KOLYVAGIN SYSTEMS (WORKING DRAFT, 10/22/00)

BARRY MAZUR AND KARL RUBIN

Contents

Introduction 2

Part I. Local cohomology groups 2 1. Local conditions 2 2. The finite/singular homomorphism 4 3. Local duality 5

Part II. Global cohomology groups and Selmer structures 6 4. Selmer modules 6 5. Global duality 9

Part III. Kolyvagin systems 10 6. Kolyvagin systems 10 7. Euler systems and Kolyvagin systems 13 8. Sheaves on graphs 13 9. Hypotheses on T, F, and P. 15 10. Choosing useful primes 17 11. Some remarks about hypothesis (H.6) 20

Part IV. Kolyvagin systems over principal artinian rings 22 12. Cohomological margins 22 13. The sheaf of stub Selmer modules 24 14. Kolyvagin systems and the stub Selmer sheaf 26 15. Kolyvagin systems over a field 28 16. Kolyvagin systems over principal artinian rings 32

Part V. Kolyvagin systems over integral domains 35 17. Kolyvagin systems over a discrete valuation ring 35 18. Kolyvagin systems over Λ 38 19. The module of Kolyvagin systems 38 20. The Kolyvagin Selmer group 39 References 41

Introduction

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).

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}(Q_{ℓ}, T) = H_{L}^{1}(K, T) ⊂ H^{1}(Q_{ℓ}, 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. (to appear) □

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) ⊗ F^{×} ≅ T^{Fr=1}.

Proof. (to appear) □

Definition 2.2. Suppose that T is free of finite rank as an R-module, and 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 isomorphisms. In particular both H_{f}^{1}(K, T) and H_{s}^{1}(K, T) are free of rank one over R.

Proof. The assertion for ϕ^{fs} follows immediately from the assertion for Q(Fr^{−1}), and then the fact that H_{f}^{1}(K, T) and H_{s}^{1}(K, T) are free of rank one follows from Lemma 2.1(i) and (ii).

It remains to show that Q(Fr^{−1}) is an isomorphism. When R is a discrete valuation ring (or a field) this is Corollary A.2.6 of [Ru]. 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. (to appear) □

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

Proof. (to appear) □

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 Example 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).

Let SS(T) denote the set of Selmer structures on T, and if Σ is a set of primes let SS(T, Σ) ⊂ SS(T) denote the set of Selmer structures F with Σ(F) ⊂ Σ.

If F is such 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). 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. (unit group example) Example 4.4. 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 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 4.5. 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 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 ∤ n \\ 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 4.6. 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}

and let

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

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

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 4.7. 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. F_{a}^{b}(c)^{∗} = (F^{∗})_{a}^{b}(c) Example 5.3. (ideal class group example, dual to Example 4.3) Theorem 5.4. Suppose that G, H are Selmer structures and G ≤ H.

(i) There are exact sequences

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

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

(ii) The images loc_{G}^{H} and loc_{H^{∗}}^{G^{∗}} 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 [Ru] Theorem 1.7.3 or [?]. □

Corollary 5.5. 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 5.4 with G = F and H = 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 5.4 with G = F (n) and H = 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 G a category, a simplicial sheaf S on X with values in G is a rule assigning:

• to each simplex σ in X, an object S(σ) of G, 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 G. 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 particular, if X is a graph, with set of vertices V and edges E, 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 4.7)

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_{tr}^{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}.

The constructions (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 6.5. 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 [Ru]) the following collections of classes will suffice. Definition 6.6. 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 6.7. 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 6.5), 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 6.8. 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^{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.

Most of the weak Kolyvagin systems constructed in this way will not be Kolyvagin systems. We will see in §17 that in this example every choice of κ_{1} extends uniquely to a Kolyvagin system. Example 6.9. 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, so κ_{n} ∈ H_{F}^{1}(Q, T/I_{n}T) ⊗ G_{n} = 0. Example 6.10. 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 as a map from G_{nm} = G_{n} ⊗ G_{m} to G_{m} 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.)

If κ is a weak Kolyvagin system for (T, F, P), then the construction above defines a weak Kolyvagin system κ^{(n)} for (T/I_{n}T, F^{n}, P(n)). Definition 6.11. Suppose κ is a Kolyvagin system for (T, F) (over a base ring R). For each prime ideal P of R, consider the induced Kolyvagin system

κ_{P} = κ ⊗_{R} R/P

for (T ⊗_{R} R/P, F ⊗_{R} R/P) over the base ring R/P. We shall say that P is in the support of κ if κ_{P} is the trivial Kolyvagin system over R/P. If P is not in the support of κ, then say that the multiplicity of κ along P is the largest integer μ = μ_{κ}(P) such that all the cohomology classes κ_{P,d} of κ_{P} vanish for all d ∈ N (P) such that ν(d) < μ. (Recall that ν(n) is the number of prime factors of n.)

7. Euler systems and Kolyvagin systems

Suppose for this section that F is a Selmer structure for T such that for every ℓ ∈ Σ(F), one of the following properties holds:

(to be filled in later)

Definition 7.1. An Euler system c for (T, F, P) is (to appear).

Let ES(T, F, P) (or simply ES(T), if there is no danger of confusion) denote the collection of Euler systems for (T, F, P). Then ES(T) is an R module and a G_{Q}-module. Theorem 7.2. There is a canonical R-module homomorphism ES(T) → KS(T) 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_{1}.

Proof. (To appear. The construction, and the fact that it gives a weak Kolyvagin system, is essentially in [Ru].) □

Example 7.3. (cyclotomic unit example)

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 says that S has trivial monodromy if 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 whenever e is an edge of X joining the vertices v and w, both maps ψ_{v}^{e} and ψ_{w}^{e} are surjective and (at least) one of them is an isomorphism.

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 for every pair v, w of vertices and and every pair P, P′ of surjective paths from v to w, 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. Proposition 8.4. Suppose S is locally cyclic.

(i) If v is a hub of S, then 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. An assignment κ : w ↦ κ_{w} is a global section if and only if for every vertex w and every surjective path P from v to w, κ_{w} = ψ_{P} (κ_{v}). Thus if v is a hub then κ_{v} determines κ and f_{v} : Γ(S) → S(v) is injective.

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

Conversely if P, P′ are surjective paths from v to w, then it follows that ψ_{P} = ψ_{P′} on image(f_{v}). Hence if f_{v} is surjective then 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. Corollary 8.6. Suppose S is locally free of rank one and X is connected. Then

(i) If v is a vertex of X, then 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, and Γ(S) is free of

rank one over R if and only if S has trivial monodromy, (iii) a global section κ ∈ Γ(S) is primitive if and only if there is a vertex w ∈ X

such that κ_{w} generates S(w).

Proof. This is just a special case of Proposition 8.4, since if S is locally free of rank one then it is locally cyclic and if it is connected then every vertex is a hub. □

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) p > 3.

(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.

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.

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} βT^{∗} ⟶ 0

0 ⟶ βT^{∗} ⟶ T^{∗} ⟶ T^{∗}/βT^{∗} ⟶ 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, βT^{∗})
0 ⟶ H^{1}(G, βT^{∗}) ⟶ H^{1}(G/Q, 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′.

Now 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 Example 6.9 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.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/IT). 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 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, 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, 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_{t} defined in condition (H.5), and let N_{t} = N (P_{t}) be the corresponding set of positive integers as in Definition 4.6: all squarefree products of primes in P_{t}. Lemma 9.6. (i) Suppose τ ∈ G_{Q} satisfies (H.2) and t ∈ Z^{+}. If ℓ ∉ Σ(F)

and the Frobenius class of ℓ in Gal(Q(T, μ_{p^{t}})/Q) is the conjugacy class of τ, then ℓ ∈ P_{t}.

(ii) Suppose R is principal and artinian of length k. If ℓ ∈ P_{k} then H_{f}^{1}(Q_{ℓ}, T), H_{s}^{1}(Q_{ℓ}, T), and H_{f}^{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. (to appear) □

Lemma 9.7. Suppose R is principal and artinian, and (T, F, P) satisfies (H.0) through (H.6). Then so does (T^{∗}, F^{∗}, P).

Proof. The only nontrivial conditions to check are (H.2) and (H.6). It is not difficult to show that if (H.2) holds for T, then it holds for T^{∗} with the same τ, and (H.6) for T^{∗} follows from Lemma 9.3. □

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. In this section we let T be 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). 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.

Proof. Increasing k if necessary, and replacing R by R/𝔪^{k} and T by T/𝔪^{k}T, we may suppose without loss of generality 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 Ω = Q(T, T^{∗}, μ_{p^{k}}). By (H.3), the restriction maps

H^{1}(Q, T) → H^{1}(Ω, T) = Hom(G_{Ω}, T), H^{1}(Q, T^{∗}) → H^{1}(Ω, T^{∗}) = Hom(G_{Ω}, T^{∗})

are injective. In particular the restrictions of the c_{i} are all nonzero. For each i define F_{i} to be the abelian extension of Ω such that G_{F_{i}} is the kernel of the restriction of c_{i} to G_{Ω}.

Define H_{1}, H_{2} ⊂ G_{Ω} by

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

where τ is as in (H.2). 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_{Ω} − H_{i} and ℓ is a rational prime whose Frobenius conjugacy class in Gal(F_{i}/Q) is the class of τγ, then ℓ ∈ P_{k} and (c_{i})_{ℓ} ≠ 0.

Let μ be Haar measure on G_{Ω} normalized so that μ(G_{Ω}) = 1. As in the proof of Lemma 5.2.1 of [Ru], μ(H_{i}) ≤ 1/p.

Suppose first that p > 3. Then

μ(H_{1} ∪ H_{2} ∪ H_{3} ∪ H_{4}) ≤ \frac{4}{p} < 1,

so we can find γ ∈ G_{Ω} − (H_{1} ∪ H_{2} ∪ H_{3} ∪ H_{4}). Let S be the set of rational primes whose Frobenius conjugacy class in Gal(F_{1}F_{2}F_{3}F_{4}/Q) is the class of τγ. Then S satisfies the conclusions of the proposition.

Now suppose that hypothesis (H.4b) holds. It follows from (H.4b) that F_{1}F_{2} ∩ F_{3}F_{4} = Ω. Also, it follows from the irreducibility of T (hypothesis (H.1)) that if c_{2} is not a multiple of c_{1} then F_{1} ∩ F_{2} = Ω, and similarly for c_{3} and c_{4}.

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

Proposition 10.2. Suppose that R is artinian and principal and (H.4a) holds, i.e., T/𝔪T ≅ T̸^{∗}[𝔪] as F_{p}[[G_{Q}]]-modules.

Suppose also that C ⊂ H^{1}(Q, T) is a finite submodule, d ∈ H^{1}(Q, T^{∗})[𝔪] is nonzero, and ϕ : C → R is a homomorphism.

Then for every k ∈ Z^{+} 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) and d_{ℓ} ≠ 0

where loc_{ℓ} is localization at ℓ.

The rest of this section is devoted to the proof of Proposition 10.2. 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. □

Now 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.

Suppose k is large enough so that 𝔪^{k} = 0 (recall that R is artinian in Proposition 10.2), and let F = Q(T, μ_{p^{k}}) (a finite extension of Q). 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. □

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_{1}) = η^{−1} ∘ ϕ

so in particular ker(loc_{ℓ}) = ker(ϕ).

Now repeat the argument above with T replaced by T^{∗} and C replaced by D = Rd. 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 d_{ℓ} ≠ 0 in H^{1}(Q_{ℓ}, T^{∗}).

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 even 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 the proposition. □

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
π^{j−i}↓  _{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 4.4) 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 free by our assumption on the H_{F}^{1}(Q_{ℓ}, T). Therefore H^{1}(G, T)/H_{F}^{1}(Q, T) is 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}(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.

Since g_{2} is also injective it follows that f is injective, and hence both rows of the diagram are exact. Thus 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 independent 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 = T/(τ − 1)T is free of rank one over R and similarly 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} and for all n ∈ N the ideal I_{n} vanishes, and H(n) = H_{F (n)}^{1}(Q, T) ⊗ G_{n}. Note that since R is principal artinian, T^{∗} is also a free R-module; this is not true for general R.

12. Cohomological margins

Definition 12.1. For every n ∈ N = N_{k} define

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

Note that the second definition is really equivalent to the first, since F (n)^{∗} = F^{∗}(n) by Example 5.2. Also, these definitions apply equally well when T is replaced by T/𝔪^{i} for i ∈ Z^{+}. Proposition 12.2. If 0 ≤ i ≤ k then

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

Proof. The two equalities follow directly from Lemmas 9.3, 9.4, and 11.4. □

Proposition 12.3. The differences λ(n, T) − λ(n, T^{∗}) are independent of n ∈ N.

Proof. This follows immediately from Corollary 5.5 and Lemma 9.6(ii). □

Definition 12.4. By the cohomological margin of T with Selmer structure F, we mean the integer χ(T) := λ(n, T) − λ(n, T^{∗}) for any n ∈ N, noting that it is independent of n.

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

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

Corollary 12.5. For every n ∈ N, (i) λ(n, T) ≥ χ(T) and λ(n, T̄) ≥ χ(T̄),

(ii) λ(n, T) = χ(T) ⟺ λ(n, T̄) = χ(T̄) ⟺ H_{F (n)^{∗}}^{1}(Q, T^{∗}) = 0

Proof. The first assertion is immediate from the definitions and Proposition 12.3. Similarly the equivalences

λ(n, T) = χ(T) ⟺ H_{F (n)^{∗}}^{1}(Q, T^{∗}) = 0,
λ(n, T̄) = χ(T̄) ⟺ H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = 0,

are clear, so (ii) follows by Lemma 9.3. □

Recall that ν(n) denote the number of prime divisors of n ∈ Z^{+}. Lemma 12.6. Suppose nℓ ∈ N with ℓ prime.

(i) |λ(nℓ, T̄) − λ(n, T̄)| ≤ 1 and |λ(nℓ, T) − λ(n, T)| ≤ k.

(ii) If λ(nℓ, T) = λ(n, T)−k then H_{F (nℓ)}^{1}(Q, T) is the kernel of the localization

map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T). (iii) If λ(nℓ, T) = λ(n, T) + k then H_{F (n)}^{1}(Q, T) is the kernel of the localization

map H_{F (nℓ)}^{1}(Q, T) → H_{s}^{1}(Q_{ℓ}, T). (iv) Suppose that the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective.

Then λ(nℓ, T^{∗}) ≤ λ(n, T^{∗}) and localization gives a surjection

H_{F (nℓ)}^{1}(Q, T) ↠ 𝔪^{λ(n,T^{∗})−λ(nℓ,T^{∗})}H_{s}^{1}(Q_{ℓ}, T).

(v) If 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, then λ(nℓ, T̄) = λ(n, T̄)−1 and λ(nℓ, T̄^{∗}) = λ(n, T̄^{∗})−1. (vi) If λ(n, T̄) = χ(T̄) and the localization map H_{F (n)}^{1}(Q, T)[𝔪] → H_{f}^{1}(Q_{ℓ}, T)

is nonzero, then λ(nℓ, T̄) = χ(T̄). (vii) If λ(nℓ, T̄) = χ(T̄) and the localization map H_{F (nℓ)}^{1}(Q, T)[𝔪] → H_{s}^{1}(Q_{ℓ}, T)

is nonzero, then λ(n, T̄) = χ(T̄). (viii) |λ(n, T̄) − λ(1, T̄)| ≤ ν(n) and |λ(n, T̄^{∗}) − λ(1, T̄^{∗})| ≤ ν(n).

Proof. From the definitions we have injections

H_{F (n)}^{1}(Q, T)/H_{F_{ℓ}(n)}^{1}(Q, T) ↪^{ϕ_{1}} H_{f}^{1}(Q_{ℓ}, T),

H_{F (nℓ)}^{1}(Q, T)/H_{F_{ℓ}(n)}^{1}(Q, T) ↪^{ϕ_{2}} H_{s}^{1}(Q_{ℓ}, T).

Thus

λ(nℓ, T) = λ(n, T) − length(image(ϕ_{1})) + length(image(ϕ_{2})).

By Lemma 9.6(ii), both H_{f}^{1}(Q_{ℓ}, T) and H_{s}^{1}(Q_{ℓ}, T) are free of rank one (and hence have length k), and (i), (ii), (iii), and (iv) follow immediately. (For (iv) we use that λ(n, T^{∗}) − λ(nℓ, T^{∗}) = λ(n, T) − λ(nℓ, T) by Lemma 12.3.)

Using Lemmas 9.3 and 9.4, the last four assertions are really statements about T̄, so to simplify notation we may just suppose that k = 1, i.e., R is a field and T = T̄.

By Theorem 5.4 the image of ϕ_{2} is the orthogonal complement of the image of the localization map H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) ⟶^{ϕ_{3}} H_{f}^{1}(Q_{ℓ}, T^{∗}). If the map H_{F (n)^{∗}}^{1}(Q, T^{∗}) → H_{f}^{1}(Q_{ℓ}, T^{∗}) is nonzero, then so is ϕ_{3}, and so ϕ_{2} must be zero. This proves (v) for λ(nℓ, T̄), and the statement for λ(nℓ, T̄^{∗}) follows by Proposition 12.3.

If ϕ_{1} is nonzero, then we have λ(nℓ, T̄) ≤ λ(n, T̄). But if λ(n, T̄) = χ(T̄), then Corollary 12.5(i) shows that λ(nℓ, T̄) ≥ λ(n, T̄). This proves (vi), and the proof of (vii) is the same.

Finally, (viii) for T̄ follows from (i) by induction on ν(n), and then we deduce it for T̄^{∗} from Lemma 12.3. □

Corollary 12.7. Suppose χ(T̄) ≥ 0. For every j ≥ k there are integers m ∈ N_{j} with ν(m) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}) and λ(m, T^{∗}) = λ(m, T̄^{∗}) = 0.

Proof. Suppose n ∈ N and λ(n, T̄^{∗}) > 0. Since χ(T) ≥ 0, Proposition 12.3 shows that λ(n, T) > 0 as well. It follows easily from Proposition 10.1 and Lemma 12.6(v) that there is an ℓ ∈ P_{j} such that λ(nℓ, T̄^{∗}) = λ(n, T̄^{∗}) − 1. Starting with n = 1 and proceeding inductively, after λ(1, T̄^{∗}) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T̄^{∗}) steps we reach an integer satisfying the conclusions of the corollary. □

Definition 12.8. If m ∈ N and λ(m, T^{∗}) = 0 we say that m is a core vertex (of the graph X of Definition 6.2). Corollary 12.9. χ(T) > 0 ⟺ χ(T̄) > 0.

Proof. Suppose first that χ(T̄) ≥ 0, and let m be a core vertex. Then by Corollary 12.5(ii) and Lemma 9.4,

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

and the corollary follows in this case.

Now suppose that χ(T̄) < 0. By Lemma 9.7, T^{∗} satisfies all necessary hypotheses and (using Lemmas 9.3 and 9.4)

χ(T^{∗}/𝔪T^{∗}) = χ(T̄^{∗}) = −χ(T̄) > 0.

Thus we can apply the argument above to T^{∗} to show that χ(T^{∗}) > 0, and hence χ(T) = −χ(T^{∗}) < 0. □

13. The sheaf of stub Selmer modules

Proposition 13.1. Suppose χ(T) ≥ 0. For every core vertex m and for every n ∈ N, there is a noncanonical isomorphism

H_{F (n)}^{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 (using Proposition 12.3 for the second equality and Lemmas 9.3 and 9.4 for the last)

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

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

Thus the desired noncanonical isomorphism exists. □

Corollary 13.2. If χ(T) ≥ 0 and n, m ∈ N are both core vertices, then there is a (noncanonical) isomorphism H_{F (n)}^{1}(Q, T) ≅ H_{F (m)}^{1}(Q, T).

Proof. Immediate. □

Definition 13.3. If χ(T) ≥ 0, fix an R-module M which is isomorphic over R to H_{F (m)}^{1}(Q, T) for some, and hence for all, core vertices m. If χ(T) < 0, set M = 0. We call M the core Selmer module for T with Selmer structure F.

When χ(T) ≥ 0, Corollary 12.5(ii) shows that

length(M) = χ(T), dim_{𝕜}(M[𝔪]) = dim_{𝕜}(M/𝔪M) = χ(T̄),

and by Proposition 13.1 there is a noncanonical isomorphism

H_{F (n)}^{1}(Q, T) ≅ M ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗})  (9)

for every n ∈ N. Definition 13.4. Recall that H(n) = H_{F (n)}^{1}(Q, T) ⊗ G_{n} is the Selmer stalk of Definition 6.2. If n ∈ N, the stub Selmer submodule H′(n) ⊂ H(n) is the submodule defined by

H′(n) := 𝔪^{λ(n,T^{∗})} H(n) ⊂ H(n).

It follows from Proposition 13.1 that if n ∈ N, we have a noncanonical R-module isomorphism H′(n) ≅ 𝔪^{λ(n,T^{∗})} M, and therefore the isomorphism type of the stub Selmer module H′(n) is dependent only on the integer λ(n, T^{∗}).

We define a subsheaf of the Selmer sheaf H = H_{T} = H_{(T,F,P)} of Definition 6.2 as follows. Definition 13.5. By the sheaf of stub Selmer modules H′ = H′_{T} ⊂ H_{T} we mean the sub-simplicial sheaf of R-modules on the graph X (P) whose stalks at vertices n ∈ N are given by the stub Selmer modules H′(n), and whose edge stalks and vertex-to-edge maps coincide with those of the Selmer sheaf H_{T}. Lemma 13.6. Suppose κ ∈ KS(T) and n ∈ N. If ℓ ∈ P is such that

(a) localization at ℓ maps H′(n) to zero,

(b) κ_{nℓ} ∈ H′(nℓ), then (κ_{n})_{ℓ} = 0.

Proof. As in the proof of Lemma 12.6, we have

λ(nℓ, T) = λ(n, T) − length(image(ϕ_{1})) + length(image(ϕ_{2})).

where ϕ_{1} and ϕ_{2} are the localization maps

H_{F (n)}^{1}(Q, T)/H_{F_{ℓ}(n)}^{1}(Q, T) ↪^{ϕ_{1}} H_{f}^{1}(Q_{ℓ}, T),

H_{F (nℓ)}^{1}(Q, T)/H_{F_{ℓ}(n)}^{1}(Q, T) ↪^{ϕ_{2}} H_{s}^{1}(Q_{ℓ}, T).

Condition (a) on ℓ says that the image of ϕ_{1} is annihilated by 𝔪^{λ(n,T^{∗})}, so the image of ϕ_{1} has length at most λ(n, T^{∗}). Therefore by Proposition 12.3 we have length(image(ϕ_{2})) ≤ λ(nℓ, T^{∗}). Thus by (b), (κ_{nℓ})_{ℓ,s} = ϕ_{2}(κ_{nℓ}) = 0. Now by definition of a Kolyvagin system, we conclude that (κ_{n})_{ℓ} = 0. □

Theorem 13.7. If χ(T) ≤ 0 then KS(T) = 0.

Proof. Since χ(T) ≤ 0 we have λ(n, T) ≤ λ(n, T^{∗}) for every n ∈ N. In particular H′(n) = 0 for every n, so condition (a) of Lemma 13.6 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 12.2, and there is nothing to prove.

Choose n ∈ N with λ(n, T̄) > 0, and suppose that κ_{n} ≠ 0. Since χ(T) ≤ 0, we have χ(T̄) ≤ 0 by Corollary 12.9, so 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 12.6(v) that λ(nℓ, T̄) < λ(n, T̄). Hence our induction hypothesis applies, and so κ_{nℓ} = 0. In particular we can apply Lemma 13.6 to conclude that (κ_{n})_{ℓ} = 0, which contradicts (a). Hence κ_{n} = 0. □

14. Kolyvagin systems and the stub Selmer sheaf

The main result of this section is that, under suitable hypotheses, a Kolyvagin system, a priori a global section of the Selmer sheaf H, is actually a global section of the subsheaf H′. Definition 14.1. Suppose R̃ is a local principal artinian ring such that R is a quotient of R̃. We say that (T̃, F̃, P) is a lifting of (T, F, P) to R̃ if T̃ is an R̃[[G_{Q}]]- module, (T̃, F̃, P) is a Selmer triple over R̃, T = T̃ ⊗R̃ R, and F = F̃ ⊗R̃ R. Theorem 14.2. Suppose κ ∈ KS(T) and one of the following two conditions holds:

(a) (H.4a) holds, i.e., T̄ ≅ ¯T̸^{∗} as F_{p}[[G_{Q}]]-modules.

(b) The Kolyvagin system κ is sufficiently liftable in the following sense. There is a local principal artinian ring R̃ of length k̃ ≥ 2k − 1 and a lifting (T̃, F̃, P) of (T, F, P) to R̃ such that:

• (T̃, F̃, P) satisfies (H.0) through (H.6) and P ⊂ Pk̃.

• The core Selmer module M̃ of the Selmer triple (T̃, F̃, P) has the property that 𝔪̃^{(k̃−k)}M̃ is free over R, where 𝔪̃ is the maximal ideal of R̃.

• κ is in the image of the natural map KS(T̃) → KS(T).

Then κ is a global section of the subsheaf H′_{T} of stub Selmer modules (i.e., for all n ∈ N we have κ_{n} ∈ H′(n)). Remark 14.3. Hypothesis (a) of Theorem 14.2 is reasonably general, but unfortunately it leaves out the example R = Z/p^{k}Z, T = E[p^{k}] with an elliptic curve E. It is for this reason that we go to the extra trouble to prove the theorem under hypothesis (b).

If R is a field, i.e., k = 1, then every Kolyvagin system is sufficiently liftable because we can take k̃ = k, R̃ = R, etc.

We will prove Theorem 14.2 below (considering the two hypotheses separately) after the following corollary. The 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 [Ru]. In Theorem 16.8 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 14.4. Suppose κ ∈ KS(T) and either hypothesis (a) or (b) of Theorem 14.2 hold. Then

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

Proof. This is just a restatement of Theorem 14.2 for n = 1. □

Proof of Theorem 14.2 under hypothesis (a). By Theorem 13.7 (and Corollary 12.9) we may assume that χ(T̄) > 0. We will prove that κ_{n} ∈ H′(n) by induction on λ(n, T̄^{∗}). When λ(n, T̄^{∗}) = 0 we have λ(n, T^{∗}) = 0 (Proposition 12.2) so there is nothing to prove.

Suppose now that λ(n, T̄^{∗}) > 0, and suppose κ_{n} ∉ H′(n). (In particular we are supposing that κ_{n} ≠ 0.)

Since χ(T̄) > 0, it follows from Proposition 13.1 that

H(n)[𝔪] ⊄ H′(n).

Therefore 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 we can fix a prime ℓ ∈ P, prime to n, such that (a) c_{ℓ} = 0 for every c ∈ H′(n),

(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 12.6(v), we have λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}), so by the induction

hypothesis we have

κ_{nℓ} ∈ H′(nℓ).

Therefore by Lemma 13.6, (κ_{n})_{ℓ} = 0, which contradicts condition (b). This contradiction shows that κ_{n} ∈ H′(n). □

Proof of Theorem 14.2 under hypothesis (b). Keep the notation of hypothesis (b)

The proof will be by induction on both k and λ(n, T̄^{∗}). If k = 0 then everything is trivial, and if λ(n, T̄^{∗}) = 0 then λ(n, T^{∗}) = 0 (Proposition 12.2) so H′(n) = H(n) and again there is nothing to prove. So suppose that λ(n, T̄^{∗}) > 0. Case 1: λ(n, T^{∗}) < k.

Whenever 0 ≤ d < d′ ≤ k̃, cohomology of the exact sequence

0 → T̃/𝔪^{d′−d}T̃ → T̃/𝔪^{d′} T̃ → T̃/𝔪^{d}T̃ → 0

together with Lemma 9.4 shows that

ker(H_{T/𝔪^{d′} T} → H_{T/𝔪^{d}T}) = H_{T/𝔪^{d′} T} [𝔪^{d′−d}].  (10)

Let i = λ(n, T^{∗}), and consider the image κ^{(i)} of κ in KS(T/𝔪^{i}T). By Lemma 9.3 we have

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

so λ(n, T^{∗}[𝔪^{i}]) = λ(n, T^{∗}) = i. Thus by our induction hypothesis applied to the R/𝔪^{i}-module T/𝔪^{i}T, we have

κ_{n}^{(i)} ∈ H′_{T/𝔪^{i}T} (n) = 𝔪^{i}H_{T/𝔪^{i}T} (n) = 0.

By (10) (with d′ = k, d = i) it follows that κ_{n} ∈ H_{T} (n)[𝔪^{k−i}].

Let \overline{HT̃}(n) denote the image of HT̃(n) in H_{T} (n). If we now apply (10) with d′ = k̃, d = k and use (9), we see that

\overline{HT̃}(n) ≅ (M̃ ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}))/(M̃ ⊕ H_{F (n)^{∗}}^{1}(Q, T^{∗}))[𝔪^{k̃−k}].

Since k̃−k ≥ k−1 ≥ λ(n, T^{∗}), we have H_{F (n)^{∗}}^{1}(Q, T^{∗})[𝔪^{k̃−k}] = H_{F (n)^{∗}}^{1}(Q, T^{∗}). By

assumption M̃/M̃[𝔪^{k̃−k}] is free over R, and hence so is \overline{HT̃}(n). Since κ_{n} ∈ \overline{HT̃}(n) and 𝔪^{k−i}κ_{n} = 0, we conclude that

κ_{n} ∈ 𝔪^{i}\overline{HT̃}(n) ⊂ 𝔪^{i}H_{T} (n) = H′_{T} (n),

as desired. Case 2: λ(n, T^{∗}) ≥ k. In this case we need to prove that κ_{n} = 0.

Suppose κ_{n} ≠ 0. Using Proposition 10.1 we can fix a prime ℓ such that (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 12.6(v) that λ(nℓ, T̄^{∗}) < λ(n, T̄^{∗}). Since λ(n, T^{∗}) ≥ k we have H′_{T} (n) = 𝔪^{λ(n,T^{∗})}H_{T} (n) = 0, and exactly as in the proof of Theorem 14.2 we conclude from our induction hypothesis and Lemma 13.6 that (κ_{nℓ})_{ℓ} = 0. But this contradicts (a), so we must have κ_{n} = 0. □

Remark 14.5. Note that the proof above shows that under hypothesis (b) of Theorem 14.2, κ_{n} ∈ 𝔪^{λ(n,T^{∗})}\overline{HT̃}(n) which can be smaller than H′_{T} (n).

15. Kolyvagin systems over a field

In this section we suppose that R is a field, i.e., R = 𝕜, k = 1, and T̄ = T. Corollary 15.1. If dim_{𝕜} H_{F}^{1}(Q, T) ≤ dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}) (i.e., the cohomological margin χ(T) ≤ 0), then KS(T) = 0.

More generally, for arbitrary χ(T), if κ ∈ KS(T) and λ(n, T) ≠ χ(T), then κ_{n} = 0.

Proof. The first statement is just Theorem 13.7 in this setting. The second follows from Theorem 14.2(b) with k̃ = k = 1, since if λ(n, T) ≠ χ(T) then λ(n, T^{∗}) > 0 (Corollary 12.5(ii)). □

We suppose for the rest of this section that the cohomological margin χ(T) = 1. Definition 15.2. 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 integers n ∈ N such that λ(n, T) = 1 (or equivalently, λ(n, T^{∗}) = 0). We join n and nℓ by an edge in X^{0} if and only if the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is not zero (or equivalently, surjective).

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}. In other words, H^{0}(n) = H_{F (n)}^{1}(Q, T) ⊗ G_{n}, if e is the edge joining n and nℓ then H^{0}(e) = H_{s}^{1}(Q_{ℓ}, T) ⊗ G_{nℓ}, the map H^{0}(nℓ) → H^{0}(e) is localization at ℓ, and the map H^{0}(n) → H^{0}(e) is localization at ℓ followed by ϕ_{ℓ}^{fs}.

Note that 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). Lemma 15.3. The restriction map on global sections H(X) → H^{0}(X^{0}) is an isomorphism.

Proof. Suppose κ is a global section of H. If n is a vertex of X which is not a vertex of X^{0}, then Corollary 15.1 shows that κ_{n} = 0. Thus the restriction map H(X) → H^{0}(X^{0}) is injective.

Conversely, if κ is a global section of H^{0}, we can extend κ to X by defining kappa_{n} = 0 if n is a vertex of X which is not a vertex of X^{0}. It follows from Lemma 12.6(vi) and (vii) that this extension is a global section of H. □

Lemma 15.4. 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, T) → H_{f}^{1}(Q_{ℓ}, T) is nonzero. (iii) The localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is an isomorphism. (iv) The localization map H_{F (nℓ)}^{1}(Q, T) → H_{s}^{1}(Q_{ℓ}, T) is nonzero.

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

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

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

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

the remaining equivalences follow. □

Lemma 15.5. The sheaf H^{0} is locally free of rank one.

Proof. By definition if n is a vertex of X^{0} then H^{0}(n) is free of rank one. If e is an edge joining n and nℓ in X^{0}, then by Lemmas 15.4 and 9.6(ii) the maps from H^{0}(n) and H^{0}(nℓ) to H^{0}(e) are both isomorphisms. □

Theorem 15.6. The graph X^{0} is connected.

Theorem 15.6 will be proved later in this section. We first discuss its consequences and give some examples. Theorem 15.7. Suppose χ(T) = 1.

(i) If n ∈ N and λ(n, T) = 1, then the map KS(T) → H(n) defined by κ ↦ κ_{n} is injective. (ii) dim_{𝕜} KS(T) ≤ 1. (iii) KS(T) is nonzero (and hence one-dimensional) if and only if the sheaf X^{0}

has trivial monodromy. (iv) If κ ∈ KS(T) is nonzero, then κ_{n} ≠ 0 if and only if λ(n, T) = 1.

Proof. By Lemma 15.3 we can identify KS(T) with Γ(H^{0}). Theorem 15.6 says that X^{0} is connected, and Lemma 15.5 says that H^{0} is locally free of rank one. Now the theorem is immediate from Corollary 8.6 applied to H^{0} (along with Corollary 15.1 for the “only if” part of (iv)). □

Theorem 15.8. Suppose χ(T) = 1 and κ ∈ KS(T) is nonzero. Then

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

Proof. By Lemma 12.7, there is an n ∈ N such that ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}). On the other hand, if m ∈ N and ν(m) < dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}), then Lemma 12.6(viii) shows that λ(m, T^{∗}) > 0, and hence by Theorem 15.7(iv) κ_{m} = 0. This proves the theorem. □

Example 15.9. (To appear: cyclotomic units, elliptic curves)

The rest of this section is devoted to the proof of Theorem 15.6.

Lemma 15.10. 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. Then the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is zero, so by Lemma 15.4, H_{F (nℓ)}^{1}(Q, T) = H_{F (n)}^{1}(Q, T).

By Corollary 12.5(ii) we have H_{F (n)^{∗}}^{1}(Q, T^{∗}) = 0. Applying Theorem 5.4 with G = F_{ℓ}(n) and H = F (n) we conclude that dim_{𝕜} H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) = 1.

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

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

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

are both isomorphisms. By Lemma 12.6(v) 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, T^{∗}) = 0, so applying Theorem 5.4 with G = F_{ℓ}^{q}(n) and H = F^{q}(n) shows that the localization map H_{F^{q}(n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is nonzero. Since the image of H_{F (n)}^{1}(Q, T) in H_{f}^{1}(Q_{ℓ}, T) is zero, and (since λ(n, T) = λ(nq, T) = 1) H_{F^{q}(n)}^{1}(Q, T) = H_{F (n)}^{1}(Q, T) ⊕ H_{F (nq)}^{1}(Q, T), we conclude that the image of H_{F (nq)}^{1}(Q, T) in H_{f}^{1}(Q_{ℓ}, T) 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 15.11. Suppose that n is a vertex of X^{0} and ν(n) ≥ dim_{𝕜} H_{F}^{1}(Q, T). 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, T) → H_{f}^{1}(Q_{ℓ}, T) is nonzero. By Lemma 12.6(vii) λ(n/ℓ, T) = 1, and by Lemma 15.4 there is an edge joining n and n/ℓ, and we are done.

Case 2: for every ℓ dividing n, the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is zero. Then

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

Since this intersection of at least dim_{𝕜} H_{F}^{1}(Q, T) subspaces of H_{F}^{1}(Q, T) 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, T) = H_{F_{g}}^{1}(Q, T).

If λ(g, T) = 1 then by Lemma 15.10 there is a path from n to g, and again we are done. So we may assume that λ(g, T) > 1. By Lemma 12.6(i) we have λ(g, T) = 2, and by Corollary 12.5(ii) we also have that H_{F (g)^{∗}}^{1}(Q, T^{∗}) ≠ 0.

Choose nonzero elements c ∈ H_{F (n)}^{1}(Q, T) ⊂ H_{F (g)}^{1}(Q, T), d ∈ H_{F (g)^{∗}}^{1}(Q, T^{∗}), 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 12.6(v) we have λ(gℓ, T) = 1, so gℓ is a vertex of X^{0}. By Lemma 12.6(ii) we also have

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

Since the class c belongs to the first kernel but not the second, these two kernels are different. It follows that the localization maps

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

are both nonzero, so by Lemmas 12.6(vi) and 15.4 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, T) → H_{s}^{1}(Q_{r}, T) is nonzero. If not, then

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

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 15.12. Suppose that n and m are vertices of X^{0} and ν(n) = ν(m) = dim_{𝕜} H_{F}^{1}(Q, T) − 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, T) − 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. By Lemma 12.6(i) and (viii) we must have λ(n/q, T) = λ(m/r, T) = 2. As in the proof of Proposition 15.11 it follows that

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

Choose nonzero elements c ∈ H_{F (n)}^{1}(Q, T), c′ ∈ H_{F (m)}^{1}(Q, T), d ∈ H_{F (n/q)^{∗}}^{1}(Q, T^{∗}), and d′ ∈ H_{F (m/r)^{∗}}^{1}(Q, T^{∗}). 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 15.11, it follows that

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

and nℓ/q to nℓ,

• λ(mℓ/r, T) = λ(mℓ, T) = 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(nm)) + 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. □

Proof of Theorem 15.6. Suppose n and m are vertices of X^{0}. By Lemma 12.6(viii), we must have ν(n), ν(m) ≥ dim_{𝕜} H_{F}^{1}(Q, T) − 1. Applying Proposition 15.11 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, T) − 1. By Proposition 15.12 there is a path from n′ to m′, and the proof is complete. □

16. Kolyvagin systems over principal artinian rings

Return now to the general case of a principal local artinian ring R of length k ≥ 1. By Theorem 13.7, if χ(T) ≤ 0 then KS(T) = 0. So we assume for this section that χ(T) > 0.

We also assume for the rest of this section that there is a local principal artinian ring R̃, and a Selmer triple (T̃, F̃, P) lifting (T, F, P) to R̃, satisfying both of the following conditions. Let 𝔪̃ denote the maximal ideal of R̃ and k̃ = length(R̃).

• Either (H.4a) holds, i.e., T̄ ≅ ¯T̸^{∗} as F_{p}[[G_{Q}]]-modules, or k̃ ≥ 2k − 1.

• (T̃, F̃, P) satisfies (H.0) through (H.6) and P ⊂ Pk̃.

• Let M and M̃ be the core Selmer modules attached to T and T̃, respectively, and let r(T) = dim_{𝕜}(𝔪^{k−1}M). Then 𝔪̃^{(k̃−k)}M̃ is free of rank r(T) over R.

Remark 16.1. If (H.4a) holds and the core Selmer module of T is free over R, then both conditions above hold with R̃ = R, T̃ = T. Definition 16.2. Define a subgraph X̄ = X̄(T, F, P) of the canonical graph X (P) of Definition 6.2, by taking the same set N of vertices, but we join n and nℓ by an edge in X̄ if and only if the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective.

We define a subsheaf \overline{H′ ; T̃} of the restriction to X̄ of the Selmer sheaf H_{T} as follows. As in the proof of Theorem 14.2, for n ∈ N we let \overline{HT̃}(n) be the image of HT̃(n) = H_{F (n)}^{1}(Q, T̃) ⊗ G_{n} in H_{T} (n) = H_{F (n)}^{1}(Q, T) ⊗ G_{n}, and we set

\overline{H′ ; T̃}(n) = 𝔪^{λ(n,T^{∗})}\overline{HT̃}(n).

If e is an edge joining n and nℓ, we put

\overline{H′ ; T̃}(e) = 𝔪^{λ(n,T^{∗})}H_{s}^{1}(Q_{ℓ}, T) ⊗ G_{nℓ},

the map \overline{H′ ; T̃}(n) → \overline{H′ ; T̃}(e) is localization at ℓ followed by ϕ_{ℓ}^{fs}, and the map \overline{H′ ; T̃}(nℓ) → \overline{H′ ; T̃}(e) is localization at ℓ. (The image of the second localization map lies in \overline{H′ ; T̃}(e) thanks to Lemma 12.6(iv), so this is well-defined.) Theorem 16.3. The image of KS(T̃) in KS(T) is contained in the module of global sections Γ(\overline{H′ ; T̃}) ⊂ Γ(H_{T}).

Proof. This is just a restatement of Theorem 14.2, combined with Remark 14.5. □

Theorem 16.4. (i) If n ∈ N then \overline{H′ ; T̃}(n) is a free R/𝔪^{k−λ(n,T^{∗})}-module of

rank r(T). (ii) Suppose r(T) = 1. Then the sheaf \overline{H′ ; T̃} is locally cyclic, and every n ∈ N

with λ(n, T^{∗}) = 0 is a hub.

This theorem will be proved below. Corollary 16.5. (i) If r(T) = 0 then the image of KS(T̃) in KS(T) is zero.

(ii) If r(T) = 1 then the image of KS(T̃) in KS(T) is a cyclic R-module. (iii) Suppose r(T) = 1 and κ is in the image of KS(T̃) → KS(T). If there is some n ∈ N such that \overline{H′ ; T̃}(n) ≠ 0 and κ_{n} generates \overline{H′ ; T̃}(n), then κ_{n} generates \overline{H′ ; T̃}(n) for every n ∈ N.

Proof. The corollary follows from Theorem 16.3, Theorem 16.4 and Proposition 8.4. □

Corollary 16.6. Suppose that r(T) = 1 and κ is in the image of KS(T̃) → KS(T). Then the following are equivalent.

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

(ii) κ_{n} generates \overline{H′ ; T̃}(n) for every n ∈ N.

(iii) The global section of Γ(\overline{H′ ; T̃}) 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 16.5(iii), and (ii) ⟺ (iii) is Definition 8.5.

Suppose n ∈ N. By Lemma 9.4, 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 \overline{H′ ; T̃}(n), so (iv) ⇒ (i). Conversely, suppose (iii) holds and n is a core vertex of X (i.e., λ(n, T^{∗}) = 0; these exist by Lemma 12.7). Then by Theorem 16.4(i), 𝔪^{k−1}κ_{n} ≠ 0, so the image of κ_{n} in H^{1}(Q, T̄) is nonzero. Thus (ii) ⇒ (iv). □

Definition 16.7. We say that κ is primitive if it satisfies the equivalent conditions of Corollary 16.6.

The following theorem is a sharpening of the more general Corollary 14.4. Theorem 16.8. Suppose r(T) = 1 and κ is in the image of KS(T̃) → KS(T). If the image of κ in KS(T̄) is nonzero, and κ_{1} ≠ 0, 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 16.6, κ_{1} generates \overline{H′ ; T̃}(1) ⊂ 𝔪^{length(H_{F^{∗}}^{1} (Q,T^{∗}))}H_{F}^{1}(Q, T), and by Theorem 16.4(i), length(\overline{H′ ; T̃}(1) = k − length(H_{F^{∗}}^{1}(Q, T^{∗}). □

We can formulate a more precise version of Theorem 16.8, which determines the R-module structure of H_{F^{∗}}^{1}(Q, T^{∗}) in terms of a primitive Kolyvagin system. Definition 16.9. If κ ∈ KS(T) and i ≥ 0, define

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

Theorem 16.10. Suppose r(T) = 1, κ is in the image of KS(T̃) → KS(T), the image of κ in KS(T̄) is nonzero, and κ_{1} ≠ 0. Then

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

and

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

Proof. Write H_{F^{∗}}^{1}(Q, T^{∗}) ≅ ⊕_{i}R/𝔪^{d_{i}} with nonnegative integers d_{1} ≥ d_{2} ≥ · · ·. We need to show that ∂^{(i)} = ∑_{j>i} d_{j}. By Corollary 16.6 and Theorem 16.4(i), we have

∂^{(i)}(κ) = min{λ(n, T^{∗}) : n ∈ N , ν(n) = i}.

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

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

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

(converse: to appear) [Sketch: Choose ℓ so H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective and the image of H_{F (n)^{∗}}^{1}(Q, T^{∗}) → H_{f}^{1}(Q_{ℓ}, T^{∗}) is as big as the largest cyclic summand R/𝔪^{d} of H_{F (n)^{∗}}^{1}(Q, T^{∗}). Then H_{F (nℓ)^{∗}}^{1}(Q, T^{∗}) = ker(H_{F (n)^{∗}}^{1}(Q, T^{∗}) → H_{f}^{1}(Q_{ℓ}, T^{∗})), so H_{F (n)^{∗}}^{1}(Q, T^{∗}) ≅ H_{F (nℓ)^{∗}}^{1}(Q, T^{∗}) ⊕ R/𝔪^{d}, and arguing by induction we can complete the proof.] □

The rest of this section is devoted to the proof of Theorem 16.4.

For n ∈ N, let \overline{H(n)} be the image of H_{F (n)}^{1}(Q, T̃) in H_{F (n)}^{1}(Q, T), and let

\overline{H′(n)} = 𝔪^{λ(n,T^{∗})}\overline{H(n)}. Thus \overline{H′ ; T̃}(n) = \overline{H′(n)} ⊗ G_{n}.

Lemma 16.11. If n ∈ N then \overline{H(n)} is a free R/𝔪^{k−λ(n,T^{∗})}-module of rank r(T).

Proof. From Lemma 9.4 and Proposition 13.1 we see that

\overline{H(n)} ≅ H_{F (n)}^{1}(Q, T̃)/(H_{F (n)}^{1}(Q, T̃)[𝔪̃^{k̃−k+λ(n,T^{∗})}]) ≅ 𝔪̃^{k̃−k+λ(n,T^{∗})}M̃,

so the lemma follows from our hypothesis on M̃. □

Proposition 16.12. Suppose nℓ ∈ N and the localization map H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective. Then localization gives surjective maps

\overline{H′(n)} ↠ 𝔪^{λ(n,T^{∗})}H_{f}^{1}(Q_{ℓ}, T), \overline{H′(nℓ)} ↠ 𝔪^{λ(n,T^{∗})}H_{s}^{1}(Q_{ℓ}, T).

Proof. We may assume that λ(n, T^{∗}) ≤ k−1, or else there is nothing to prove. Then by Lemma 9.3, the natural map H_{F (n)}^{1}(Q, T^{∗}) → H_{F (n)}^{1}(Q, T̃^{∗}) is an isomorphism, so λ(n, T̃^{∗})) = λ(n, T^{∗}).

Fix a generator π of 𝔪̃. Then the image of \overline{H′(n)} in H_{F (n)}^{1}(Q, T̃) under the map induced by π^{k̃−k} : T ↪ T̃ is 𝔪^{k̃−k+λ(n,T^{∗})}H_{F (n)}^{1}(Q, T̃).

By Proposition 13.1 and the definition of r(T), dim_{𝕜}(𝔪^{k−1}H_{F (n)}^{1}(Q, T)) = r(T).

On the other hand, by Lemma 16.11 we also have dim_{𝕜}(𝔪^{k−1−λ(n,T^{∗})}\overline{H′(n)}) = r(T). Hence

𝔪^{k−1}H_{F (n)}^{1}(Q, T) = 𝔪^{k−1−λ(n,T^{∗})}\overline{H′(n)}.

Thus, since H_{f}^{1}(Q_{ℓ}, T) and H_{f}^{1}(Q_{ℓ}, T̃) are free of rank one over R and R̃, respectively, and the map H_{f}^{1}(Q_{ℓ}, T) → H_{f}^{1}(Q_{ℓ}, T̃) induced by π^{k̃−k} is injective, we have

H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is surjective
⟺ 𝔪^{k−1}H_{F (n)}^{1}(Q, T) → H_{f}^{1}(Q_{ℓ}, T) is nonzero

⟺ 𝔪^{k−1−λ(n,T^{∗})}\overline{H′(n)} → H_{f}^{1}(Q_{ℓ}, T) is nonzero

⟺ 𝔪^{k̃−1}H_{F (n)}^{1}(Q, T̃) → H_{f}^{1}(Q_{ℓ}, T̃) is nonzero
⟺ H_{F (n)}^{1}(Q, T̃) → H_{f}^{1}(Q_{ℓ}, T̃) is surjective.

Now the commutative diagram

𝔪^{k̃−k+λ(n,T^{∗})}H_{F (n)}^{1}(Q, T̃) ⊂ H_{F (n)}^{1}(Q, T̃) ⟶ H_{f}^{1}(Q, T̃) ⟶^{∼} R̃
≅↑  ↑  ↑  ↑_{π^{k̃−k}}
\overline{H′(n)}  ⊂ H_{F (n)}^{1}(Q, T) ⟶ H_{f}^{1}(Q_{ℓ}, T) ⟶^{∼} R

shows that the image of \overline{H′(n)} under localization is 𝔪^{λ(n,T^{∗})}H_{f}^{1}(Q_{ℓ}, T).

The proof of surjectivity of the second map is similar. Lemma 12.6(iv) shows that λ(nℓ, T̃^{∗}) ≤ λ(n, T̃^{∗})) and

H_{F (nℓ)}^{1}(Q, T̃) ↠ 𝔪̃^{λ(n,T^{∗})−λ(nℓ,T^{∗})}H_{s}^{1}(Q_{ℓ}, T̃)

is surjective (where we use that, just as for n above, λ(nℓ, T̃^{∗}) = λ(nℓ, T^{∗})). As above the image of \overline{H′(nℓ)} in H_{F (nℓ)}^{1}(Q, T̃) is 𝔪^{k̃−k+λ(nℓ,T^{∗})}H_{F (nℓ)}^{1}(Q, T̃), so a diagram like the one above completes the proof. □

Proof of Theorem 16.4. The first assertion is immediate from Lemma 16.11. When r(T) = 1, Lemma 16.11 shows that the stalks of \overline{H′ ; T̃} are cyclic, and Proposition 16.12 shows that the vertex-to-edge maps are surjective, so \overline{H′ ; T̃} is locally cyclic.

It remains to show that if n ∈ N and λ(n, T^{∗}) = 0, then n is a hub of \overline{H′ ; T̃}.

(to appear)

□

Part V. Kolyvagin systems over integral domains

17. Kolyvagin systems over a discrete valuation ring

For this section we assume that R is a discrete valuation ring. We assume that T satisfies 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 KS(T/𝔪^{k}T) for every k, and use this information to study KS(T).

Again we write T̄ = T/𝔪T.

Definition 17.1. Writing Frac(R) for the field of fractions of R, we define the rank of an R-module 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}).

We define the cohomological margin in this setting (compare Definition 12.4) by

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

Lemma 17.2. For k ∈ Z^{+} consider the natural restriction map

ψ_{k} : KS(T, P) → KS(T/𝔪^{k}T, P ∩ 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}, I_{n})T). In particular I_{n} ⊂ 𝔪^{k}, so n ∈ N_{k} and ψ_{k}(κ) ≠ 0. □

Corollary 17.3. If χ(T̄) ≤ 0 then KS(T) = 0.

Proof. By Theorem 13.7 (and Corollary 12.9) KS(T/𝔪^{k}T, P ∩ P_{k}) = 0, so this is immediate from Lemma 17.2. □

Proposition 17.4. (i) |χ(T/𝔪^{k}T) − kχ(T)| is bounded independent of k. (ii) If χ(T̄) ≥ 0 then 0 ≤ χ(T/𝔪^{k}T) ≤ kχ(T̄) and 0 ≤ χ(T) ≤ χ(T̄). (iii) If χ(T̄) ≤ 0 then kχ(T̄) ≤ χ(T/𝔪^{k}T) ≤ 0 and χ(T̄) ≤ χ(T) ≤ 0. (iv) If χ(T̄) = χ(T) then χ(T/𝔪^{k}T) = kχ(T).

(v) If χ(T) ≥ 0 then there is a torsion R-module A with corank(A) = χ(T) and dim_{𝕜} A[𝔪] = χ(T̄) such that for every k ∈ Z^{+} and every n ∈ N ∩ N_{k} there is a noncanonical isomorphism

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

Proof. (To appear) Sketch: Let d = rank_{R}H_{F}^{1}(Q, T) and d^{∗} = corank_{R}H_{F^{∗}}^{1}(Q, T^{∗}). Then length(H_{F}^{1}(Q, T/𝔪^{k}T))−kd and length(H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{k}]))−kd^{∗} are bounded. This proves (i).

Suppose χ(T̄) > 0. Choose n so H_{F (n)^{∗}}^{1}(Q, T^{∗}) = H_{F (n)^{∗}}^{1}(Q, T̄^{∗}) = 0. Then

dim_{𝕜}(H_{F (n)}^{1}(Q, T/𝔪^{k}T)[𝔪]) = dim_{𝕜} H_{F (n)}^{1}(Q, T̄) = χ(T̄),

so length(H_{F (n)}^{1}(Q, T/𝔪^{k}T)) ≤ kχ(T̄). Also

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

so this proves (ii). □

By Corollary 17.3 and Proposition 17.4(ii), if χ(T) < 0 then KS(T) = 0. So we will assume from now on that χ(T) ≥ 0. Definition 17.5. Let A(T) be the R-module of Proposition 17.4(v), and let c(T) be the exponent of A(T)/A(T)_{div}, where A(T)_{div} is the maximal divisible R-submodule of A(T). Corollary 17.6. Suppose that j ∈ Z^{+}, k ≥ 2j − 1 and k ≥ j + c(T).

(i) If χ(T) = 0 then the image of KS(T) → KS(T/𝔪^{j}T, P ∩ P_{k}) is zero. (ii) If χ(T) = 1 then the image of KS(T) → KS(T/𝔪^{j}T, P ∩ P_{k}) is a cyclic

R-module.

Proof. This is Corollary 16.5 applied to the image of KS(T) in KS(T/𝔪^{j}T, P ∩ P_{k}). □

Remark 17.7. We would like to use Corollary 17.6 to study KS(T). Unfortunately there is a complication, since in that corollary we need to restrict from P to P ∩ P_{k} for appropriate k. Proposition 17.9 below shows that when χ(T̄) = 1, this restriction map is harmless.

Fix a generator π of the maximal ideal 𝔪 of R. If i, j ∈ Z^{+} write [π^{j}] : H^{1}(Q, T/𝔪^{i}T) → H^{1}(Q, T/𝔪^{i+j}T) for the map induced by π^{j} : T/𝔪^{i} ↪ T/𝔪^{i+j}. Proposition 17.8. If i, j ∈ Z^{+} and P ⊂ P_{i+j} then there is an exact sequence

0 ⟶ KS(T/𝔪^{i}T) ⟶^{[π^{j}]} KS(T/𝔪^{i+j}T) ⟶ KS(T/𝔪^{j}T).

Proof. The maps in question are all well-defined. We have (T/𝔪^{j}T)^{G_{Q}} = 0 by Lemma 9.2, so we get a cohomology exact sequence

0 ⟶ H^{1}(Q, T/𝔪^{i}T) ⟶^{[π^{j}]} H^{1}(Q, T/𝔪^{i+j}T) ⟶ H^{1}(Q, T/𝔪^{j}T). (11)

Since P ⊂ P_{i+j} we have I_{n} ⊂ 𝔪^{i+j} for every n ∈ N, and so Lemmas 11.4 and 9.4 show that (11) remains exact when H^{1} is replaced throughout by H_{F (n)}^{1}. The proposition follows easily. □

Proposition 17.9. Suppose that χ(T̄) = 1.

If j ∈ Z^{+} and I ⊂ R is an ideal, then the restriction map KS(T/IT, P) → KS(T/IT, P ∩ P_{j}) is injective.

Proof. Suppose κ ∈ KS(T/IT, P) is nonzero; we want to show that its restriction in KS(T/IT, P ∩ P_{j}) is nonzero. Let k be maximal such that the image of κ in KS(T/𝔪^{k−1}T, P) is zero; clearly k is finite and I ⊂ 𝔪^{k}.

Thus there is an n ∈ N such that κ_{n} ≠ 0 in H^{1}(Q, T/(𝔪^{k}, I_{n})T), but κ_{n} = 0 in H^{1}(Q, T/(𝔪^{k−1}, I_{n})T). In particular we have I_{n} ⊂ 𝔪^{k}, so the restriction of κ is nonzero in KS(T/𝔪^{k}T, P ∩ P_{k}).

The exact sequence of Proposition 17.8 (applied with i = 1, j = k − 1) shows that κ is the image of a (necessarily nonzero) κ̄ ∈ KS(T/𝔪T, P ∩P_{k}). By Theorem 15.7(iv) and Corollary 12.7, the restriction of κ̄ in KS(T/𝔪T, P ∩ P_{j}) is nonzero, and hence the restriction of κ in KS(T/IT, P ∩ P_{j}) is nonzero. □

Theorem 17.10. Suppose that χ(T̄) = 1.

(i) If χ(T) = 0 then KS(T) = 0.

(ii) If χ(T) = 1 then KS(T) is free of rank zero or one over R.

Proof. Let ψ_{k} be the natural map KS(T, P) → KS(T/𝔪^{k}T, P ∩ P_{k}).

Suppose first that χ(T) = 0. By Corollary 17.6, for every k ∈ Z^{+} we can find a k′ ≥ k such that the image of KS(T) in KS(T/𝔪^{k}T, P ∩ P_{k′}) is zero. Then by Proposition 17.9 (this is why we need χ(T̄) = 1) we have ψ_{k} = 0 for every j, and so Lemma 17.2 shows that KS(T) = 0.

Now suppose that χ(T) = 1. Then as above Corollary 17.6 and Proposition 17.9 show that for every k ∈ Z^{+} the image of ψ_{k}(KS(T)) is cyclic. By Lemma 17.2 we get an injection

KS(T) ↪ \underleftarrow{lim}_{k}(KS(T)/ ker(ψ_{k})).

Since each group on the right is a cyclic R-module, so is KS(T). The fact that KS(T) has no R-torsion follows from Proposition 17.9 and the commutative diagram

KS(T, P ∩ P_{i+j}) ⟶^{π^{j}}  KS(T, P ∩ P_{i+j})
↓  ↓
0 ⟶ KS(T/𝔪^{i}T, P ∩ P_{i+j}) ⟶^{[π^{j}]} KS(T/𝔪^{i+j}T, P ∩ P_{i+j})

in which the bottom row is exact by Proposition 17.8. □

Definition 17.11. Suppose χ(T) = χ(T̄) = 1 and KS(T) ≠ 0. We define the Kolyvagin rank of T to be

rk_{KS}(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) define (compare Definition 16.9)

∂^{(i)}(κ) = max{j : κ_{n} ∈ 𝔪^{j}H_{F (n)}^{1}(Q, T/I_{n}T) for every n ∈ N with ν(n) = i}.

Then ∂^{(0)}(κ) = max{j : κ_{1} ∈ 𝔪^{j}H_{F}^{1}(Q, T)}, and if i < rk_{KS}(T) then ∂^{(i)}(κ) = ∞. Theorem 17.12. Suppose χ(T) = χ(T̄) = 1 and KS(T) ≠ 0.

(i) corank_{R}(H_{F^{∗}}^{1}(Q, T^{∗})) = rk_{KS}(T).

(ii) If κ ∈ KS(T) then length(H_{F^{∗}}^{1}(Q, T^{∗})/H_{F^{∗}}^{1}(Q, T^{∗})_{div}) ≤ ∂^{(rk_{KS}(T))}(κ). (iii) If κ ∈ KS(T) and the image of κ in KS(T̄) is nonzero, then equality holds

in (ii), and more precisely

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

Proof. (to appear) □

Example 17.13. (To appear: cyclotomic unit example) Example 17.14. (To appear: Kato example: if L(E, 1) ≠ 0 and …, then p-adic BSD ⟺ Kato ES is primitive)

18. Kolyvagin systems over Λ

(To appear)

19. The module of Kolyvagin systems

In this section R is an integral domain (as well as, as always, complete, noetherian, and local). Again we assume that (H.1), (H.2), (H.3), (H.4), (H.5), and (H.6) of §9 are all satisfied. Conjecture 19.1. If rank_{R}H_{F}^{1}(Q, T) − corank_{R}H_{F^{∗}}^{1}(Q, T^{∗}) ≤ 1 then

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

Remark 19.2. (recall what we’ve proved towards this so far) Example 19.3. (example to show that the hypothesis on rank_{R} − corank_{R} is necessary) Example 19.4. R = Λ Example 19.5. elliptic curves Example 19.6. Gauss sums Example 19.7. Flach?

20. The Kolyvagin Selmer group

Suppose again for this section that T satisfies (H.1), (H.2), (H.3), (H.4), (H.5), and (H.6) of section §9. For (H.5) we fix a k ∈ Z^{+} such that P ⊃ P_{k}. Also fix for this section a Kolyvagin system κ for (T, F, P). Definition 20.1. We define the Kolyvagin Selmer groups H_{κ}^{1}(Q, T) and H_{κ}^{1}(Q, T^{∗}) attached to κ as follows.

Suppose j ≥ k. Write ι_{j} for the natural maps H_{F}^{1}(Q, T/𝔪^{h}T) → H_{F}^{1}(Q, T/𝔪^{j}T) with any j ≤ h ≤ ∞. We define the Kolyvagin submodule K_{j}(T/𝔪^{j}T) to be the submodule of H^{1}(Q, T/𝔪^{j}T) generated by

{ι_{j}(κ_{n}) ⊗ Hom(G_{n}, R/𝔪^{j}) : n ∈ N_{j}}.

(Recall that κ_{n} ∈ H^{1}(Q, T/I_{n}T) ⊗ G_{n}, and if n ∈ N_{j} then I_{n} ⊂ 𝔪^{j}, so ι_{j}(κ_{n}) ⊗ Hom(G_{n}, R/𝔪^{j}) ⊂ H^{1}(Q, T/𝔪^{j}T).)

We can use the local pairings to define a pairing

⟨ , ⟩_{P_{j}} : H^{1}(Q, T/𝔪^{j}T) × ∏_{ℓ∈P_{j}} H_{f}^{1}(Q_{ℓ}, T^{∗}[𝔪^{j}]) ⟶ Q_{p}/Z_{p}

by

⟨c, (d_{ℓ})⟩_{P_{j}} = ∑_{ℓ∈P_{j}} ⟨c_{ℓ}, d_{ℓ}⟩_{ℓ}.

Since c_{ℓ} ∈ H_{f}^{1}(Q_{ℓ}, T) for all but finitely many ℓ, Example 3.2 shows that all but finitely many terms in the sum are zero, so this is well-defined.

Define

H_{κ,j}^{1}(Q, T/𝔪^{j}T) = K_{j}(T/𝔪^{j}T) ∩ H_{F}^{1}(Q, T/𝔪^{j}T),

and let H_{κ,j}^{1}(Q, T^{∗}[𝔪^{j}]) ⊂ ∏_{ℓ∈P_{j}} H_{f}^{1}(Q_{ℓ}, T^{∗}[𝔪^{j}]) be the orthogonal complement of K_{j}(T/𝔪^{j}T) under ⟨ , ⟩_{P_{j}}. Equivalently, we have an exact sequence

0 ⟶ H_{κ,j}^{1}(Q, T/𝔪^{j}T) ⟶ K_{j}(T/𝔪^{j}T) ⟶ ⨁_{ℓ∈P_{j}}H_{s}^{1}(Q_{ℓ}, T/𝔪^{j}T)

⟶ Hom(H_{κ,j}^{1}(Q, T^{∗}[𝔪^{j}]), Q_{p}/Z_{p}) ⟶ 0

where the center map is the sum of the localization maps, and the next map is induced by the pairing ⟨ , ⟩_{P_{j}}.

Finally, define

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

where the maps H_{κ,j}^{1}(Q, T^{∗}[𝔪^{j}]) → H_{κ,j+1}^{1}(Q, T^{∗}[𝔪^{j+1}]) in the direct limit are induced by local maps H_{f}^{1}(Q_{ℓ}, T^{∗}[𝔪^{j}]) → H_{f}^{1}(Q_{ℓ}, T^{∗}[𝔪^{j+1}]) which are zero if ℓ ∈ P_{j} − P_{j+1}, and induced by T^{∗}[𝔪^{j}] ↪ T^{∗}[𝔪^{j+1}] if ℓ ∈ P_{j+1}.

Note that κ_{1} gives a canonical element of H_{κ}^{1}(Q, T).

Proposition 20.2. With notation as above, the natural maps

\underleftarrow{lim} H_{F}^{1}(Q, T/𝔪^{j}T) ≅ H_{F}^{1}(Q, T), H_{F^{∗}}^{1}(Q, T^{∗}[𝔪^{j}]) ⟶ ∏_{ℓ∈P_{j}} H_{f}^{1}(Q_{ℓ}, T^{∗}[𝔪^{j}])

induce injections

H_{κ}^{1}(Q, T) ↪ H_{F}^{1}(Q, T), H_{F^{∗}}^{1}(Q, T^{∗}) ↪ H_{κ}^{1}(Q, T^{∗}).

Proof. (To appear) □

Theorem 20.3. Suppose R = 𝕜 is a field, dim_{𝕜}(H_{F}^{1}(Q, T))−dim_{𝕜}(H_{F^{∗}}^{1}(Q, T^{∗})) = 1, and κ ≠ 0. Then the natural map H_{F^{∗}}^{1}(Q, T^{∗}) → H_{κ}^{1}(Q, T^{∗}) is an isomorphism.

Proof. We use the notation of §15. Fix a generator of G_{n} for each n so that for this proof we can view κ_{n} ∈ H_{F (n)}^{1}(Q, T).

By Theorem 15.7(iv) we can fix n ∈ N_{k} such that ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}) and λ(n) = 1. We will show that the projection map

H_{κ}^{1}(Q, T^{∗}) ⟶ ∏_{q|n} H_{f}^{1}(Q, T^{∗})  (12)

is injective, and hence

dim_{𝕜}(H_{κ}^{1}(Q, T^{∗})) ≤ ∑_{q|n} H_{f}^{1}(Q, T^{∗}) = ν(n) = dim_{𝕜}(H_{F^{∗}}^{1}(Q, T^{∗}))

the middle equality by Lemma 9.6(ii). Since the map H_{F^{∗}}^{1}(Q, T^{∗}) → H_{κ}^{1}(Q, T^{∗}) is injective by Proposition 20.2, it must be an isomorphism.

It remains to prove the injectivity of (12). Suppose d belongs to the kernel of (12). Via the natural inclusion H_{κ}^{1}(Q, T^{∗}) ⊂ ∏_{ℓ∈P_{k}} H_{f}^{1}(Q_{ℓ}, T^{∗}) we will write d = (d_{ℓ}) where d_{ℓ} ∈ H_{f}^{1}(Q_{ℓ}, T^{∗}) for ℓ ∈ P_{k} and d_{q} = 0 for every q dividing n. We need to show that d_{ℓ} = 0 for every ℓ not dividing n. Fix such an ℓ.

Since κ ≠ 0, Theorem 15.7(iii) shows that κ_{n} ≠ 0. Case 1: (κ_{n})_{ℓ} ≠ 0. We have

0 = ⟨κ_{nℓ}, d⟩_{P_{k}} = ⟨(κ_{nℓ})_{ℓ}, d_{ℓ}⟩_{ℓ} = ⟨ϕ_{ℓ}^{fs}(κ_{n}), d_{ℓ}⟩_{ℓ}.

Since κ_{n} ≠ 0 and ϕ_{ℓ}^{fs} is an isomorphism, we conclude that d_{ℓ} = 0. Case 2: (κ_{n})_{ℓ} = 0. Recall that F_{ℓ}(n) is the Selmer structure F (n) modified so that H_{F_{ℓ}(n)}^{1}(Q_{ℓ}, T) = 0. Since λ(n) = 1, κ_{n} generates H_{F (n)}^{1}(Q, T), and we conclude that H_{F (n)}^{1}(Q, T) = H_{F_{ℓ}(n)}^{1}(Q, T). Therefore global duality (Theorem 5.4 applied with G = F_{ℓ}(n) and H = F (n)) gives an exact sequence

0 ⟶ H_{F (n)^{∗}}^{1}(Q, T^{∗}) ⟶ H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}) ⟶ H_{s}^{1}(Q_{ℓ}, T^{∗}) ⟶ 0

Fix a nonzero element δ ∈ H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}).

Applying Proposition 10.1 we can find a prime q ∈ P_{k} prime to nℓ such that (a) (κ_{n})_{q} ≠ 0,

(b) δ_{q} ≠ 0. Condition (b) tells us that δ ∉ H_{F_{ℓ}^{q}(n)^{∗}}^{1}(Q, T^{∗}), so H_{F_{ℓ}^{q}(n)^{∗}}^{1}(Q, T^{∗}) is a proper subspace of H_{F_{ℓ}(n)^{∗}}^{1}(Q, T^{∗}). Therefore global duality (Theorem 5.4 applied with G = F_{ℓ}^{q}(n) and H = F_{ℓ}(n)) tells us that H_{F_{ℓ}(n)}^{1}(Q, T) = H_{F_{ℓ}^{q}(n)}^{1}(Q, T).

Consider κ_{nq}. It follows from condition (a) on q and the definition of a Kolyvagin system that (κ_{nq})_{q,s} ≠ 0. Hence κ_{nq} ∉ H_{F_{ℓ}(n)}^{1}(Q, T) = H_{F_{ℓ}^{q}(n)}^{1}(Q, T). In other words (κ_{nq})_{ℓ} ≠ 0.

It follows from condition (a) on q and Case 1 that d_{q} = 0. Thus as in Case 1 we have

0 = ⟨κ_{nqℓ}, d⟩_{P_{k}} = ⟨(κ_{nqℓ})_{ℓ}, d_{ℓ}⟩_{ℓ} = ⟨ϕ_{ℓ}^{fs}(κ_{nq}), d_{ℓ}⟩_{ℓ}.

Exactly as in Case 1, it follows that d_{ℓ} = 0. □

Theorem 20.4. Suppose R is a discrete valuation ring. Under reasonable hypotheses (to be filled in when the proof is written out), the natural map H_{F^{∗}}^{1}(Q, T^{∗}) → H_{κ}^{1}(Q, T^{∗}) is an isomorphism.

Proof. (To appear. Will use Theorem 20.3 and a bit more.) □

Theorem 20.5. Suppose R = 𝕜 is a field, dim_{𝕜}(H_{F}^{1}(Q, T))−dim_{𝕜}(H_{F^{∗}}^{1}(Q, T^{∗})) = 1, hypothesis (H.4b) is satisfied (along with (H.1), (H.2), and (H.3)), and κ ≠ 0. Then the natural map H_{κ}^{1}(Q, T) → H_{F}^{1}(Q, T) is an isomorphism.

Proof. (Sketch; full proof to appear)

We will show that H_{κ,j}^{1}(Q, T) → H_{F}^{1}(Q, T) is surjective for every j ≥ k, and then the theorem follows immediately.

Fix j ≥ k and suppose c ∈ H_{F}^{1}(Q, T). Using a variant of Proposition 10.1 (which at the moment I see how to prove only under (H.4b)), we find an n ∈ N_{j} satisfying ν(n) = dim_{𝕜} H_{F^{∗}}^{1}(Q, T^{∗}) and λ(n) = 1, and such that c ∈ H_{F (n)}^{1}(Q, T). Then Theorem 15.7(iii) shows that κ_{n} generates the one-dimensional space c ⊗ G_{n}, and hence c ∈ H_{κ,j}^{1}(Q, T). □

References

[BK] Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift (Vol. I), P. Cartier, et al., eds., Prog. in Math. 86, Boston: Birkhäuser (1990) 333–400.

[Ka] Kato, K.: To appear.

[Ko] Kolyvagin, V.: Euler systems, in: The Grothendieck Festschrift (Vol. II), P. Cartier et al., eds., Prog. in Math 87, Boston: Birkhäuser (1990) 435–483.

[PR] Perrin-Riou, B.: Systèmes d’Euler p-adiques et théorie d’Iwasawa, Ann. Inst. Fourier (Grenoble) 48 (1998) 1231–1307.

[Ru] Rubin, K.: Euler Systems. Annals of Math. Studies 147, Princeton: Princeton University Press (2000).

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