Controlling Selmer groups in the higher core rank case
aa r X i v : . [ m a t h . N T ] D ec CONTROLLING SELMER GROUPS IN THEHIGHER CORE RANK CASE
BARRY MAZUR AND KARL RUBIN
Abstract.
We define Kolyvagin systems and Stark systems attached to p -adic representations in the case of arbitrary “core rank” (the core rank is ameasure of the generic Selmer rank in a family of Selmer groups). Previouswork dealt only with the case of core rank one, where the Kolyvagin and Starksystems are collections of cohomology classes. For general core rank, they arecollections of elements of exterior powers of cohomology groups. We showunder mild hypotheses that for general core rank these systems still controlthe size and structure of Selmer groups, and that the module of all Kolyvagin(or Stark) systems is free of rank one. Contents
Introduction 2
Part 1. Cohomology groups and Selmer structures
41. Local cohomology groups 42. Global cohomology groups and Selmer structures 53. Selmer structures and the core rank 74. Running hypotheses 85. Examples 9
Part 2. Stark systems and the structure of Selmer groups
Part 3. Kolyvagin systems
Date : March 20, 2018.2010
Mathematics Subject Classification.
Primary 11G40, 11F80; Secondary 11R23, 11R34.This material is based upon work supported by the National Science Foundation under grantsDMS-1302409 and DMS-1065904.
Introduction
Let K be a number field and G K := Gal( ¯ K/K ) its Galois group. Let R be eithera principal artinian local ring, or a discrete valuation ring, and T an R [ G K ]-modulethat is free over R of finite rank. Let T ∗ := Hom( T, µ ∞ ) be its Cartier dual.A cohomology class c in H ( G K , T ) provides (after localization and cup-product)a linear functional L c,v on H ( G K v , T ∗ ) for any place v of K . Thanks to the dualitytheorems of class field theory, these L c,v , when summed over all places v of K , give alinear functional L c that annihilates the adelic image of H ( G K , T ∗ ). By imposinglocal conditions on the class c , we get a linear functional that annihilates a Selmergroup in H ( G K , T ∗ ). Following this thread, a systematic construction of classes c can be of used to control the size of Selmer groups. Even better, a sufficiently fullcollection (a system ) of classes c can sometimes be used to completely determinethe structure of the relevant Selmer groups.We have just described a very vague outline of the strategy of controlling Selmergroups of Galois representations T ∗ , by systems of cohomology classes for T . Inpractice there are variants of this strategy. First, we will control the local conditionsthat we impose on our cohomology classes. That is, we will require our classes tolie in certain Selmer groups for T . But more importantly, in general one encounterssituations where sufficiently many of the relevant Selmer groups for T are free over R of some (fixed) rank r ≥
1. We call r the core rank of T ; see Definition 3.4below. In the natural cases that we consider, all relevant Selmer groups contain afree module of rank equal to the core rank r , and r is maximal with respect to thisproperty.If R is a discrete valuation ring and our initial local conditions are what we call unramified (see Definition 5.1 and Theorem 5.4), then under mild hypotheses thecore rank r of T is given by the simple formula r = X v |∞ corank H ( G K v , T ∗ ) . So, for example, if T is the p -adic Tate module of an abelian variety of dimension d over K , then the core rank is d [ K : Q ].To deal with the case where r is greater than 1 we will ask for elements in the r -th exterior powers (over R ) of those Selmer groups, so that for every r we will beseeking systems of classes in R -modules that are often free of rank one over R .One of the main aims of this article is to extend the more established theoryof core rank r = 1 (see for example [MR1]) to the case of higher core rank. Wedeal with two types of systems of cohomology classes: Stark systems (collections ofclasses generalizing the units predicted by Stark-type conjectures) and
Kolyvaginsystems (generalizing Kolyvagin’s original formulation). Our Stark systems aresimilar to the “unit systems” that occur in the recent work of Sano [S]. There is athird type,
Euler systems (see for example [PR2] or [Ru2]), which we do not dealwith in this paper. When r = 1, Euler systems provide the crucial link ([MR1,Theorem 3.2.4]) between Kolyvagin or Stark systems and L -values. We expect thatwhen r > ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 3
The Euler systems that have been already constructed in the literature, or thatare conjectured to exist, are motivic: they come from arithmetic objects such as cir-cular units or more generally the conjectural Stark units; or—in another context—Heegner points; or elements of K -theory. Euler systems are ‘vertically configured’in the sense that they provide classes in many abelian extensions of the base num-ber field, and the classes cohere via norm projection from one abelian extension toa smaller one when modified by the multiplication of appropriate ‘Euler factors’(hence the terminology ‘Euler system’).On the other hand, the Stark and Kolyvagin systems are ‘horizontally configured’in the sense that they consist only of cohomology classes over the base number field,but conform to a range of local conditions. The local conditions for Stark systemsare more elementary and—correspondingly—the Stark systems are somewhat easierto handle than Kolyvagin systems. In contrast, the local conditions for Kolyvaginsystems connect more directly with the changes of local conditions that arise fromtwisting the Galois representation T by characters.One of the main results of this paper (Theorem 12.4) is that—under suitablehypotheses, but for general core rank—there is an equivalence between Stark sys-tems and special Kolyvagin systems that we call stub Kolyvagin systems , and, upto a scalar unit, there is a unique ‘best’ Stark (equivalently: stub Kolyvagin) sys-tem (Theorems 6.10 and 7.4). We show, as mentioned in the title of this article,that the corresponding Selmer modules are controlled by (either of) these systems(Theorems 8.9 and 13.4), in the sense that there is a relatively simple descriptionof the elementary divisors (and hence the isomorphy type) of the Selmer group of T starting with any Stark or stub Kolyvagin system. When the core rank is one,every Kolyvagin system is a stub Kolyvagin system [MR1, Theorem 4.4.1].Although we have restricted our scalar rings R to be either principal artinianlocal rings or complete discrete valuation rings with finite residue field, it is naturalto wish to extend the format of our systems of cohomology classes to encompassGalois representations T that are free of finite rank over more general completelocal rings, so as to be able to deal effectively with deformational questions. Layout of the paper.
In Part 1 (sections 1–5) we recall basic facts that we willneed about local and global cohomology groups, and define our abstract Selmergroups and the core rank. In Part 2 (sections 6–8) we define Stark systems andinvestigate the relations between Stark systems and the structure of Selmer groups.Part 3 (sections 9–14) deals with Kolyvagin systems, and the relation betweenKolyvagin systems and Stark systems.The results of [MR1] were restricted to the case where the base field K is Q . Inmany cases the proofs for general K are the same, and in those cases we will feelfree to use results from [MR1] without further comment. Notation.
Fix a rational prime p . Throughout this paper, R will denote a com-plete, noetherian, local principal ideal domain with finite residue field of character-istic p . Let m denote the maximal ideal of R . The basic cases to keep in mind are R = Z /p n Z or R = Z p .If K is a field, ¯ K will denote a fixed separable closure of K and G K := Gal( ¯ K/K ).If A is an R -module and I is an ideal of R , we will write A [ I ] for the submodule of A killed by I . If A is a G K -module, we write K ( A ) for the fixed field in ¯ K of thekernel of the map G K → Aut( A ). BARRY MAZUR AND KARL RUBIN
If a group H acts on a set X , then the subset of elements of X fixed by H isdenoted X H .If n is a positive integer, µ n will denote the group of n -th roots of unity in ¯ K . Part Cohomology groups and Selmer structures Local cohomology groups
For this section K will be a local field (archimedean or nonarchimedean). If K is nonarchimedean let O be the ring of integers in K , F its residue field, K ur ⊂ ¯ K the maximal unramified subfield of ¯ K , and I the inertia group Gal( ¯ K/K ur ), so G F = G K / I = Gal( K ur /K ).Fix an R -module T endowed with a continuous G K -action. By H ∗ ( K, T ) := H ∗ ( G K , T ) we mean cohomology computed with respect to continuous cochains. Definition 1.1. A local condition on T (over K ) is a choice of an R -submodule of H ( K, T ). If we refer to the local condition by a symbol, say F , we will denote thecorresponding R -submodule H F ( K, T ) ⊂ H ( K, T ) . If I is an ideal of R , then a local condition on T induces local conditions on T /IT and T [ I ] by taking H F ( K, T /IT ) and H F ( K, T [ I ]) to be the image andinverse image, respectively, of H F ( K, T ) under the maps induced by T ։ T /IT, T [ I ] ֒ → T. One can similarly propagate the local condition F canonically to arbitrary subquo-tients of T , and if R → R ′ is a homomorphism of complete noetherian local PID’s,then F induces a local condition on the R ′ -module T ⊗ R R ′ . Definition 1.2.
Suppose K is nonarchimedean and T is unramified (i.e., I actstrivially on T ). Define the finite (or unramified ) local condition by H ( K, T ) := ker (cid:2) H ( K, T ) → H ( K ur , T ) (cid:3) = H ( K ur /K, T ) . More generally, if L is a Galois extension of K we define the L -transverse localcondition by H L -tr ( K, T ) := ker (cid:2) H ( K, T ) → H ( L, T ) (cid:3) = H ( L/K, T G L ) . Suppose for the rest of this section that the local field K is nonarchimedean, the R -module T is of finite type, and the action of G K on T is unramified.Fix a totally tamely ramified cyclic extension L of K such that [ L : K ] annihilates T . We will write simply H ( K, T ) for H L -tr ( K, T ) ⊂ H ( K, T ). Lemma 1.3. (i)
The composition H ( K, T ) ֒ → H ( K, T ) ։ H ( K, T ) /H ( K, T ) is an isomorphism, so there is a canonical splitting H ( K, T ) = H ( K, T ) ⊕ H ( K, T ) . There are canonical functorial isomorphisms (ii) H ( K, T ) ∼ = T / (Fr − T , (iii) H ( K, T ) ∼ = Hom( I , T Fr=1 ) , H ( K, T ) ⊗ Gal(
L/K ) ∼ = T Fr=1 .Proof.
Assertion (i) is [MR1, Lemma 1.2.4]. The rest is well known; see for example[MR1, Lemma 1.2.1]. (cid:3)
ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 5
Definition 1.4.
Suppose that T is free of finite rank as an R -module, and thatdet(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 − Q ( x ) = P ( x ) in R [ x ] . By the Cayley-Hamilton theorem, P (Fr − ) annihilates T , so Q (Fr − ) T ⊂ T Fr=1 .We define the finite-singular comparison map φ fs on T to be the composition, usingthe isomorphisms of Lemma 1.3(ii,iii), H ( K, T ) ∼ −→ T / (Fr − T Q (Fr − ) −−−−−→ T Fr=1 ∼ −→ H ( K, T ) ⊗ Gal(
L/K ) . Lemma 1.5.
Suppose that T is free of finite rank over R , and that T / (Fr − T isa free R -module of rank one. Then det(1 − Fr | T ) = 0 and the map φ fs : H ( K, T ) −→ H ( K, T ) ⊗ Gal(
L/K ) of Definition 1.4 is an isomorphism. In particular both H ( K, T ) and H ( K, T ) are free of rank one over R .Proof. This is [MR1, Lemma 1.2.3]. (cid:3)
Definition 1.6.
Define the dual of T to be the R [[ G K ]]-module T ∗ := Hom( T, µ p ∞ ) . We have the (perfect) local Tate cup product pairing h , i : H ( K, T ) × H ( K, T ∗ ) −→ H ( K, µ p ∞ ) ∼ −−→ Q p / Z p . A local condition F for T determines a local condition F ∗ for T ∗ , by taking H F ∗ ( K, T ∗ ) to be the orthogonal complement of H F ( K, T ) under the Tate pairing h , i . Proposition 1.7.
With notation as above, we have: (i) H ( K, T ) and H ( K, T ∗ ) are orthogonal complements under h , i . (ii) H ( K, T ) and H ( K, T ∗ ) are orthogonal complements under h , i .Proof. The first assertion is (for example) Theorem I.2.6 of [Mi]. Both assertionsare [MR1, Lemma 1.3.2]. (cid:3) Global cohomology groups and Selmer structures
For the rest of this paper, K will be a number field and T will be a finitelygenerated free R -module with a continuous action of G K , that is unramified outsidea finite set of primes. Global notation.
Let ¯ K ⊂ C be the algebraic closure of K in C , and for eachprime q of K fix an algebraic closure K q of K q containing ¯ K . This determines achoice of extension of q to ¯ K . Let D q := Gal( K q /K q ), which we identify with aclosed subgroup of G K := Gal( ¯ K/K ). In other words D q is a particular decompo-sition group at q in G K , and H ( D q , T ) = H ( K q , T ). Let I q ⊂ D q be the inertiagroup, and Fr q ∈ D q / I q the Frobenius element. If T is unramified at q , then D q / I q acts on T , and hence so does Fr q . If we choose a different decomposition group at q , then the action of Fr q changes by conjugation in G K . We will write loc q for thelocalization map H ( K, T ) → H ( K q , T ). BARRY MAZUR AND KARL RUBIN If q is a prime of K , let K ( q ) denote the p -part of the ray class field of K modulo q (i.e., the maximal p -power extension of K in the ray class field), and K ( q ) q thecompletion of K ( q ) at the chosen prime above q . If q is principal then K ( q ) q /K q is cyclic and totally tamely ramified.If q is principal, T is unramified at q , and [ K ( q ) q : K q ] T = 0, the transversesubmodule of H ( K q , T ) is the submodule H ( K q , T ) := H K ( q ) q -tr ( K q , T ) = ker (cid:2) H ( K q , T ) → H ( K ( q ) q , T ) (cid:3) of Definition 1.2. Definition 2.1. A Selmer structure F on T is a collection of the following data: • a finite set Σ( F ) of places of K , including all infinite places, all primesabove p , and all primes where T is ramified, • for every q ∈ Σ( F ) (including archimedean places), a local condition (inthe sense of Definition 1.1) on T over K q , i.e., a choice of R -submodule H F ( K q , T ) ⊂ H ( K q , T ) . If F is a Selmer structure, we define the Selmer module H F ( K, T ) ⊂ H ( K, T )to be the kernel of the sum of restriction maps H ( K Σ( F ) /K, T ) −→ L q ∈ Σ( F ) (cid:0) H ( K q , T ) /H F ( K q , T ) (cid:1) where K Σ( F ) denotes the maximal extension of K that is unramified outside Σ( F ).In other words, H F ( K, T ) consists of all classes which are unramified (or equiva-lently, finite) outside of Σ( F ) and which locally at q belong to H F ( K q , T ) for every q ∈ Σ( F ).For examples of Selmer structures see [MR1]. Note that if F is a Selmer structureon T and I is an ideal of R , then F induces canonically (see Definition 1.1) Selmerstructures on the R/I -modules
T /IT and T [ I ], that we will also denote by F . Definition 2.2.
Suppose now that T is free over R , q ∤ p ∞ is prime, and T isunramified at q . If q is not principal, let I q := R . If q is principal, let I q ⊂ R be thelargest power of m (i.e., m k with k ≥ K ( q ) q : K q ] R ⊂ I q and T / ((Fr q − T + I q T ) is free of rank one over R/I q .Let P denote a set of prime ideals of K , disjoint from Σ( F ). Typically P willbe a set of positive density. Define a filtration P ⊃ P ⊃ P ⊃ · · · by P k = { q ∈ P : I q ⊂ m k } for k ≥
1. Let N := N ( P ) denote the set of squarefree products of primes in P (with the convention that the trivial ideal 1 ∈ N ). Let I := 0 and if n ∈ N , n = 1,define I n := X q | n I q ⊂ R. Definition 2.3.
Suppose F is a Selmer structure, and a , b , n are pairwise relativelyprime ideals of K with n ∈ N and I n T = 0. Define a new Selmer structure F ba ( n )by • Σ( F ba ( n )) := Σ( F ) ∪ { q : q | abn } , ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 7 • H F ba ( n ) ( K q , T ) := H F ( K q , T ) if q ∈ Σ( F ) , q | a ,H ( K q , T ) if q | b ,H ( K q , T ) if q | n . In other words, F ba ( c ) consists of F together with the strict condition at primesdividing a , the unrestricted condition at primes dividing b , and the transversecondition at primes dividing n .If any of a , b , n are the trivial ideal, we may suppress them from the notation.For example, we will be especially interested in Selmer groups of the form H F n ( K, T ) : no restriction at q dividing n , same as F elsewhere, H F ( n ) ( K, T /I n T ) : transverse condition at q dividing n , same as F elsewhere.If m | n ∈ N , the definition leads to an exact sequence(2.4) 0 −→ H F m ( K, T ) −→ H F n ( K, T ) −→ L q | ( n / m ) H ( K q , T ) /H ( K q , T ) . Definition 2.5.
The dual of T is the R [[ G K ]]-module T ∗ := Hom( T, µ p ∞ ) . Forevery q we have the local Tate pairing h , i q : H ( K q , T ) × H ( K q , T ∗ ) −→ Q p / Z p as in § T determines a local condition on T ∗ (Definition1.6), a Selmer structure F for T determines a Selmer structure F ∗ for T ∗ . Namely,take Σ( F ∗ ) := Σ( F ), and for q ∈ Σ( F ) take H F ∗ ( K q , T ∗ ) to be the local conditioninduced by F , i.e., the orthogonal complement of H F ( K q , T ) under h , i q .3. Selmer structures and the core rank
Suppose for this section that the R is a principal local ring. We continue toassume for the rest of this paper that T is free of finite rank over R , in addition tobeing a G K -module. Definition 3.1.
A Selmer structure F on T is is called cartesian if for every q ∈ Σ( F ), the local condition F at q is “cartesian on the category of quotients of T ” as defined in [MR1, Definition 1.1.4]. Remark 3.2. If F is cartesian then for every k the induced Selmer structureon the R/ m k -module T / m k T is cartesian. If R is a field (i.e., m = 0) thenevery Selmer structure on T is cartesian. If R is a discrete valuation ring and H ( K q , T ) /H F ( K q , T ) is torsion-free for every q ∈ Σ( F ), then F is cartesian (see[MR1, Lemma 3.7.1(i)]). Proposition 3.3.
Suppose R is a principal artinian local ring of length k (i.e., m k = 0 and m k − = 0 ), F is a cartesian Selmer structure on T , and T G K =( T ∗ ) G K = 0 .If n ∈ N and I n = 0 then: (i) the exact sequence −→ T / m i T −→ T −→ T / m k − i T → BARRY MAZUR AND KARL RUBIN induces an isomorphism H F ( n ) ( K, T / m i T ) ∼ −→ H F ( n ) ( K, T )[ m i ] and anexact sequence −→ H F ( n ) ( K, T )[ m i ] −→ H F ( n ) ( K, T ) −→ H F ( n ) ( K, T / m k − i T ) . (ii) the inclusion T ∗ [ m i ] ֒ → T ∗ induces an isomorphism H F ( n ) ∗ ( K, T ∗ [ m i ]) ∼ −−→ H F ( n ) ∗ ( K, T ∗ )[ m i ] . (iii) there is a unique integer r , independent of n , such that there is a non-canonical isomorphism H F ( n ) ( K, T ) ∼ = H F ( n ) ∗ ( K, T ∗ ) ⊕ R r if r ≥ ,H F ( n ) ( K, T ) ⊕ R − r ∼ = H F ( n ) ∗ ( K, T ∗ ) if r ≤ . Proof.
These assertions are [MR1, Lemma 3.5.4], [MR1, Lemma 3.5.3], and [MR1,Theorem 4.1.5], respectively. (cid:3)
Definition 3.4.
Suppose F is a cartesian Selmer structure on T . If R is artinian,then the core rank of ( T, F ) is the integer r of Proposition 3.3(iii). If R is a discretevaluation ring, then the core rank of ( T, F ) is the core rank of ( T / m k T, F ) forevery k >
0, which by Proposition 3.3 is independent of k .We will denote the core rank by χ ( T, F ), or simply χ ( T ) when F is understood.For n ∈ N , let ν ( n ) denote the number of primes dividing n . Corollary 3.5.
Suppose R is artinian, χ ( T ) ≥ , n ∈ N , and I n = 0 . Let λ ( n ) := length( H F ( n ) ∗ ( K, T ∗ )) and µ ( n ) := length( H F ∗ ) n ( K, T ∗ )) . There arenoncanonical isomorphisms (i) H F ( n ) ( K, T ) ∼ = H F ( n ) ∗ ( K, T ∗ ) ⊕ R χ ( T ) , (ii) H F n ( K, T ) ∼ = H F ∗ ) n ( K, T ∗ ) ⊕ R χ ( T )+ ν ( n ) , (iii) m λ ( n ) ∧ χ ( T ) H F ( n ) ( K, T ) ∼ = m λ ( n ) , (iv) m µ ( n ) ∧ χ ( T )+ ν ( n ) H F n ( K, T ) ∼ = m µ ( n ) .Proof. The first isomorphism is just Proposition 3.3(iii). For (ii), observe that theSelmer structure F n is cartesian by [MR1, Lemma 3.7.1(i)], so applying Proposition3.3(iii) to ( T, F n ) we have H F n ( K, T ) ∼ = H F ∗ ) n ( K, T ∗ ) ⊕ R χ ( T, F n ) . To completethe proof of (ii) we need only show that χ ( T, F n ) = χ ( T ) + ν ( n ), and this followswithout difficulty from Poitou-Tate global duality (see for example [MR1, Theorem2.3.4]).Assertions (iii) and (iv) follow directly from (i) and (ii), respectively. (cid:3) Running hypotheses
Definition 4.1. By Selmer data we mean a tuple ( T, F , P , r ) where • T is a G K -module, free of finite rank over R , unramified outside finitelymany primes, • F is a Selmer structure on T , • P is a set of primes of K disjoint from Σ( F ), • r ≥ ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 9
Definition 4.2. If L is a finite Galois extension of K and τ ∈ G K , define P ( L, τ ) := { primes q / ∈ Σ( F ) : q is unramified in L/K and Fr q is conjugate to τ in Gal( L/K ) } . Fix Selmer data ( T, F , P , r ) as in Definition 4.1. Let ¯ T = T / m T , so ¯ T ∗ = T ∗ [ m ].If R is artinian, let M denote the smallest power of p such that M R = 0. If R is adiscrete valuation ring, let M := p ∞ . Let H denote the Hilbert class field of K , and H M := H ( µ M , ( O × K ) /M ). Let k denote the residue field R/ m . In order to obtainthe strongest results, we will usually make the following additional assumptions.(H.1) ¯ T G K = ( ¯ T ∗ ) G K = 0 and ¯ T is an absolutely irreducible k [[ G K ]]-module,(H.2) there is a τ ∈ Gal( ¯ K/ H M ) and a finite Galois extension L of K in H M such that T / ( τ − T is free of rank one over R and P ( L, τ ) ⊂ P ,(H.3) H ( H M ( T ) /K, T / m T ) = H ( H M ( T ) /K, T ∗ [ m ]) = 0,(H.4) either ¯ T = ¯ T ∗ as k [[ G K ]]-modules, or p > F is cartesian (Definition 3.1),(H.6) r = χ ( T ) >
0, where χ ( T ) is the core rank of T .(Only) when R is artinian, we will also sometimes assume(H.7) I q = 0 for every q ∈ P . Remark 4.3.
Note that if the above properties hold for ( T, F , P , r ), then theyalso hold if R is replaced by R/ m k and T by T / m k , for k ≥
0. If R is artinian and(H.1) through (H.6) hold, then Lemma 4.5 below shows that (H.1) through (H.7)hold if we replace L by H M and P by P ( H M , τ ). Remark 4.4.
Assumption (H.5) is needed to have a well-defined notion of corerank. Assumption (H.2) is needed to provide is with a large selection of primes q such that T / (Fr q − , m k ) is free of rank one, for large k .We deduce from assumption (H.3) that restriction from K to H M ( T ) is injectiveon the Selmer group; this allows us to view Selmer classes in Hom( G H M ( T ) , T ).Assumptions (H.1) and (H.4) then allow us to satisfy various Cebotarev conditionssimultaneously. Lemma 4.5.
Suppose R is artinian and τ is as in (H.2) . If q ∈ P ( H M , τ ) , then I q = 0 .Proof. Since Fr q fixes H , q is principal. By class field theory we have(4.6) Gal( K ( q ) q /K q ) ∼ = ( O K / q ) × / image( O × K ) . Since τ acts trivially on µ M , so does Fr q , so | ( O K / q ) × | is cyclic of order divisibleby M . Since τ acts trivially on ( O × K ) /M , so does Fr q , so the reduction of O × K iscontained in (( O K / q ) × ) M . By (4.6) we conclude that [ K ( q ) q : K q ] is divisible by M , so [ K ( q ) q : K q ] R = 0. We also have that T / (Fr q − T ∼ = T / ( τ − T is free ofrank one over R , so the lemma follows from the definition of I q . (cid:3) Examples
A canonical Selmer structure.Definition 5.1.
When R is a discrete valuation ring, we define a canonical unram-ified Selmer structure F ur on T by • Σ( F ur ) := { q : T is ramified at q } ∪ { p : p | p } ∪ { v : v | ∞} , • if q ∈ Σ( F ur ) and q ∤ p ∞ then H F ur ( K q , T ) := ker (cid:2) H ( K q , T ) → H ( K ur q , T ⊗ Q p ) (cid:3) , • if p | p then define the universal norm subgroup H ( K p , T ) u := ∩ K p ⊂ L ⊂ K ur p Cor
L/K p H ( L, T ) , intersection over all finite unramified extensions L of K p . Define H F ur ( K p , T ) := H ( K p , T ) u , sat , the saturation of H ( K p , T ) u in H ( K p , T ), i.e., H ( K p , T ) /H F ur ( K p , T )is R -torsion-free and H F ur ( K p , T ) /H ( K p , T ) u has finite length, • if v | ∞ then H F ur ( K v , T ) := H ( K v , T ) . In other words, H F ur ( K, T ) is the Selmer group of classes that (after multiplica-tion by some power of p ) are unramified away from p , and universal norms in theunramified Z p -extension above p .Note that the Selmer structure F ur satisfies (H.5) by Remark 3.2. Lemma 5.2. If p | p then corank R H F ∗ ur ( K p , T ∗ ) = corank R H ( K p , T ∗ ) .Proof. By the Lemma in [PR1, § Z p -extension of K p ), H F ∗ ur ( K p , T ∗ ) is the maximal divisible submodule of the image of the (injec-tive) inflation map H ( K ur p /K p , ( T ∗ ) G K ur p ) −→ H ( K p , T ∗ ) . We have H ( K ur p /K p , ( T ∗ ) G K ur p ) ∼ = ( T ∗ ) G K ur p / ( γ − T ∗ ) G K ur p where γ is a topological generator of Gal( K ur p /K p ). Thus we have an exact sequence0 −→ H ( K p , T ∗ ) −→ ( T ∗ ) G K ur p γ − −−−→ ( T ∗ ) G K ur p −→ H ( K ur p /K p , ( T ∗ ) G K ur p ) −→ (cid:3) Corollary 5.3. If p | p and H ( K p , T ∗ ) has finite length, then H F ur ( K p , T ) = H ( K p , T ) .Proof. By Lemma 5.2 H F ∗ ur ( K p , T ∗ ) has finite length, so H ( K p , T ) /H F ur ( K p , T )has finite length. But by definition H ( K p , T ) /H F ur ( K p , T ) is R -torsion-free, so H F ur ( K p , T ) = H ( K p , T ). (cid:3) Theorem 5.4.
Suppose R is a discrete valuation ring. Then χ ( T, F ur , P ) = X v |∞ corank R ( H ( K v , T ∗ )) . Proof.
For every k > T k = T / m k T . If f, g are functions of k ∈ Z + , we willwrite f ( k ) ∼ g ( k ) to mean that | f ( k ) − g ( k ) | is bounded independently of k . Bydefinition of core rank (see Definition 3.4 and Proposition 3.3(iii)), the theorem willfollow if we can show that(5.5) length( H F ur ( K, T k )) − length( H F ∗ ur ( K, T ∗ k )) ∼ k X v |∞ corank R ( H ( K v , T ∗ )) . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 11
By [MR1, Proposition 2.3.5] (which is essentially [Wi, Lemma 1.6]), for every k ∈ Z + (5.6) length( H F ur ( K, T k )) − length( H F ∗ ur ( K, T ∗ k ))= length( H ( K, T k )) − length( H ( K, T ∗ k ))+ X v ∈ Σ( F ur ) (length( H ( K v , T ∗ k )) − length( H F ∗ ur ( K v , T ∗ k ))) . By hypothesis (H.1), H ( K, T k ) = H ( K, T ∗ k ) = 0. If v | ∞ , thenlength( H ( K v , T ∗ k )) ∼ k corank R ( H ( K v , T ∗ )) , length( H F ∗ ur ( K v , T ∗ k )) ∼ . Suppose q ∈ Σ( F ), q ∤ p ∞ . Let I q denote an inertia group above q in G K . By[Ru2, Lemma 1.3.5], we havelength( H F ∗ ur ( K q , T ∗ k )) ∼ length(( T ∗ k ) I q / (Fr q − T ∗ k ) I q ) . On the other hand, the exact sequence0 → H ( K q , T ∗ k ) → ( T ∗ k ) I q Fr q − −−−−→ ( T ∗ k ) I q → ( T ∗ k ) I q / (Fr q − T ∗ k ) I q → H ( K q , T ∗ k )) = length(( T ∗ k ) I q / (Fr q − T ∗ k ) I q ) . Thus the term for v = q in (5.6) is bounded independent of k .Now suppose p | p . By Lemma 5.2, corank R H F ∗ ( K p , T ∗ ) = corank R H ( K p , T ∗ ).By definition H F ∗ ur ( K p , T ∗ k ) is the inverse image of H F ∗ ur ( K p , T ∗ ) under the naturalmap H ( K p , T ∗ k ) → H ( K p , T ∗ )[ m k ]. A simple exercise shows that the kernel andcokernel of this map have length bounded independent of k , so we see thatlength( H F ∗ ur ( K p , T ∗ k )) ∼ k corank R H F ∗ ur ( K p , T ∗ ) = k corank R H ( K p , T ∗ ) . Thus the term for v = p in (5.6) is bounded independent of k .Combining these calculations proves (5.5), and hence the theorem. (cid:3) Multiplicative groups.
Suppose K is a number field and ρ is a character of G K of finite order. For simplicity we will assume that p > ρ is nontrivial, and ρ takes values in Z × p . (Everything that follows holds more generally, only assumingthat ρ has order prime to p , but we would have to tensor everything with theextension Z p [ ρ ] where ρ takes its values.)Let T := Z p (1) ⊗ ρ − , a free Z p -module of rank one with G K acting via theproduct of ρ − and the cyclotomic character. Let E be the cyclic extension of K cutout by ρ , i.e., such that ρ factors through an injective homomorphism Gal( E/K ) ֒ → Z × p . Let P = { primes q of K : q ∤ p and ρ is unramified at q } . A simple exercise in Galois cohomology (see for example [MR1, § § H ( K, T ) ∼ = ( E × ⊗ Z p ) ρ where the superscript ρ means the subgroup on which Gal( E/K ) acts via ρ , andfor every prime q , H ( K q , T ) ∼ = ( E × q ⊗ Z p ) ρ where E q = E ⊗ K K q is the product of the completions of E above q . With theseidentifications, the unramified Selmer structure of Definition 5.1 is given by H F ur ( K q , T ) := ( O × E, q ⊗ Z p ) ρ for every q , where O E, q is the ring of integers of E q . Proposition 5.7.
Let
Cl( E ) denote the ideal class group of E . There are naturalisomorphisms H F ur ( K, T ) ∼ = ( O × E ⊗ Z p ) ρ , H F ∗ ur ( K, T ∗ ) ∼ = Hom(Cl( E ) ρ , Q p / Z p ) and for every k ≥ an exact sequence −→ ( O × E / ( O × E ) p k ) ρ −→ H F ur ( K, T /p k T ) −→ Cl( E )[ p k ] ρ −→ and an isomorphism H F ∗ ur ( K, T ∗ [ p k ]) ∼ = Hom(Cl( E ) ρ , Z /p k Z ) . Proof.
See for example [MR1, Proposition 6.1.3]. (cid:3)
Suppose in addition now that ρ = ω , and either ρ = ω or p >
3, where ω : G K → Z × p is the Teichm¨uller character giving the action of G K on µ p . Thenconditions (H.1), (H.3), and (H.4) of § F ur satisfies (H.5) as well, and condition (H.2) holds with τ = 1 and L = E . Finally, if there is at least one real place v of K such that ρ is trivial oncomplex conjugation at v , then the following corollary shows that condition (H.6)holds. Corollary 5.8.
The core rank χ ( T, F ur ) is χ ( T ) = dim F p ( O × E / ( O × E ) p ) ρ = rank Z p ( O × E ⊗ Z p ) ρ = |{ archimedean v : ρ ( σ v ) = 1 }| where σ v ∈ Gal(
E/K ) is the complex conjugation at v .Proof. The first equality follows from Proposition 5.7 and the definition of corerank, and the second because ρ = ω . The third equality is well-known (using that ρ = 1); see for example [T, Proposition I.3.4]. (cid:3) Thus if
E/K is an extension of totally real fields and ρ = 1, then χ ( T, F ur ) =[ K : Q ] by Corollary 5.8, and all conditions (H.1) through (H.6) are satisfied.If K = Q , then χ ( T ) = 1, and a Kolyvagin system (see §
10) can be constructedfrom the Euler system of cyclotomic units (see [MR1]).For a general totally real field K , if we assume the version of Stark’s Conjecturedescribed in [Ru1], then the so-called “Rubin-Stark” elements predicted by thatconjecture can be used to construct both an Euler system and a Stark system (see § Abelian varieties.
Suppose A is an abelian variety of dimension d definedover the number field K . Let P = { primes q of K : q ∤ p and A has good reduction at q } . Let T be the Tate module T p ( A ) := lim ←− A [ p k ]. Then T is a free Z p -module of rank2 d with a natural action of G K , and T ∗ = ˇ A [ p ∞ ] where ˇ A is the dual abelian varietyto A .Let F be the Selmer structure on T given by H F ( K v , T ) = H ( K v , T ) for every v . Then F is the unramified Selmer structure F ur given by Definition 5.1. (For v ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 13 dividing p , this follows from the Lemma in [PR1, § v not dividing p it follows from the fact that H ( K v , T ) is finite.) Further, F is the usual Selmerstructure attached to an abelian variety, with the local conditions at primes above p relaxed (see for example [Ru2, § −→ H F ( K, T ∗ ) −→ Sel p ∞ ( ˇ A/K ) −→ ⊕ p | p H ( K p , ˇ A [ p ∞ ]) . Suppose now that p >
3, and that the image of G K in Aut( A [ p ]) ∼ = GL d ( F p ) islarge enough so that conditions (H.1), (H.2), and (H.3) of § G K contains GSp d ( F p ). Condition (H.4) holdssince p >
3, and F satisfies (H.5) by Remark 3.2. The following consequence ofTheorem 5.4 shows that condition (H.6) holds as well. Proposition 5.9.
The core rank of T is given by χ ( T ) = d [ K : Q ] . Proof.
By Theorem 5.4, we have χ ( T ) = X v |∞ corank Z p H ( K v , ˇ A [ p ∞ ]) . If v is a real place, then corank Z p H ( K v , ˇ A [ p ∞ ]) = d , and if v is a complex placethen corank Z p H ( K v , ˇ A [ p ∞ ]) = corank Z p ˇ A [ p ∞ ] = 2 d . Thus X v |∞ corank Z p H ( K v , ˇ A [ p ∞ ]) = X v |∞ d [ K v : R ] = d [ K : Q ] . (cid:3) If K = Q and d = 1 (i.e., A is an elliptic curve), then Proposition 5.9 shows that χ ( T ) = 1. In this case Kato has constructed an Euler system for T , from whichone can produce a Kolyvagin system ([MR1, Theorem 3.2.4]). Part Stark systems and the structure of Selmer groups Stark systems
Suppose for this section that R is a principal artinian ring of length k , so m k = 0and m k − = 0. Fix Selmer data ( T, F , P , r ) as in Definition 4.1. We assumethroughout this section that (H.7) of § I q = 0 for every q ∈ P .Recall that ν ( n ) denotes the number of prime factors of n . Definition 6.1.
For every n ∈ N , define W n := ⊕ q | n Hom( H ( K q , T ) , R ) ,Y n := ∧ r + ν ( n ) H F n ( K, T ) ⊗ ∧ ν ( n ) W n , where as usual the exterior powers are taken in the category of R -modules.Then W n is a free R -module of rank ν ( n ), since each H ( K q , T ) is free of rankone (Lemma 1.5). If we fix an ordering n = q i · · · q ν ( n ) of the primes dividing n ,and a generator h i of Hom( H ( K q i , T ) , R ) for every i , then h ∧ · · · ∧ h ν ( n ) is agenerator of the free, rank-one R -module ∧ ν ( n ) W n .For the structure of Y n when r is the core rank of T , see Lemma 6.9 below. Definition 6.2.
For every q ∈ P , define the transverse localization maploc tr q : H ( K, T ) loc q −−→ H ( K q , T ) ։ H ( K q , T ) , where the second map is projection (using the direct sum decomposition of Lemma1.3(i)) with kernel H ( K q , T ). If n ∈ N and q | n , then(6.3) ker (cid:0) loc tr q (cid:12)(cid:12) H F n ( K, T ) (cid:1) = H F n / q ( K, T ) . In exactly the same way, we can define a map loc f q by using the finite projectionand the isomorphism φ fs q of Definition 1.4loc f q : H ( K, T ) loc q −−→ H ( K q , T ) ։ H ( K, T ) φ fs q −−→ H ( K q , T ) ⊗ Gal( K ( q ) q /K q ) , and then(6.4) ker (cid:0) loc f q (cid:12)(cid:12) H F n ( K, T ) (cid:1) = H F n / q ( q ) ( K, T ) . Definition 6.5.
Suppose n ∈ N and m | n . By (6.3) we have an exact sequence0 −→ H F m ( K, T ) −→ H F n ( K, T ) ⊕ loc tr q −−−−→ L q | ( n / m ) H ( K q , T )and it follows that the square(6.6) H F m ( K, T ) (cid:31) (cid:127) / / ⊕ loc tr q (cid:15) (cid:15) H F n ( K, T ) ⊕ loc tr q (cid:15) (cid:15) L q | m H ( K q , T ) (cid:31) (cid:127) / / L q | n H ( K q , T )is cartesian. Let Ψ n , m : Y n −→ Y m be the map of Proposition A.3(i) attached to this diagram.Concretely, Ψ n , m is given as follows. Fix a factorization n = q · · · q t , with m = q · · · q s , and a generator h i of Hom( H ( K q i , T ) , R ) for every i . Let n i = Q j ≤ i q j .These choices lead to a map \ h s +1 ◦ loc tr q s +1 ◦ · · · ◦ \ h t ◦ loc tr q t : ∧ r + t H F n ( K, T ) −→ ∧ r + s H F m ( K, T )(where \ h i ◦ loc tr q i : ∧ i H F n i ( K, T ) → ∧ i − H F n i − ( K, T ) is given by Proposition A.1)and an isomorphism ∧ ν ( n ) W n ∼ −→ ∧ ν ( m ) W m given by h ∧· · ·∧ h t h ∧· · ·∧ h s . Thetensor product of these two maps is the map Ψ n , m : Y n −→ Y m , and is independentof the choices made. Proposition 6.7.
Suppose n ∈ N , n ′ | n , and n ′′ | n ′ . Then Ψ n ′ , n ′′ ◦ Ψ n , n ′ = Ψ n , n ′′ .Proof. This is Proposition A.3(iii). (cid:3)
Definition 6.8.
Thanks to Proposition 6.7, we can define the R -module SS r ( T ) = SS r ( T, F , P ) of Stark systems of rank r to be the inverse limit SS r ( T ) := lim ←−−− n ∈N Y n with respect to the maps Ψ n , m . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 15
We call these collections Stark systems because a fundamental example is givenby elements predicted by a generalized Stark conjecture [MR2, Ru1].Let Y ′ n = m length( H F∗ ) n ( K,T ∗ )) Y n . Lemma 6.9.
Suppose that hypotheses (H.1) through (H.7) of § r is the core rank of T . Then: (i) Y ′ n is a cyclic R -module of length max { k − length( H F ∗ ) n ( K, T ∗ )) , } . (ii) There are n ∈ N such that H F ∗ ) n ( K, T ∗ ) = 0 . (iii) If H F ∗ ) n ( K, T ∗ ) = 0 then Y n is free of rank one over R . (iv) If H F ∗ ) n ( K, T ∗ ) = 0 and m | n , then Ψ n , m ( Y n ) = Y ′ m . Proof.
Assertions (i) and (iii) follow directly from Corollary 3.5(iv).Since H F ∗ ( K, T ∗ ) is finite, we can choose generators c , . . . , c t of H F ∗ ( K, T ∗ )[ m ].For each i , use [MR1, Proposition 3.6.1] to choose q i ∈ N such that loc q i ( c i ) = 0,and let n = Q i q i . Then H F ∗ ) n ( K, T ∗ ) = 0, so (ii) holds.Proposition A.3(ii) applied to the diagram (6.6) shows thatΨ n , m ( Y n ) = m length( H F m ( K,T )) − ( r + ν ( m )) k Y m . Corollary 3.5(ii) shows thatlength( H F m ( K, T )) − ( r + ν ( m )) k = length( H F ∗ ) m ( K, T ∗ ))which proves (iv). (cid:3) Theorem 6.10.
Suppose that hypotheses (H.1) through (H.7) of § R -module SS r ( T ) is free of rank one, and for every n ∈ N , the image ofthe projection map SS r ( T ) → Y n is Y ′ n .Proof. Using Lemma 6.9(ii), choose an d ∈ N such that H F ∗ ) d ( K, T ∗ ) = 0. Then H F ∗ ) n ( K, T ) = 0 for every n ∈ N divisible by d . Now the theorem follows fromLemma 6.9(iv). (cid:3) Stark systems over discrete valuation rings
For this section we assume that R is a discrete valuation ring, and we fix Selmerdata ( T, F , P , r ) as in Definition 4.1. We assume throughout this section thathypotheses (H.1) through (H.6) of § k > P k := { q ∈ P : I q ∈ m k } , and N k is the set of squarefree products of primes in P k . By Remark 4.3, theSelmer data ( T / m k T, F , P k , r ) satisfies (H.1) through (H.7) over the ring R/ m k .In this section we will define the module SS r ( T ) of Stark systems of rank r over T ,and use the results of § SS r ( T / m k T ) to study SS r ( T ). Definition 7.1.
For every n ∈ N , define W n := ⊕ q | n Hom( H ( K q , T /I n T ) , R/I n ) ,Y n := ∧ r + ν ( n ) H F n ( K, T /I n T ) ⊗ ∧ ν ( n ) W n ,Y ′ n := m length( H F∗ ) n ( K,T ∗ [ I n ])) Y n . A Stark system of rank r for T (more precisely, for ( T, F , P )) is a collection { ǫ n ∈ Y n : n ∈ N } such that if n ∈ N and m | n , thenΨ n , m ( ǫ n ) = ¯ ǫ m where ¯ ǫ m is the image of ǫ m in Y m ⊗ R/I n , and Ψ n , m : Y n → Y m ⊗ R/I n is the mapof Definition 6.5 applied to T /I n T and R/I n . Denote by SS r ( T ) = SS r ( T, F , P )the R -module of Stark systems for T . Lemma 7.2. If j ≤ k , then the projection map T / m k T → T / m j T and restrictionto P k induce a surjection and an isomorphism, respectively SS r ( T / m k T, P k ) / / / / SS r ( T / m j T, P k ) SS r ( T / m j T, P j ) ∼ o o Proof.
Let n ∈ N k be such that H F ∗ ) n ( K, T ∗ [ m ]) = 0. Then by Theorem 6.10,projecting to Y n gives a commutative diagram with vertical isomorphisms SS r ( T / m k T, P k ) / / ∼ = (cid:15) (cid:15) SS r ( T / m j T, P k ) ∼ = (cid:15) (cid:15) SS r ( T / m j T, P j ) o o ∼ = (cid:15) (cid:15) Y n ⊗ R/ m k / / / / Y n ⊗ R/ m j Y n ⊗ R/ m j = o o Since the bottom maps are a surjection and an isomorphism, so are the top ones. (cid:3)
Proposition 7.3.
The natural maps T ։ T / m k and P k ֒ → P induce an isomor-phism SS r ( T, P ) ∼ −−→ lim ←− SS r ( T / m k T, P k ) where the inverse limit is with respect to the maps of Lemma 7.2.Proof. Suppose ǫ ∈ SS r ( T ) is nonzero. Then we can find an n such that ǫ n = 0 in Y n . If n = 1 then I n = 0, and we let k be such that m k = I n . If n = 1 choose k sothat ǫ = 0 in ∧ r H F ( K, T / m k T ). In either case I n ⊂ m k , and the image of ǫ in SS r ( T / m k T, P k ) is nonzero. Thus the map in the proposition is injective.Now suppose { ǫ ( k ) } ∈ lim ←− SS r ( T / m k , P k ). If n ∈ N and n = 1, let j be suchthat I n = m j and define ǫ n := ǫ ( j ) n ∈ Y n . If n = 1, define ǫ = lim k →∞ ǫ ( k )1 ∈ lim k →∞ ∧ r H F ( K, T / m k T ) = ∧ r H F ( K, T ) = Y . It is straightforward to verify that this defines an element ǫ ∈ SS r ( T, P ) that mapsto ǫ ( k ) ∈ SS r ( T / m k T, P k ) for every k . Thus the map in the proposition is surjectiveas well. (cid:3) Theorem 7.4.
Suppose R is a discrete valuation ring and hypotheses (H.1) through(H.6) hold. Then the R -module of Stark systems of rank r , SS r ( T, P ) , is free ofrank one, generated by a Stark system ǫ whose image in SS r ( T / m T, P ) is nonzero.The map SS r ( T, P ) → SS r ( T / m k , P k ) is surjective for every k .Proof. By Theorem 6.10, SS r ( T / m k T, P k ) is free of rank one over R/ m k for every k . The maps SS r ( T / m k +1 T, P k +1 ) → SS r ( T / m k T, P k ) are surjective by Lemma7.2, so the theorem follows from Proposition 7.3. (cid:3) ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 17 Structure of the dual Selmer group
In this section R is either a principal artinian local ring or a discrete valuationring. We let k := length( R ), so k is finite in the artinian case and k = ∞ in thediscrete valuation ring case.Fix Selmer data ( T, F , P , r ). We continue to assume that hypotheses (H.1)through (H.6) are satisfied, and if R is artinian we assume that (H.7) is satisfied aswell. Recall that if n ∈ N then ν ( n ) denotes the number of prime divisors of n . Definition 8.1.
Define functions µ, λ, ϕ ǫ ∈ Maps( N , Z ≥ ∪ {∞} ) • µ ( n ) = length( H F ∗ ) n ( K, T ∗ )), • λ ( n ) = length( H F ( n ) ∗ ( K, T ∗ )),and if ǫ ∈ SS r ( T ) is a Stark system • ϕ ǫ ( n ) = max { j : ǫ n ∈ m j Y n } .Define ∂ : Maps( N , Z ≥ ∪ {∞} ) → Maps( Z ≥ , Z ≥ ∪ {∞} ) by ∂f ( i ) = min { f ( n ) : n ∈ N and ν ( n ) = i } . Definition 8.2.
The order of vanishing of a nonzero Stark system ǫ ∈ SS r ( T ) isord( ǫ ) := min { ν ( n ) : n ∈ N , ǫ n = 0 } = min { i : ∂ϕ ǫ ( i ) = ∞} . We say ǫ ∈ SS r ( T ) is primitive if its image in SS r ( T / m T ) is nonzero. We alsodefine the sequence of elementary divisors d ǫ ( i ) := ∂ϕ ǫ ( i ) − ∂ϕ ǫ ( i + 1) , i ≥ ord( ǫ ) . Note that ∂ϕ ǫ ( i ) = ∞ if i < ord( ǫ ); Theorems 8.6 and 8.9 below show that theconverse is true as well, so the d ǫ ( i ) are well-defined and finite. Proposition 8.3.
Suppose R is artinian, and H F ∗ ( K, T ∗ ) ∼ = ⊕ i ≥ R/ m e i with e ≥ e ≥ · · · . Then for every t ≥ , ∂λ ( t ) = ∂µ ( t ) = X i>t e i . Proof.
Suppose n ∈ N and ν ( n ) = t . Consider the map H F ∗ ( K, T ∗ ) −→ L q | n H ( K q , T ∗ ) . The right-hand side is free of rank t over R , and R is principal, so the image is aquotient of H F ∗ ( K, T ∗ ) generated by (at most) t elements. Hence the image haslength at most P i ≤ t e i , so the kernel has length at least P i>t e i . But by definitionthis kernel is H F ∗ ) n ( K, T ∗ ), which is contained in H F ( n ) ∗ ( K, T ∗ ), so(8.4) λ ( n ) ≥ µ ( n ) ≥ X i>t e i . We will prove by induction on t that n can be chosen so that ν ( n ) = t and H F ( n ) ∗ ( K, T ∗ ) ∼ = ⊕ i>t R/ m e i . For such an n equality holds in (8.4), and the lemmafollows. When t = 0 we can just take n = 1.Suppose we have an n with ν ( n ) = t − H F ( n ) ∗ ( K, T ∗ ) ∼ = ⊕ i>t − R/ m e i .Since χ ( T ) >
0, Corollary 3.5 shows that m k − H F ( n ) ( K, T ) = 0. Fix a nonzeroelement c ∈ m k − H F ( n ) ( K, T ) ⊂ H F ( n ) ( K, T )[ m ]. If e t > c ′ ∈ m e t − H F ( n ) ∗ ( K, T ∗ ) ⊂ H F ( n ) ∗ ( K, T ∗ )[ m ]. By [MR1, Proposition q ∈ P such that thelocalization loc q ( c ) = 0 and, if e t >
0, such that loc q ( c ′ ) = 0 as well.Since H ( K q , T ) is free of rank one over R , and (by our choice of q ) the local-ization of m k − H F ( n ) ( K, T ) at q is nonzero, it follows that the localization map H F ( n ) ( K, T ) → H ( K q , T ) is surjective. Similarly, we have that H F ( n ) ∗ ( K, T ∗ ) hasexponent m e t , and if e t > m e t − H F ( n ) ∗ ( K, T ∗ ) at q isnonzero, so H F ( n ) ∗ ( K, T ∗ ) /H F q ( n ) ∗ ( K, T ∗ ) ∼ = loc q ( H F ( n ) ∗ ( K, T ∗ )) ∼ = R/ m e t and therefore H F q ( n ) ∗ ( K, T ∗ ) ∼ = ⊕ i>t R/ m e i . By [MR1, Theorem 4.1.7(ii)] we have H F ( nq ) ∗ ( K, T ∗ ) = H F q ( n ) ∗ ( K, T ∗ ), so nq ∈ N has the desired property. (cid:3) Proposition 8.5.
Suppose R is artinian of length k , and ǫ ∈ SS r ( T ) . Fix s ≥ such that ǫ generates m s SS r ( T ) , and nonnegative integers e ≥ e ≥ · · · such that H F ∗ ( K, T ∗ ) ∼ = ⊕ i R/ m e i . Then for every t ≥ , ∂ϕ ǫ ( t ) = ( s + P i>t e i if s + P i>t e i < k, ∞ if s + P i>t e i ≥ k. Proof.
It is enough to prove the proposition when s = 0, and the general casewill follow. So we may assume that ǫ generates SS r ( T ). By Theorem 6.10 andLemma 6.9(i), we have that ǫ n generates Y ′ n = m µ ( n ) Y n , which is cyclic of lengthmax { k − µ ( n ) , } . Hence ǫ n ∈ m µ ( n ) Y n , and ǫ n ∈ m µ ( n )+1 Y n if and only if µ ( n ) ≥ k .Therefore ∂ϕ ǫ ( t ) = ( ∂µ ( t ) if ∂µ ( t ) < k, ∞ if ∂µ ( t ) ≥ k. Now the proposition follows from the calculation of ∂µ ( t ) in Lemma 8.3. (cid:3) Theorem 8.6.
Suppose R is artinian, ǫ ∈ SS r ( T ) , and ǫ = 0 . Then ∂ϕ ǫ (0) ≥ ∂ϕ ǫ (1) ≥ ∂ϕ ǫ (2) ≥ · · · ,d ǫ (0) ≥ d ǫ (1) ≥ d ǫ (2) ≥ · · · ≥ , and H F ∗ ( K, T ∗ ) ∼ = L i ≥ R/ m d ǫ ( i ) . Proof.
Let s be such that ǫ generates m s SS r ( T ). If ǫ = 0 then ∂ϕ ǫ (0) < k , soin Proposition 8.5 we have ∂ϕ ǫ ( t ) = s + P i>t e i for every t . The theorem followsdirectly. (cid:3) If R is a discrete valuation ring then F will denote the field of fractions of R ,and if M is an R -module we define • rank R M := dim F M ⊗ F , • corank R M := rank R Hom R ( M, F/R ), • M div is the maximal divisible submodule of M . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 19
Proposition 8.7.
Suppose R is a discrete valuation ring, and ǫ ∈ SS r ( T ) generates m s SS r ( T ) . Let a := corank R ( H F ∗ ( K, T ∗ )) and write H F ∗ ( K, T ∗ ) / ( H F ∗ ( K, T ∗ )) div ∼ = L i>a R/ m e i with e a +1 ≥ e a +2 ≥ · · · . Then ∂ϕ ǫ ( t ) = ( ∞ if t < a,s + P i>t e i if t ≥ a. Proof.
Let e = · · · = e a := ∞ . Since H F ∗ ( K, T ∗ ) = lim −→ H F ∗ ( K, T ∗ [ m k ]) , Proposition 3.3(ii) applied to all the
T / m k T shows that for every k ∈ Z + we have(8.8) H F ∗ ( K, T ∗ [ m k ]) = H F ∗ ( K, T ∗ )[ m k ] ∼ = L i ≥ R/ m min { k,e i } . For every k ≥ ǫ ( k ) denote the image of ǫ in SS r ( T / m k T, P k ). Fix s ≥ ǫ generates m s SS r ( T ). Then by Theorem 7.4, ǫ ( k ) generates m s SS r ( T / m k T )for every k .Fix t , and choose n ∈ N with ν ( n ) = t . Let k be such that I n = m k . By(8.8) and Proposition 8.5 we have that ǫ ( k ) n = 0 if t < a , and ǫ ( k ) n ∈ m s + P i>t e i Y n if t > a . But ǫ ( k ) n = ǫ n ∈ Y n , so we conclude that ∂ϕ ǫ ( t ) = ∞ if t < a , and ∂ϕ ǫ ( t ) ≥ s + P i>t e i if t ≥ a .Now suppose t ≥ a , and fix k > s + P i>t e i . By Proposition 8.5 we can find n ∈ N with I n ⊂ m k such that ǫ ( k ) n / ∈ m s +1+ P i>t e i Y n . Since ǫ ( k ) n is the image of ǫ n , we have that ǫ n / ∈ m s +1+ P i>t e i Y n . This shows that ∂ϕ ǫ ( t ) ≤ s + P i>t e i , andthe proof is complete. (cid:3) Theorem 8.9.
Suppose R is a discrete valuation ring, ǫ ∈ SS r ( T ) and ǫ = 0 .Then: (i) the sequence ∂ϕ ǫ ( t ) is nonincreasing, finite for t ≥ ord( ǫ ) , and nonnega-tive, (ii) the sequence d ǫ ( i ) is nonincreasing, finite for i ≥ ord( ǫ ) , and nonnegative, (iii) ord( ǫ ) and the d ǫ ( i ) are independent of the choice of nonzero ǫ ∈ SS r ( T ) , (iv) corank R ( H F ∗ ( K, T ∗ )) = ord( ǫ ) , (v) H F ∗ ( K, T ∗ ) / ( H F ∗ ( K, T ∗ )) div ∼ = ⊕ i ≥ ord( ǫ ) R/ m d ǫ ( i ) , (vi) length R ( H F ∗ ( K, T ∗ ) / ( H F ∗ ( K, T ∗ )) div ) = ∂ϕ ǫ (ord( ǫ )) − ∂ϕ ǫ ( ∞ ) , where ∂ϕ ǫ ( ∞ ) := lim t →∞ ∂ϕ ǫ ( t )(vii) ǫ is primitive if and only if ∂ ( ∞ ) ( ǫ ) = 0 , (viii) length( H F ∗ ( K, T ∗ )) is finite if and only if ǫ = 0 , (ix) length( H F ∗ ( K, T ∗ )) ≤ ∂ϕ ǫ (0) = max { s : ǫ ∈ m s ∧ r H F ( K, T ) } , withequality if and only if ǫ is primitive.Proof. The theorem follows directly from Proposition 8.7. (cid:3)
Part Kolyvagin systems Sheaves and monodromy
In this section we recall some concepts and definitions from [MR1].
Definition 9.1. If X is a graph, a sheaf S (of R -modules) on X is a rule assigning: • to each vertex v of X , an R -module S ( v ) (the stalk of X at v ), • to each edge e of X , an R -module S ( e ), • to each pair ( e, v ) where v is an endpoint of the edge e , an R -module map ψ ev : S ( v ) → S ( 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 ev ( κ v ) = ψ ev ′ ( κ v ′ ) in S ( e ). We write Γ( S ) forthe R -module of global sections of S . Definition 9.2.
We say that a sheaf S on a graph X is locally cyclic if all the R -modules S ( v ), S ( e ) are cyclic and all the maps ψ ev are surjective.If S is locally cyclic then a surjective path (relative to S ) from v to w is a path( v = v , v , . . . , v k = w ) in X such that for each i , if e i is the edge joining v i and v i +1 , then ψ e i v i +1 is an isomorphism. We say that the vertex v is a hub of S if forevery vertex w there is an S -surjective path from v to w .Suppose now that the sheaf S is locally cyclic. If P = ( v , v , . . . , v k ) is asurjective path in X , we can define a surjective map ψ P : S ( v ) → S ( v k ) by ψ P := ( ψ e k − v k ) − ◦ ψ e k − v k − ◦ ( ψ e k − v k − ) − ◦ · · · ◦ ( ψ e v ) − ◦ ψ e v since all the inverted maps are isomorphisms. We will say that S has trivialmonodromy if whenever v, w, w ′ are vertices, P, P ′ are surjective paths ( v, . . . , w )and ( v, . . . , w ′ ), and w, w ′ are joined by an edge e , then ψ ew ◦ ψ P = ψ ew ′ ◦ ψ P ′ ∈ Hom( S ( v ) , S ( e )). In particular for every pair v, w of vertices and and every pair P, P ′ of surjective paths from v to w , we require that ψ P = ψ P ′ ∈ Hom( S ( v ) , S ( w )). Proposition 9.3.
Suppose S is locally cyclic and v is a hub of S . (i) The map f v : Γ( S ) → S ( v ) defined by κ κ v is injective, and is surjectiveif and only if S has trivial monodromy. (ii) 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. This is [MR1, Proposition 3.4.4]. (cid:3)
Definition 9.4.
A global section κ ∈ Γ( S ) will be called primitive if for everyvertex v , κ ( v ) ∈ S ( v ) is a generator of the R -module S ( v ).It follows from Proposition 9.3 that a locally cyclic sheaf S with a hub has aprimitive global section if and only if S has trivial monodromy.10. Kolyvagin systems and the Selmer sheaf
Fix Selmer data ( T, F , P , r ) as in Definition 4.1. Recall that we have defineda Selmer structure F ( n ) for every n ∈ N (Definition 2.3) by modifying the localcondition at primes dividing n , and that K ( q ) is the p -part of the ray class field of K modulo q . Definition 10.1.
For every n ∈ N , define G n := N q | n Gal( K ( q ) q /K q ) . Each Gal( K ( q ) q /K q ) is cyclic with order contained in I n , so G n ⊗ ( R/I n ) is free ofrank one over R/I n . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 21 If q is a prime dividing n , then ( T /I n T ) / (Fr q − T /I n T ) is free of rank oneover R/I n , so we can apply the results of § H ( K q , T /I n T ). In particular wewill write φ fs q : H ( K q , T /I n T ) −→ H ( K q , T /I n T ) ⊗ G q for the finite-singular isomorphism of Definition 1.4 applied to K q .If q is a prime, nq ∈ N , and r ≥
1, then we can compare ∧ r H F ( n ) ( K, T /I n T ) ⊗ G n and ∧ r H F ( nq ) ( K, T /I nq T ) ⊗ G nq using the exterior algebra of Appendix A. Namely,applying Proposition A.1 with the localization maps of Definition 6.2loc f q : H F ( n ) ( K, T /I nq T ) −→ H ( K q , T /I nq T ) φ fs q −−→ H ( K q , T /I nq T ) ⊗ G q , loc tr q : H F ( nq ) ( K, T /I nq T ) −→ H ( K q , T /I nq T )gives the top and bottom maps, respectively, in the following diagram:(10.2) ( ∧ r H F ( n ) ( K, T /I n T )) ⊗ G n d loc f q ⊗ ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ H ( K q , T /I nq T ) ⊗ ( ∧ r − H F q ( n ) ( K, T /I nq T )) ⊗ G nq ( ∧ r H F ( nq ) ( K, T /I nq T )) ⊗ G nq d loc tr q ⊗ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ Definition 10.3.
Define a graph X := X ( P ) by taking the set of vertices of X tobe N := N ( P ) (Definition 2.2), and whenever n , nq ∈ N (with q prime) we join n and nq by an edge.The Selmer sheaf associated to ( T, F , P , r ) is the sheaf S = S ( T, F , P ,r ) of R -modules on X defined as follows. Let • S ( n ) := ( ∧ r H F ( n ) ( K, T /I n T )) ⊗ G n for n ∈ N ,and if e is the edge joining n and nq define • S ( e ) := H ( K q , T /I nq T ) ⊗ ( ∧ r − H F q ( n ) ( K, T /I nq T )) ⊗ G nq , • ψ e n : S ( n ) → S ( e ) is the upper map of (10.2), • ψ e nq : S ( nq ) → S ( e ) is the lower map of (10.2).We call S ( n ) := ∧ r H F ( n ) ( K, T /I n T ) ⊗ G n the Selmer stalk at n . Definition 10.4. A Kolyvagin system for ( T, F , P , r ) (or simply a Kolyvagin sys-tem of rank r for T , if F and P are fixed) is a global section of the Selmer sheaf S .We write KS r ( T, F , P ), or simply KS r ( T ) when there is no risk of confusion, forthe R -module of Kolyvagin systems Γ( S ).Concretely, a Kolyvagin system for ( T, F , P , r ) is a collection of classes { κ n ∈ ( ∧ r H F ( n ) ( K, T /I n T )) ⊗ G n : n ∈ N } such that if q is prime and nq ∈ N , the images of κ n and κ nq coincide in the diagram(10.2). Remark 10.5.
The definition of Kolyvagin system given in [MR1] corresponds tothe definition above with r = 1. Stub Kolyvagin systems
Suppose until the final result of this section that R is a principal artinian ringof length k . Fix Selmer data ( T, F , P , r ) as in Definition 4.1 such that hypotheses(H.1) through (H.7) of § r = χ ( T ) is the core rank of T .Recall that for n ∈ N we defined λ ( n ) := length R ( H F ( n ) ∗ ( K, T ∗ )) ∈ Z ≥ ∪ {∞} . We say that a vertex n ∈ N is a core vertex if λ ( n ) = 0. Proposition 11.1.
The following are equivalent: (i) n is a core vertex for T , (ii) H F ( n ) ( K, T ) is free of rank r over R , (iii) S ( n ) is free of rank one over R , (iv) n is a core vertex for T / m T .Proof. We have (i) ⇐⇒ (ii) by Corollary 3.5, and (i) ⇐⇒ (iv) by Proposition3.3(ii). It is easy to see that (ii) ⇐⇒ (iii). (cid:3) Proposition 11.2. If n , nq ∈ N and e is the edge joining them, then ψ e n ( m λ ( n ) S ( n )) = ψ e nq ( m λ ( nq ) S ( nq )) ⊂ S ( e ) . Proof.
By Proposition A.1(ii) and Definition 10.3 of ψ e n and ψ e nq , we have ψ e n ( S ( n )) = φ fs q (loc q ( H F ( n ) ( K, T ))) ⊗ ∧ r − H F q ( n ) ( K, T ) ⊗ G n ,ψ e nq ( S ( nq )) = loc q ( H F ( nq ) ( K, T )) ⊗ ∧ r − H F q ( n ) ( K, T ) ⊗ G nq . By [MR1, Lemma 4.1.7], global duality shows that m λ ( n ) φ fs q (loc q ( H F ( n ) ( K, T ))) = m λ ( nq ) loc q ( H F ( nq ) ( K, T )) ⊗ G q and the proposition follows. (cid:3) We define a subsheaf S ′ of the Selmer sheaf S as follows. Definition 11.3.
The sheaf of stub Selmer modules S ′ = S ′ ( T, F , P ,r ) ⊂ S is thesubsheaf of S defined by • S ′ ( n ) := m λ ( n ) S ( n ) = m λ ( n ) ( ∧ r H F ( n ) ( K, T )) ⊗ G n ⊂ S ( n ) if n ∈ N , • S ′ ( e ) is the image of S ′ ( n ) in S ( e ) under the vertex-to-edge map of S , if n is a vertex of the edge e (this is well-defined by Proposition 11.2),and the vertex-to-edge maps are the restrictions of those of the sheaf S . Definition 11.4. A stub Kolyvagin system is a global section of the sheaf S ′ . Welet KS ′ r ( T ) = KS ′ r ( T, F , P ) := Γ( S ′ ) ⊂ KS r ( T ) denote the R -module of stubKolyvagin systems. Remark 11.5.
It is shown in [MR1, Theorem 4.4.1] that when the core rank χ ( T ) = 1, we have KS ′ ( T ) = KS ( T ). In other words, in that case for everyKolyvagin system κ ∈ KS ( T ) and n ∈ N , we have κ n ∈ m λ ( n ) H F ( n ) ( K, T ) ⊗ G n . Theorem 11.6. (i)
There are core vertices.
ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 23 (ii)
Suppose n , n ′ are core vertices. Then there is a path n = n e n e · · · e t n t = n ′ in X such that every n i is a core vertex and all of the maps ψ e i +1 n i and ψ e i n i are isomorphisms. (iii) The stub subsheaf S ′ is locally cyclic, and every core vertex is a hub. Forevery vertex n ∈ N , there is a core vertex n ′ ∈ N divisible by n . Theorem 11.6 will be proved in §
14. In the remainder of this section we derivesome consequences of it.
Theorem 11.7. (i)
The module KS ′ r ( T ) of stub Kolyvagin systems is free ofrank one over R , and for every core vertex n the specialization map KS ′ r ( T ) −→ S ′ ( n ) = ( ∧ r H F ( n ) ( K, T )) ⊗ G n given by κ κ n is an isomorphism. (ii) There is a Kolyvagin system κ ∈ KS ′ r ( T ) such that κ n generates S ′ ( n ) forevery n ∈ N . (iii) The locally cyclic sheaf S ′ has trivial monodromy.Proof. This follows from Proposition 9.3, using Theorem 11.6(i,iii). (cid:3)
For the next theorem we take R to be a discrete valuation ring. Theorem 11.8.
Suppose that R is a discrete valuation ring, and hypotheses (H.1) through (H.6) are satisfied for the Selmer data ( T, F , P , r ) . For k > let P k ⊂ P be as in Definition 2.2.The natural maps T ։ T / m k and P k ֒ → P induce an isomorphism KS ′ r ( T, P ) ∼ −−→ lim ←− KS ′ r ( T / m k T, P k ) . The R -module KS ′ r ( T, P ) , is free of rank one, generated by a Kolyvagin system κ whose image in KS ′ r ( T / m T ) is nonzero. The maps KS ′ r ( T, P ) → KS ′ r ( T / m k , P k ) are surjective.Proof. This can be proved easily directly from Theorem 11.7, as in the proofs ofProposition 7.3 and Theorem 7.4 for Stark systems. See also [MR1, Proposition5.2.9]. (cid:3)
Remark 11.9.
When r = χ ( T ) >
1, it is not generally true that KS ′ r ( T ) = KS r ( T ). For example, suppose R is principal artinian of length k >
1, and sup-pose m ∈ N is such that H F ( m ) ( K, T ) ∼ = R r ⊕ ( R/ m ) r , with corresponding basis c , . . . , c r , d , . . . , d r . Let g m be a generator of G m .For every q ∈ P and every i , loc q ( d i ) is killed by m , so it is divisible by m k − in the free R -module H ( K q , T ). It follows that if we define κ := { κ n } where κ n := ( ( d ∧ · · · ∧ d r ) ⊗ g m if n = m , n = m , then κ is a Kolyvagin system, but κ m / ∈ S ′ ( m ) so κ / ∈ KS ′ r ( T ). Kolyvagin systems and Stark systems
Suppose that R is a principal artinian ring, and fix Selmer data ( T, F , P , r ) as inDefinition 4.1 such that I q = 0 for every q ∈ P . Recall the R module Y n of Definition6.1, and let loc f q : H ( K, T ) → H ( K, T ) ⊗ G q and loc tr q : H ( K, T ) → H ( K, T )be the maps of Definition 6.2.
Definition 12.1.
Suppose n ∈ N . By (6.4) we have an exact sequence0 −→ H F ( n ) ( K, T ) −→ H F n ( K, T ) ⊕ loc f q −−−−→ L q | n H ( K q , T ) ⊗ G q and it follows that the square H F ( n ) ( K, T ) (cid:31) (cid:127) / / (cid:15) (cid:15) H F n ( K, T ) ⊕ loc f q (cid:15) (cid:15) (cid:31) (cid:127) / / L q | n H ( K q , T ) ⊗ G q is cartesian. Proposition A.3(i,iv) attaches to this diagram a map ∧ r + ν ( n ) H F n ( K, T ) ⊗ ∧ ν ( n ) Hom( ⊕ q | n H ( K q , T ) ⊗ G q , R ) −→ ∧ r H F ( n ) ( K, T ) . Tensoring both sides with G n defines a mapΠ n : Y n −→ ∧ r H F ( n ) ( K, T ) ⊗ G n . See the proof of Proposition 12.3 below for an explicit description of the mapΠ n . Recall that if m | n ∈ N , then Ψ n , m : Y n → Y m is the map of Definition 6.5. Lemma 12.2.
Suppose that hypotheses (H.1) through (H.7) of § r is the core rank of T . If H F ∗ ) n ( K, T ∗ ) = 0 and m | n , then (Π m ◦ Ψ n , m )( Y n ) = m length( H F ( m ) ∗ ( K,T ∗ )) S ( m ) = S ′ ( m ) . Proof. If H F ∗ ) n ( K, T ∗ ) = 0 then H F n ( K, T ) is free of rank r + ν ( n ) over R byCorollary 3.5(ii). By (6.3) and (6.4) we have( ∩ q | m ker(loc f q | H F n ( K, T ))) ∩ ( ∩ q | ( n / m ) ker(loc tr q | H F n ( K, T ))) = H F ( m ) ( K, T ) . Now the lemma follows from Proposition A.3(ii,iii) applied to the cartesian square H F ( m ) ( K, T ) (cid:31) (cid:127) / / (cid:15) (cid:15) H F n ( K, T ) ⊕ q | m loc f q ⊕ q | ( n / m ) loc tr q (cid:15) (cid:15) (cid:31) (cid:127) / / L q | m ( H ( K q , T ) ⊗ G q ) L q | ( n / m ) H ( K q , T ) (cid:3) Proposition 12.3.
Suppose ǫ = { ǫ n : n ∈ N } is a Stark system of rank r for T .Let Π( ǫ ) denote the collection { ( − ν ( n ) Π n ( ǫ n ) : n ∈ N } . Then: (i) Π( ǫ ) ∈ KS r ( T ) . (ii) If hypotheses (H.1) through (H.7) of § Π( ǫ ) ∈ KS ′ r ( T ) . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 25
Proof.
By definition Π n ( ǫ n ) ∈ ∧ r H F ( n ) ( K, T ) ⊗ G n , so we only need to check thecompatibility (10.2).Suppose nq ∈ N , with n = q · · · q ν ( n ) , and for every i let h i be a generator ofHom( H ( K q i , T ) , R ) and similarly for h and q . Let ̟ tr q ,h := \ h ◦ loc tr q : ∧ t H F nq ( K, T ) → ∧ t − H F n ( K, T ) ,̟ f q ,h := \ h ◦ loc f q : ∧ t H F nq ( K, T ) → ∧ t − H F n ( q ) ( K, T ) ⊗ G q be the maps given by Proposition A.1, for t >
0, and similarly for ̟ tr q i ,h i and ̟ f q i ,h i .Let ǫ nq = d nq ⊗ ( h ∧· · ·∧ h ν ( n ) ∧ h ) with d nq ∈ ∧ r + ν ( nq ) H F nq ( K, T ), and similarly ǫ n = d n ⊗ ( h ∧ · · · ∧ h ν ( n ) ). By definition of Ψ nq , n we have d n = ̟ tr q ,h ( d nq ). If e denotes the edge joining n and nq , then( h ⊗ ψ e nq (Π nq ( ǫ nq ))) = ̟ tr q ,h (( ̟ f q ,h ◦ · · · ◦ ̟ f q ν ( n ) ,h ν ( n ) ◦ ̟ f q ,h )( d nq ))= ( − ν ( n )+1 ( ̟ f q ,h ◦ · · · ◦ ̟ f q ν ( n ) ,h ν ( n ) ◦ ̟ f q ,h ◦ ̟ tr q ,h )( d nq )= ( − ν ( n )+1 ( ̟ f q ,h ◦ · · · ◦ ̟ f q ν ( n ) ,h ν ( n ) ◦ ̟ f q ,h )( d n )= − ̟ f q ,h (( ̟ f q ,h ◦ · · · ◦ ̟ f q ν ( n ) ,h ν ( n ) )( d n ))= − ( h ⊗ ψ e n (Π n ( ǫ n ))) . Since h is an isomorphism, it follows that ψ e nq (Π nq ( ǫ nq )) = − ψ e n (Π n ( ǫ n )), so thecollection { ( − ν ( n ) Π n ( ǫ n ) } is a Kolyvagin system. This proves (i), and (ii) followsfrom Lemma 12.2 (using Lemma 6.9(ii)). (cid:3) Theorem 12.4.
If hypotheses (H.1) through (H.7) of § R -modulemap Π : SS r ( T ) → KS ′ r ( T ) of Proposition 12.3 is an isomorphism.Proof. By Lemma 12.2 and Theorem 6.10, for every n the composition SS r ( T ) Π −−→ KS ′ r ( T ) −→ S ′ ( n )is surjective. Since SS ′ r ( T ) and KS ′ r ( T ) are both free of rank one over R (Theorems6.10 and 11.7(i)), it follows that Π is an isomorphism. (cid:3) Stub Kolyvagin systems and the dual Selmer group
Suppose for this section that R is either a principal artinian local ring or adiscrete valuation ring. We let k := length( R ), so k is finite in the artinian caseand k = ∞ in the discrete valuation ring case.Fix Selmer data ( T, F , P , r ) satisfying hypotheses (H.1) through (H.6), and if R is artinian satisfying (H.7) as well. In this section we prove analogues for stubKolyvagin systems of the results of § κ is primitive if it is primitive as a global section of the stubSelmer sheaf S ′ (Definition 9.4), i.e., if κ generates the R -module KS ′ r ( T ), orequivalently, if κ n generates m λ ( n ) ( ∧ r H F ( n ) ( K, T )) ⊗ G n for every n ∈ N . Corollary 13.1.
Suppose R is a principal artinian ring of length k , and κ ∈ KS ′ r ( T ) . (i) If κ = 0 then length( H F ∗ ( K, T ∗ )) ≤ k − length( Rκ ) = max { i : κ ∈ m i ∧ r H F ( K, T ) } . (ii) If κ is primitive and κ = 0 , then equality holds in (i) . (iii) If κ is primitive and κ = 0 , then length( H F ∗ ( K, T ∗ )) ≥ k .Proof. By Corollary 3.5(iii), S ′ (1) = m λ (1) ∧ r H F ( K, T ) is a cyclic R -module oflength max { , k − length( H F ∗ ( K, T ∗ )) } . Since κ ∈ S ′ (1) by definition, (i) follows.If κ is primitive, then κ generates S ′ (1), which proves (ii) and (iii). (cid:3) The following definition is the analogue for Kolyvagin systems of Definitions 8.1and 8.2 for Stark systems.
Definition 13.2.
Suppose κ ∈ KS r ( T ) is a Kolyvagin system. Define ϕ κ ∈ Maps( N , Z ≥ ∪ {∞} ) by ϕ κ ( n ) := max { j : κ n ∈ m j H F ( n ) ∗ ( K, T ) } . The order ofvanishing of κ isord( κ ) := min { ν ( n ) : n ∈ N , κ n = 0 } = min { i : ∂ϕ κ ( i ) = ∞} . We also define the sequence of elementary divisors d κ ( i ) := ∂ϕ κ ( i ) − ∂ϕ κ ( i + 1) , i ≥ ord( κ ) . Proposition 13.3.
Suppose that κ ∈ KS ′ r ( T ) , ǫ ∈ SS r ( T ) , and κ = Π( ǫ ) . Then ord( κ ) = ord( ǫ ) , ∂ϕ κ ( i ) = ∂ϕ ǫ ( i ) for every i , and d κ ( i ) = d ǫ ( i ) for every i .Proof. Suppose first that R is artinian of length k . Since Π is an isomorphism(Theorem 12.4), we may assume without loss of generality that κ and ǫ generate KS ′ r ( T ) and SS r ( T ), respectively. Recall that µ ( n ) := length( H F ∗ ) n ( K, T ∗ )).For every n ∈ N , Theorem 11.7(ii) shows that κ n generates m λ ( n ) S ( n ), andTheorem 6.10 shows that ǫ n generates m µ ( n ) Y n . Thus ∂ϕ κ ( i ) = ( ∂λ ( i ) if ∂λ ( i ) < k, ∞ if ∂λ ( i ) ≥ k, ∂ϕ ǫ ( i ) = ( ∂µ ( i ) if ∂µ ( i ) < k, ∞ if ∂µ ( i ) ≥ k. By Proposition 8.3, ∂λ ( i ) = ∂µ ( i ) for every i , and all the equalities of the Proposi-tion follow.The case where R is a discrete valuation ring follows from the artinian case asin the proof of Proposition 8.7. (cid:3) Theorem 13.4.
Suppose R is a discrete valuation ring, κ ∈ KS ′ r ( T ) and κ = 0 .Then: (i) the sequence ∂ϕ κ ( t ) is nonincreasing, and finite for t ≥ ord( κ ) , (ii) the sequence d κ ( i ) is nonincreasing, nonnegative, and finite for i ≥ ord( κ ) , (iii) ord( κ ) and the d κ ( i ) are independent of the choice of nonzero κ ∈ KS ′ r ( T ) , (iv) corank R ( H F ∗ ( K, T ∗ )) = ord( κ ) , (v) H F ∗ ( K, T ∗ ) / ( H F ∗ ( K, T ∗ )) div ∼ = ⊕ i ≥ ord( κ ) R/ m d κ ( i ) , (vi) length R ( H F ∗ ( K, T ∗ ) / ( H F ∗ ( K, T ∗ )) div ) = ∂ϕ κ (ord( κ )) − ∂ϕ κ ( ∞ ) , where ∂ϕ κ ( ∞ ) := lim t →∞ ∂ϕ κ ( t )(vii) κ is primitive if and only if ∂ϕ κ ( ∞ ) = 0 , (viii) length( H F ∗ ( K, T ∗ )) is finite if and only if κ = 0 , (ix) length( H F ∗ ( K, T ∗ )) ≤ ∂ϕ κ (0) = max { s : κ ∈ m s ∧ r H F ( K, T ) } , withequality if and only if κ is primitive.Proof. By Theorem 12.4, there is a (unique) ǫ ∈ SS r ( T ) such that Π( ǫ ) = κ . ByProposition 13.3, all the invariants of Definition 13.2 attached to κ are equal to thecorresponding invariants of ǫ . Now the theorem follows from Theorem 8.9. (cid:3) ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 27
Proof of Theorem 11.6
Keep the notation of §
11, so R is principal and artinian of length k , hypotheses(H.1) through (H.7) hold. In particular we assume that r = χ ( T ), the core rank of T . Lemma 14.1.
The sheaf S ′ is locally cyclic.Proof. By Corollary 3.5(iii), for every n ∈ N the stalk S ′ ( n ) is a cyclic R -module.By Definition 11.3 and Proposition 11.2 the vertex-to-edge maps ψ e n are all surjec-tive, and so the edge stalks S ′ ( e ) are all cyclic as well. (cid:3) Lemma 14.2.
Suppose n is a core vertex, and q ∈ P does not divide n . Let e denote the edge joining n and nq . Then the following are equivalent: (i) loc q : H F ( n ) ( K, T )[ m ] → H ( K q , T ) is nonzero, (ii) nq is a core vertex and both maps ψ e n : S ( n ) → S ( e ) , ψ e nq : S ( nq ) → S ( e ) are isomorphisms.Proof. Suppose that (i) holds. Since I q = 0 by (H.7), Lemma 1.3(ii) shows that H ( K q , T ) is free of rank one over R . Since n is a core vertex, H F ( n ) ( K, T ) is afree R -module of rank r . In particular H F ( n ) ( K, T )[ m ] = m k − H F ( n ) ( K, T ), andit follows that the localization map loc q : H F ( n ) ( K, T ) → H ( K q , T ) is surjective.By Proposition A.1, it follows that ψ e n is an isomorphism.Further, since loc q : H F ( n ) ( K, T ) → H ( K q , T ) is surjective, and H ( K q , T )is free of rank one over R , and H F ( n ) ∗ ( K, T ∗ ) = 0, [MR1, Lemma 4.1.6] showsthat nq is a core vertex and loc q : H F ( nq ) ( K, T ) → H ( K q , T ) is surjective. NowProposition A.1 shows that that ψ e nq is an isomorphism. Thus (ii) holds.Conversely, if ψ e n is an isomorphism then Proposition A.1 shows that the maploc q : H F ( n ) ( K, T ) → H ( K q , T ) is surjective, and since H ( K q , T ) is free of rankone over R it follows that loc q is not identically zero on H F ( n ) ( K, T )[ m ]. Thus (ii)implies (i). (cid:3) Recall that ¯ T := T / m T . Proposition 14.3.
Suppose n ∈ N and λ ( n , ¯ T ∗ ) > . Then there is a q ∈ P primeto n such that λ ( nq , ¯ T ∗ ) < λ ( n , ¯ T ∗ ) and ψ e n : S ′ ( n ) → S ′ ( e ) is an isomorphism,where e is the edge joining n and nq . Let ¯ λ ( n ) := dim k H F ( n ) ∗ ( K, ¯ T ∗ ). By Proposition 3.3(ii), we have λ ( n ) = 0 if andonly if ¯ λ ( n ) = 0. Proof.
By [MR1, Proposition 3.6.1] we can use the Cebotarev theorem to choose aprime q ∈ P such that the localization maps m k − H F ( n ) ( K, T ) → H ( K q , T ) , H F ( n ) ∗ ( K, T ∗ )[ m ] → H ( K q , T ∗ )are both nonzero. (Note that m k − H F ( n ) ( K, T ) = 0 by Corollary 3.5(iii).) Thenby Poitou-Tate global duality (see for example [MR1, Lemma 4.1.7(iv)]), we have¯ λ ( nq ) < ¯ λ ( n ). Further, we have that localization H F ( n ) ( K, T ) → H ( K q , T ) issurjective, so by Proposition A.1(ii) d loc q : ∧ r H F ( n ) ( K, T ) −→ H ( K q , T ) ⊗ ( ∧ r − H F q ( n ) ( K, T )) is surjective as well. Since S ′ ( e ) is defined to be the image of S ′ ( n ) := m λ ( n ) ( ∧ r H F ( n ) ( K, T )) ⊗ G n under the upper maps of (10.2), we deduce that S ′ ( e ) = m λ ( n ) H ( K q , T ) ⊗ ( ∧ r − H F q ( n ) ( K, T )) ⊗ G nq . Thus length R ( S ′ ( e )) ≥ k − λ ( n ) = length R ( S ′ ( n )) , the equality by Corollary 3.5(iii). Since the map S ′ ( n ) → S ′ ( e ) is surjective bydefinition, it must be an isomorphism. (cid:3) Theorem 14.4.
Suppose n , n ′ are core vertices. Then there is a path n = n e n e · · · e t n t = n ′ in X such that every n i is a core vertex and all of the maps ψ e i +1 n i and ψ e i n i areisomorphisms.Proof. When χ ( T ) = 1, this is [MR1, Theorem 4.3.12]. The general case can beproved in the same way, but instead we will prove it here by induction on r := χ ( T ).Denote by ¯ F the induced Selmer structure on ¯ T . By Proposition 3.3 and thedefinition of core vertices we see that the Selmer sheaves S ( T, F , P ) and S ( ¯ T , ¯ F , P ) havethe same core vertices and the same core rank r (see also [MR1, Theorem 4.1.3]).Since r >
0, we can fix nonzero classes c ∈ H F ( n ) ( K, ¯ T ) and c ′ ∈ H F ( n ′ ) ( K, ¯ T ).By [MR1, Proposition 3.6.1], we can use the Cebotarev theorem to choose q ∈ P ,not dividing nn ′ , such that the localizations c q and c ′ q are both nonzero.Note that the Selmer triple ( ¯ T , ¯ F q , P − { q } ) also satisfies hypotheses (H.1)through (H.6) (the only one of those conditions that depends on the Selmer struc-ture is (H.5), and (H.5) is vacuous when we work over R/ m ). By our choice of q ,both localization mapsloc q : H F ( n ) ( K, ¯ T ) → H ( K q , ¯ T ) , loc q : H F ( n ′ ) ( K, ¯ T ) → H ( K q , ¯ T )are nonzero, and H ( K q , ¯ T ) is a one-dimensional R/ m -vector space, so both mapsare surjective. Since n and n ′ are core vertices for ( ¯ T , ¯ F ), it follows thatdim R/ m H F q ( n ) ( K, ¯ T ) = dim R/ m H F q ( n ′ ) ( K, ¯ T ) = r − H F q ( n ) ∗ ( K, ¯ T ∗ ) = H F q ( n ′ ) ∗ ( K, ¯ T ∗ ) = 0 . In particular we deduce that χ ( ¯ T , ¯ F q ) = r −
1, and that n , n ′ are core verticesfor the sheaf S ¯ T , ¯ F q . By our induction hypotheses we conclude that there is a path n = n , n , . . . , n t = n ′ from n to n ′ in X such that every n i is prime to q , every n i is a core vertex for S ¯ T , ¯ F q , and every vertex-to-edge map (for S ¯ T , ¯ F q ) along the pathis an isomorphism. We will show that every n i is a core vertex for S T, F , and everyvertex-to-edge map (for S T, F ) along the path is an isomorphism. This will provethe theorem.Fix i , 0 ≤ i ≤ t . The exact sequence0 −→ H F q ( n i ) ( K, ¯ T ) −→ H F ( n i ) ( K, ¯ T ) loc q −−→ H ( K q , ¯ T )shows that dim R/ m H F ( n i ) ( K, ¯ T ) ≤ r . Then Corollary 3.5(i) (applied to ¯ T , ¯ F , and R/ m ) shows that n i is a core vertex of S ¯ T , ¯ F , and hence is a core vertex of S T, F . ONTROLLING SELMER GROUPS IN THE HIGHER CORE RANK CASE 29
Further, suppose l is a prime such that n i ± = n i l , and let e be the edgejoining those two vertices. By assumption, the maps S ¯ T , ¯ F q ( n i ) → S ¯ T , ¯ F q ( e ) and S ¯ T , ¯ F q ( n i l ) → S ¯ T , ¯ F q ( e ) are isomorphisms, so by Lemma 14.2 the localization map H F q ( n i ) ( K, ¯ T ) → H ( K l , ¯ T ) is nonzero. But H F q ( n i ) ( K, ¯ T ) ⊂ H F ( n i ) ( K, ¯ T ) = H F ( n i ) ( K, T )[ m ] , so loc l : H F ( n i ) ( K, T )[ m ] → H ( K l , T ) is nonzero, so by Lemma 14.2 both of themaps ψ en i and ψ en i ± are isomorphisms. This completes the proof. (cid:3) Corollary 14.5.
There are core vertices. More precisely: (i) for every n ∈ N there is an n ′ ∈ N prime to n , with ν ( n ′ ) = ¯ λ ( n ) , suchthat nn ′ is a core vertex, (ii) min { ν ( n ) : n is a core vertex } = dim R/ m H F ∗ ( K, T ∗ )[ m ] . Proof.
Choose n ∈ N . For every n ′ ∈ N prime to n , global duality (see for example[MR1, Lemma 4.1.7(i)]) shows that(14.6) ¯ λ ( nn ′ ) ≥ ¯ λ ( n ) − ν ( n ′ ) . Applying Proposition 14.3, we can construct n = n , n , n , . . . ∈ N inductively,with n i +1 = n i q i for some prime q i ∈ N and ¯ λ ( n i +1 ) < ¯ λ ( n i ), until we reach n d ∈ N with ¯ λ ( n d ) = 0. Then H F ( n d ) ∗ ( K, T ∗ )[ m ] = H F ( n d ) ∗ ( K, ¯ T ∗ ) = 0, so n d is a corevertex. Setting n ′ := n d / n we have ν ( n ′ ) = d ≤ ¯ λ ( n ) = dim R/ m H F ∗ ( K, T ∗ )[ m ] . By (14.6), since ¯ λ ( nn ′ ) = 0 we have ν ( n ′ ) ≥ ¯ λ ( n ), and so ν ( n ′ ) = ¯ λ ( n ). This proves(i), and applying (i) with n = 1 and (14.6) proves (ii). (cid:3) Proof of Theorem 11.6.
Theorem 11.6(i) is Corollary 14.5, and Theorem 11.6(ii) isTheorem 14.4. Lemma 14.1 says that S ′ is locally cyclic. To complete the proof ofTheorem 11.6 we need only show that every core vertex is a hub of S ′ .Fix a core vertex n , and let n ∈ N be any other vertex. We will show byinduction on ¯ λ ( n ) that there is an S ′ -surjective path from n to n .If ¯ λ ( n ) = 0, then n is also a core vertex and the desired surjective path exists byTheorem 14.4.Now suppose ¯ λ ( n ) >
0. Use Proposition 14.3 to find q ∈ P not dividing n suchthat ¯ λ ( nq ) < ¯ λ ( n ) and ψ e n : S ′ ( n ) → S ′ ( e ) is an isomorphism, where e is the edgejoining n and nq . By induction there is an S ′ -surjective path from n to nq , and ifwe adjoin to that path the edge e , we get an S ′ -surjective path from n to n . (cid:3) Appendix A. Some exterior algebra
Suppose for this appendix that R is a local principal ideal ring with maximalideal m . Proposition A.1.
Suppose → N → M ψ −→ C is an exact sequence of finitely-generated R -modules, with C free of rank one, and r ≥ . Then there is a uniquemap ˆ ψ : ∧ r M −→ C ⊗ ∧ r − N such that (i) the composition ∧ r M ˆ ψ −→ C ⊗ ∧ r − N → C ⊗ ∧ r − M is given by m ∧ · · · ∧ m r r X i =1 ( − i +1 ψ ( m i ) ⊗ ( m ∧ · · · ∧ m i − ∧ m i +1 · · · ∧ m r ) , (ii) the image of ˆ ψ is the image of ψ ( M ) ⊗ ∧ r − N → C ⊗ ∧ r − N .If M is free of rank r over R , then ˆ ψ is an isomorphism if and only if ψ is surjective.Proof. Since R is principal, we can “diagonalize” ψ and write M = Rm ⊕ N and N = Im ⊕ N where N ⊂ N , m ∈ M is such that ψ ( m ) generates ψ ( M ), and I isan ideal of R . In particular we have 0 = ψ ( N ) = Iψ ( M ).The formula of (i) gives a well-defined R -module homomorphism ˆ ψ : ∧ r M → ψ ( M ) ⊗ ∧ r − M . Consider the diagram ∧ r M ˆ ψ / / ψ ( M ) ⊗ ∧ r − M / / C ⊗ ∧ r − Mψ ( M ) ⊗ ∧ r − N η / / η O O C ⊗ ∧ r − N O O with maps induced by the inclusions ψ ֒ → C and N ֒ → M . We will show thatimage( ˆ ψ ) ⊂ image( η ) and ker( η ) ⊂ ker( η ). Then ˆ ψ := η ◦ η − ◦ ˆ ψ is welldefined and satisfies (i) and (ii).Since M = Rm ⊕ N , we have that the image ˆ ψ ( ∧ r M ) is generated by monomials ψ ( m ) ⊗ n ∧ · · · ∧ n r − with n i ∈ N , so image( ˆ ψ ) ⊂ image( η ).We also have ∧ r − N = ( Im ⊗ ∧ r − N ) ⊕ ∧ r − N , ∧ r − M = ( Rm ⊗ ∧ r − N ) ⊕ ∧ r − N . Therefore, since Iψ ( M ) = 0,ker( η ) = ker( ψ ( M ) ⊗ Im ⊗ ∧ r − N → ψ ( M ) ⊗ Rm ⊗ ∧ r − N )= ψ ( M ) ⊗ Im ⊗ ∧ r − N . We further have(A.2) η ( ψ ( M ) ⊗ Im ⊗ ∧ r − N ) = 0 . Thus ker( η ) ⊂ ker( η ), so ˆ ψ is well-defined and has properties (i) and (ii). Unique-ness follows from the fact that by (A.2) η ( ψ ( M ) ⊗ ∧ r − N ) = η ( ψ ( M ) ⊗ ∧ r − N )injects into C ⊗ ∧ r − M .The final assertion follows easily from the definition of ˆ ψ above. (cid:3) If M is an R -module, let M • := Hom( M, R ). Proposition A.3.
Suppose R is artinian and there is a cartesian diagram of R -modules M (cid:31) (cid:127) / / (cid:15) (cid:15) M h (cid:15) (cid:15) C (cid:31) (cid:127) / / C where C and C are free R -modules of finite rank, and the horizontal maps areinjective. (i) Suppose r ≥ and s i = rank R ( C i ) . There is a canonical R -module homo-morphism ∧ r + s M ⊗ ∧ s C • −→ ∧ r + s M ⊗ ∧ s C • defined as follows. If m ∈ ∧ r + s M , ψ , . . . , ψ s is a basis of C • such that ψ s +1 , . . . , ψ s is a basis of ( C /C ) • , and h i = ψ i ◦ h , then m ⊗ ( ψ ∧ · · · ∧ ψ s ) (ˆ h s +1 ◦ · · · ◦ ˆ h s )( m ) ⊗ ( ψ ∧ · · · ∧ ψ s ) with ˆ h i as in Proposition A.1. This is independent of the choice of the ψ i . (ii) If M is free of rank r + s over R , then the image of the map of (i) is m length( M ) − ( r + s )length( R ) ∧ r + s M ⊗ ∧ s C • . (iii) If M (cid:31) (cid:127) / / (cid:15) (cid:15) M (cid:15) (cid:15) C (cid:31) (cid:127) / / C is another such cartesian square, then the triangle ∧ r + s M ⊗ ∧ s C • / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ∧ r + s M ⊗ ∧ s C • ∧ r + s M ⊗ ∧ s C • ♠♠♠♠♠♠♠♠♠♠♠♠ induced by the maps of (i) commutes. (iv) Suppose there is an exact sequence → M → M → C , where C is freeof rank s over R . Then for every r ≥ , the map of (i) (with C = 0 and C = C ) is a canonical map ∧ r + s M ⊗ ∧ s C • → ∧ r M .Proof. Since the square is cartesian, and by our choice of the ψ i , we have(A.4) ker( ⊕ i>s h i ) = h − ( C ) = M . Applying Proposition A.1 repeatedly shows that the map defined in (i) takes valuesin ∧ r + s M ⊗ ∧ s C • . It is straightforward to check that this map is independentof the choice of the ψ i . This proves (i), and (iv) is just a special case of (i).Suppose now that M is free of rank r + s , and let s := s − s . Choose an R -basis η , . . . , η r + s of M • such that the span of η , . . . , η s contains h s +1 , . . . , h s ,i.e., there is an s × s matrix A = [ a ij ] with a ij ∈ R such that h s + j = P i a ij η i . Let N := ∩ si =1 ker( η i ). Then N is free over R of rank r + s , and we have a split exactsequence of free modules0 −→ N −→ M ⊕ i ≤ s η i −−−−→ R s −→ . It follows that the composition ˆ η ◦ · · · ◦ ˆ η s : ∧ r + s M → ∧ r + s N of maps given byProposition A.1 is an isomorphism.We also have ˆ h s +1 · · · ◦ ˆ h s = det( A ) ˆ η ◦ · · · ◦ ˆ η s , and N ⊂ M by (A.4). Since N is free, there is a noncanonical splitting M ∼ = N ⊕ M /N, so the map m length( M /N ) ∧ r + s N −→ m length( M /N ) ∧ r + s M induced by the inclusion N ֒ → M is surjective. Finally,det( A ) R = m length( M /N ) = m length( M ) − ( r + s )length( R ) , and combining these facts proves (ii).Assertion (iii) follows from the independence of the choice of the ψ i . Choosea basis ψ , . . . , ψ s s of C • such that ψ s +1 , . . . , ψ s is a basis of ( C /C ) • and ψ s +1 , . . . , ψ s is a basis of ( C /C ) • . Then ψ s +1 | C , . . . , ψ s | C is a basis of( C /C ) • , and (iii) just reduces to the statement that( ˆ ψ s +1 ◦ · · · ◦ ˆ ψ s ) ◦ ( ˆ ψ s +1 ◦ · · · ◦ ˆ ψ s ) = ( ˆ ψ s +1 ◦ · · · ◦ ˆ ψ s ) . (cid:3) Erratum to [MR1] . We thank Cl´ement Gomez for pointing out an error in thestatement of [MR1, Lemma 2.1.4]. The correct statement (which is all that wasused elsewhere in [MR1]) should be:
Lemma 2.1.4 . If ( T / m T ) G Q = 0 then ( T /IT ) G Q = 0 for every ideal I of R . References [MR1] B. Mazur, K. Rubin, Kolyvagin systems.
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Department of Mathematics, Harvard University, Cambridge, MA 02138 USA
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