor A ⊗ ⋀^{r} B isomorphically onto A and kills the other summands. This proves the claim in this special case.

In general, the image of Ψ and the kernel and cokernel of ψ depend only on the submodule of Hom(M, R) generated by the maps ψ_{1}, …, ψ_{r}, so we may assume that ψ_{i} = π^{a_{i}}ϕ_{i} where π is a uniformizer of R and {ϕ_{i}}_{1≤i≤r} extends to a basis of Hom(M, R). Let

ϕ = ϕ_{1} ⊕ … ⊕ ϕ_{r} : M → R^{r} Φ = ϕ_{1} ∧ … ∧ ϕ_{r} : ⋀^{r+1} M → M.

By the preceding case,

Ψ( ⋀^{r+1} M) = 𝔪^{a} · Φ( ⋀^{r+1} M) = 𝔪^{a} · ker(ϕ)

where a = a_{1} + ⋯ + a_{r} = length(coker(ψ)). Extending the ϕ_{i}’s to a basis of Hom(M, R) and taking the dual basis fixes a splitting M ≅ ker(ϕ) ⊕ R^{r}. In this decomposition the kernel of ψ is

ker(ψ) ≅ ker(ϕ) ⊕ ⨁_{i=1}^{r} 𝔪^{length(R)−a_{i}}R.

Since 𝔪^{a} kills 𝔪^{length(R)−a_{i}}R we have 𝔪^{a} ker(ϕ) = 𝔪^{a} ker(ψ) □

Let T be an R-module equipped with a continuous R-linear action of G_{Q} = Gal(Q̄/Q). Fix a Selmer triple (T, F, P) of core rank χ = χ(T) > 0 which we assume satisfies Hypotheses (H.0) through (H.6) of §9, as well as P ⊂ P_{length(R)}. If n is an integer we define ν(n) to be the number of prime divisors of n.

Theorem B.2. If n ∈ N = N (P) is a core vertex then H_{F^{n}}^{1}(Q, T) is free of rank ν(n) + χ.

Proof. If n a core vertex then H_{F(n)^{∗}}^{1}(Q, T) = 0 and so global duality (Theorem 5.3) gives an exact sequence

0 ⟶ H_{F(n)}^{1}(Q, T) ⟶ H_{F^{n}}^{1}(Q, T) ⟶ ⨁_{ℓ|n} H^{1}(Q_{ℓ}, T)/H_{tr}^{1}(Q_{ℓ}, T) ⟶ 0.

The term on the left is free of rank χ and the term on the right is free (so projective) of rank ν(n) by Lemma 9.6. □

For each ℓ ∈ P choose a generator of Gal(Q(μ_{ℓ})/Q) so that we may view the finite singular comparison map as an isomorphism

H_{f}^{1}(Q_{ℓ}, T) ⟶^{ϕ_{ℓ}^{fs}} H^{1}(Q_{ℓ}, T)/H_{f}^{1}(Q_{ℓ}, T) ≅ H_{tr}^{1}(Q_{ℓ}, T)

and choose also for each ℓ ∈ P an isomorphism H_{tr}^{1}(Q_{ℓ}, T) ⟶^{ι_{ℓ}} R. Fix a core vertex n ∈ N = N (P) and order the primes ℓ_{1}, ⋯, ℓ_{ν(n)} dividing n arbitrarily. Let loc_{i}^{f} (resp. loc_{i}^{tr}) from H^{1}(Q, T) to R be localization at ℓ_{i}, followed by projection onto the finite (resp. transverse) submodule, followed by ι_{ℓ_{i}} ∘ ϕ_{ℓ_{i}}^{fs} (resp. ι_{ℓ_{i}}). For each m | n we define functions

ψ_{i}^{(m)} = \begin{cases}loc_{i}^{f} & if ℓ_{i} | m \\ loc_{i}^{tr} & if ℓ_{i} ∤ m \end{cases}

and

ψ^{(m)} = ψ_{1}^{(m)} ⊕ ⋯ ⊕ ψ_{ν(n)}^{(m)} : H_{F^{n}}^{1}(Q, T) ⟶ R^{ν(n)}
Ψ^{(m)} = ψ_{1}^{(m)} ∧ ⋯ ∧ ψ_{ν(n)}^{(m)} : ⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T) ⟶ H_{F^{n}}^{1}(Q, T).

Proposition B.3. Let λ(m, T^{∗}) = length(H_{F(m)^{∗}}^{1}(Q, T^{∗})). Then

Ψ^{(m)}( ⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T)) = 𝔪^{λ(m,T^{∗})}H_{F(m)}^{1}(Q, T).

Proof. We have the exact sequence

0 ⟶ H_{F(m)}^{1}(Q, T) ⟶ H_{F^{n}}^{1}(Q, T) ⟶^{ψ^{(m)}} R^{ν(n)} ⟶ coker(ψ^{(m)}) ⟶ 0

immediately from the definitions, and the isomorphism

H_{F(m)}^{1}(Q, T) ≅ R^{χ} ⊕ H_{F(m)^{∗}}^{1}(Q, T^{∗})

by Theorem 12.12. From this and Theorem B.2 it follows that length(coker(ψ^{(n)})) = λ(m, T^{∗}). The claim now follows from Lemma B.1. □

Definition B.4. Choose c ∈ ⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T) and for each m | n define

κ_{m} = κ_{m}(n, c) = (−1)^{ν(m)}Ψ^{(m)}(c) ∈ H_{F(m)}^{1}(Q, T).

Proposition B.5. For mℓ_{i} | n we have loc_{ℓ_{i}}^{f}(κ_{m}) = loc_{ℓ_{i}}^{tr}(κ_{mℓ_{i}}).

Proof.

loc_{ℓ_{i}}^{f}(κ_{m}) = (−1)^{ν(m)}loc_{ℓ_{i}}^{f}(Ψ^{(m)}(c))
= (−1)^{ν(m)}(Ψ^{(m)} ∧ loc_{ℓ_{i}}^{f})(c)
= −(−1)^{ν(m)}(Ψ^{(mℓ_{i})} ∧ loc_{ℓ_{i}}^{tr})(c)
= (−1)^{ν(mℓ_{i})}loc_{ℓ_{i}}^{tr}(Ψ^{(mℓ_{i})}(c))
= loc_{ℓ_{i}}^{tr}(κ_{mℓ_{i}})

where the third equality is seen by transposing the factors loc_{ℓ_{i}}^{f}, loc_{ℓ_{i}}^{tr} which both occur in (Ψ^{(m)} ∧ loc_{ℓ_{i}}^{f}). □

The collection of all κ_{m}(n, c) with m | n therefore gives a section of the restriction of the stub Selmer sheaf H′ of Definition 14.1 to the subgraph X_{n} of X (P) whose vertices are all divisors m of n. We would like to show that if n′ = nd is another core vertex then this section can be extended to a section of H′ over X_{nd}. The section κ(n, c) depends on the choice of ordering of primes dividing n, but only up to sign. When we extend our section from X_{n} to X_{nd} we maintain the same ordering on primes dividing n but put them after the “new” primes which divide d. This convention remains in effect in all that follows. Let

loc_{i}^{tr}, loc_{i}^{f} : H^{1}(Q, T) ⟶ R

and

ψ_{i}^{(m)} = \begin{cases}loc_{i}^{f} & if ℓ_{i} | m \\ loc_{i}^{tr} & if ℓ_{i} ∤ m \end{cases}

be defined exactly as before but with our new indexing 1 ≤ i ≤ ν(n′).

Lemma B.6. Keep the notation of the previous paragraph. In the following diagram the image of the horizontal arrow contains the image of the vertical arrow.

⋀^{ν(n′)+1} H_{F^{nd}}^{1}(Q, T) loc_{1}^{tr}∧···∧loc_{ν(d)}^{tr} ⟶ ⋀^{ν(n)+1} H_{F^{nd}}^{1}(Q, T)

↑
⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T)

If the image of c′ ∈ ⋀^{ν(n′)+1} H_{F^{nd}}^{1}(Q, T) under the horizontal arrow agrees with the image of c under the vertical arrow then the section κ(n′, c′) of X_{nd} extends the section κ(n, c) of X_{n}.

Proof. Global duality and H_{(F^{n})^{∗}}^{1}(Q, T) = 0 imply that we have an exact sequence

0 ⟶ H_{F^{n}}^{1}(Q, T) ⟶ H_{F^{nd}}^{1}(Q, T) ⟶^{loc_{d}} ⨁_{ℓ|d} H^{1}(Q_{ℓ}, T)/H_{f}^{1}(Q_{ℓ}, T) ⟶ 0.

The right hand side is projective and so we may choose a free rank ν(d) summand A, complementary to H_{F^{n}}^{1}(Q, T) ⊂ H_{F^{nd}}^{1}(Q, T). We may extend our diagram to

⋀^{ν(n′)+1} H_{F^{nd}}^{1}(Q, T) loc_{1}^{tr}∧···∧loc_{ν(d)}^{tr} ⟶ ⋀^{ν(n)+1} H_{F^{nd}}^{1}(Q, T)

↑  ↑

⋀^{ν(d)} A ⊗ ⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T) loc_{1}^{tr}∧···∧loc_{ν(d)}^{tr}⊗id ⟶ ⋀^{ν(n)+1} H_{F^{n}}^{1}(Q, T)

The map ⨁ loc_{i}^{tr} : A → R^{ν(d)} being an isomorphism implies that

⋀ loc_{i}^{tr} : ⋀^{ν(d)} A ⟶ ⋀^{ν(d)} R^{ν(d)} = R

is as well, and so also is the bottom horizontal arrow. This proves the first claim. To prove the second claim, we compute for m | n

κ_{m}(n, c) = (−1)^{ν(m)}(ψ_{ν(d)+1}^{(m)} ∧ ⋯ ∧ ψ_{ν(nd)}^{(m)})(c)
= (−1)^{ν(m)}(ψ_{ν(d)+1}^{(m)} ∧ ⋯ ∧ ψ_{ν(nd)}^{(m)})((loc_{1}^{tr} ∧ ⋯ ∧ loc_{ν(d)}^{tr})(c′))
= (−1)^{ν(m)}(ψ_{ν(d))}^{(m)} ∧ ⋯ ∧ ψ_{ν(nd)}^{(m)})((ψ_{1}^{(m)} ∧ ⋯ ∧ ψ_{ν(d)}^{(m)})(c′))
= (−1)^{ν(m)}(ψ_{1}^{(m)} ∧ ⋯ ∧ ψ_{ν(nd)}^{(m)})(c′)
= κ_{m}(n′, c′).
□

Theorem B.7. For any m ∈ N the map

Γ(H′) ⟶ H′(m)

is surjective.

Proof. We have fixed a generator of G_{m}, so we have H′(m) = 𝔪^{λ(m,T^{∗})}H_{F(m)}^{1}(Q, T). Fix α ∈ 𝔪^{λ(m,T^{∗})}H_{F(m)}^{1}(Q, T) and (using Lemma 12.9(ii)) choose a core vertex n_{0}

divisible by m. By Proposition B.3 there is c_{0} ∈ ⋀^{ν(n_{0})+1} H_{F^{n_{0}}}^{1} (Q, T) such that κ_{m}(n_{0}, c_{0}) = α.

Now we choose a sequence of core vertices n_{1} | n_{2} … such that n_{0} | n_{1} and every n ∈ N divides n_{i} for some i. By Lemma B.6 we may choose for each i > 0

c_{i} ∈ ⋀^{ν(n_{i})+1} H_{F^{n_{i}}}^{1} (Q, T)

in such a way that the section κ(n_{i+1}, c_{i+1}) of X_{n_{i+1}} restricts to κ(n_{i}, c_{i}) on X_{n_{i}}. We now define the desired κ ∈ Γ(H′) by κ_{n} = κ_{n}(n_{i}, c_{i}) for i chosen sufficiently large. □

Corollary B.8. Γ(H′) has a free R-submodule of rank χ.

Proof. Take m to be a core vertex. Then H′(m) is free of rank χ, so the corollary follows immediately from Theorem B.7. □

Lemma B.9. Suppose χ > 1 and let n be a core vertex. There is an ℓ ∈ P such that nℓ is also a core vertex, H_{F_{ℓ}(n)}^{1}(Q, T) is free of rank ν(n) + χ − 1, and in the composition

⋀^{ν(n)+2} H_{F^{n}}^{1}(Q, T) ⟶^{loc_{ℓ}^{f}} ⋀^{ν(n)+1} H_{F_{ℓ}^{n}}^{1}(Q, T) \underrightarrow{⋀_{q|n} loc_{q}^{f}} H_{F_{ℓ}(n)}^{1}(Q, T)  (37)

both arrows are surjective.

Proof. Let k = length(R). By Proposition 10.2 we may choose l ∈ P so that the sequence

0 ⟶ H_{F_{ℓ}(n)}^{1}(Q, T) ⟶ H_{F(n)}^{1}(Q, T) ⟶^{loc_{ℓ}} H_{f}^{1}(Q_{ℓ}, T) ⟶ 0

is exact. Then by Lemma 12.7(ii), nℓ is a core vertex and that H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) = 0. By global duality the sequence

0 ⟶ H_{F_{ℓ}(n)}^{1}(Q, T) ⟶ H_{F_{ℓ}^{n}}^{1}(Q, T) ⟶^{⊕_{q|n}loc_{q}^{f}} ⨁_{q|n} H_{f}^{1}(Q_{q}, T) ⟶ 0

is exact and so Lemma B.1 implies that the second arrow of (37) is surjective. The surjectivity of the first arrow follows from the exactness of

0 ⟶ H_{F_{ℓ}^{n}}^{1}(Q, T) ⟶ H_{F^{n}}^{1}(Q, T) ⟶^{loc_{ℓ}} H_{f}^{1}(Q_{ℓ}, T) ⟶ 0

and the observation that if M ≅ A ⊕ R is free of rank ≥ r + 1 and π_{R} : A ⊕ R → R is projection onto the second factor then the map

⋀^{r+1} M ⟶^{π_{R}} ⋀^{r} A

is a surjection. Apply this with M = H_{F^{n}}^{1}(Q, T) and A = H_{F_{ℓ}^{n}}^{1}(Q, T). □

Proposition B.10. Suppose χ > 1 and n ∈ N. There is a κ ∈ Γ(H′) such that R · κ is free of rank one and the restriction of κ to X_{n} is trivial.

Proof. By Corollary 12.9(ii) we may assume that n is a core vertex. Fix ℓ ∈ P as in Lemma B.9 and choose c ∈ ⋀^{ν(n)+2} H_{F^{n}}^{1}(Q, T) which is taken by the composition (37) to an element α ∈ H_{F_{ℓ}(n)}^{1}(Q, T) which generates a free submodule. View c as

an element of ⋀^{ν(nℓ)+1} H_{F^{nℓ}}^{1}(Q, T) and let κ_{nℓ}(nℓ, c) be as in Definition B.4. By construction κ_{nℓ}(nℓ, c) = α, but because loc_{ℓ}^{tr} kills c, κ_{m}(nℓ, c) = 0 whenever m | n.

Thus κ(nℓ, c) gives a section of X_{nℓ} with the desired properties and we need only extend it to a section of all of X in exactly the same manner as the proof of Theorem B.7. Choose a sequence of core vertices n_{1} | n_{2} | … with nℓ = n_{1} in such a way that every m ∈ N divides some n_{i}. Choose c_{i} ∈ ⋀^{ν(n_{i})+1} H_{F^{n_{i}}}^{1} (Q, T) in such a way that the section κ(n_{i+1}, c_{i+1}) restricts to κ(n_{i}, c_{i}) on X_{n_{i}} (by Lemma B.6) and pass to the limit. □

Theorem B.11. If χ > 1 then for any integer n, Γ(H′) has a free rank-n submodule.

Proof. Construct a sequence κ^{i} ∈ Γ(H′) inductively as follows. Start with any vertex m_{1} and choose κ^{1} which vanishes on X_{m_{1}} and with R · κ^{1} ≅ R. Once κ^{i} has been constructed, choose a core vertex m_{i+1} which is divisible both by m_{i} and by a vertex at which κ^{i} generates a free submodule of the stalk. By the previous lemma we may construct κ^{i+1} whose restriction to X_{m_{i+1}} is trivial and such that R · κ^{i+1} ≅ R.

If there is a nontrivial linear relation among κ^{1}, …, κ^{n}, say

r_{1}κ^{1} + ⋯ + r_{n}κ^{n} = 0

then let r_{i} be the first nonzero coefficient. Then r_{i}κ^{i} restricted to X_{m_{i+1}} is trivial and so r_{i} = 0 by construction of m_{i+1}, a contradiction. □

Proof of Theorem 14.3. Assertions (i), (ii), and (iii) of Theorem 14.3 are Theorem B.7, Corollary B.8, and Theorem B.11, respectively. □

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Department of Mathematics, Harvard University, Cambridge, MA 02138 USA

   E-mail address: mazur@math.harvard.edu

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

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