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Finding large Selmer rank viaan arithmetic theory of local constants By Barry Mazur and Karl Rubin* p-extension of K that is unramified at all primes where E has bad reduction and that is

Annals of Mathematics

Finding large Selmer rank via an arithmetic theory of local constants

By Barry Mazur and Karl Rubin*

Finding large Selmer rank via

an arithmetic theory of local constants

By Barry Mazur and Karl Rubin*

p-extension of K that is unramified at all primes where E has bad reduction

and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal( K − /K) by inversion) We prove (under mild hypotheses

on p) that if the Z p -rank of the pro-p Selmer group S p (E/K) is odd, then

rankZp S p (E/F ) ≥ [F : K] for every finite extension F of K in K −.

Even in cases where one cannot prove that the L-function L(E/K, χ; s)

has an analytic continuation and functional equation, one still has a conjectural

functional equation with a sign ε(E/K, χ) :=

v ε(E/K v , χ v) =±1 expressed

as a product over places v of K of local ε-factors If ε(E/K, χ) = −1, then a

generalized Parity Conjecture predicts that the rank of the χ-part E(F ) χ of

the Gal(F/K)-representation space E(F ) ⊗ ¯Q is odd, and hence positive If

[F : K] is odd and F/K is unramified at all primes where E has bad reduction, then ε(E/K, χ) is independent of χ, and so the Parity Conjecture predicts that

if the rank of E(K) is odd then the rank of E(F ) is at least [F : K].

*The authors are supported by NSF grants DMS-0403374 and DMS-0457481, tively.

respec-Motivated by the analytic theory of the preceding paragraph, in this paper

we prove unconditional parity statements, not for the Mordell-Weil groups

Shafarevich-Tate conjecture implies that E(F ) χand S p(E/F ) χhave the same

rank.) More specifically, given the data (E, K/k, χ) where the order of χ is a power of an odd prime p, we define (by cohomological methods) local invariants

δ v ∈ Z/2Z for the finite places v of K, depending only on E/K v and χ v The δ v should be the (additive) counterparts of the ratios ε(E/K v , χ v )/ε(E/K v , 1) of

the local ε-factors The δv vanish for almost all v, and if Zp[χ] is the extension

of Zp generated by the values of χ, we prove (see Theorem 6.4):

Theorem A If the order of χ is a power of an odd prime p, then

rankZp S p(E/K) − rankZp [χ] S p(E/F ) χ ≡

v

δ v (mod 2).

Despite the fact that the analytic theory, which is our guide, predicts the

values of the local terms δ v, Theorem A would be of limited use if we could

not actually compute the δ v ’s We compute the δ v’s in substantial generality

in Section 5 and Section 6 This leads to our main result (Theorem 7.2), which

we illustrate here with a weaker version

Theorem B Suppose that p is an odd prime, [F : K] is a power of p, F/K is unramified at all primes where E has bad reduction, and all primes above p split in K/k If rankZp S p(E/K) is odd, then rankZp [χ] S p(E/F ) χ is odd for every character χ of G, and in particular rankZp S p (E/F ) ≥ [F : K].

If K is an imaginary quadratic field and F/K is unramified outside of p,

then Theorem B is a consequence of work of Cornut [Co] and Vatsal [V] Inthose cases the bulk of the Selmer module comes from Heegner points

Nekov´a˘r [N2, Th 10.7.17] proved Theorem B in the case where F is

con-tained in a Zp-power extension of K, under the assumption that E has ordinary

reduction at all primes above p We gave in [MR3] an exposition of a weaker

version of Nekov´a˘r’s theorem, as a direct application of a functional equationthat arose in [MR2] (which also depends heavily on Nekov´a˘r’s theory in [N2]).The proofs of Theorems A and B proceed by methods that are very differ-ent from those of Cornut, Vatsal, and Nekov´a˘r, and are comparatively short

We emphasize that our results apply whether E has ordinary or supersingular

reduction at p, and they apply even when F/K is not contained in a Z p-power

extension of K (but we always assume that F/k is dihedral).

This extra generality is of particular interest in connection with the searchfor new Euler systems, beyond the known examples of Heegner points Let

c,p be the maximal “generalized dihedral” p-extension of K (i.e.,

the maximal abelian p-extension of K, Galois over k, such that c acts on

consist of Selmer classes cF ∈ S p (E/F ) for every finite extension F of K in

K −, with certain compatibility relations between c

F and cF  when F ⊂ F  (see

for example [R] §9.4) A necessary condition for the existence of a nontrivial

Euler system is that the Selmer modules S p (E/F ) are large, as in the

conclu-sion of Theorem B It is natural to ask whether, in these large Selmer modules

S p (E/F ), one can find elements c F that form an Euler system

Outline of the proofs Suppose for simplicity that E(K) has no p-torsion.

The group ring Q[Gal(F/K)] splits into a sum of irreducible rational sentations Q[Gal(F/K)] = ⊕ L ρ L , summing over all cyclic extensions L of K

repre-in F , where ρ L ⊗ ¯ Q is the sum of all characters χ whose kernel is Gal(F/L).

Corresponding to this decomposition there is a decomposition (up to isogeny)

of the restriction of scalars ResF K E into abelian varieties over K

ResF K E ∼ ⊕ L A L

This gives a decomposition of Selmer modules

S p(E/F ) ∼=S p((Res F K E)/K) ∼=⊕ L S p(A L /K)

where for every L, S p (A L /K) ∼ = (ρL ⊗ Q p)d L for some d L ≥ 0 Theorem B will

follow once we show that dL ≡ rankZp S p(E/K) (mod 2) for every L More

precisely, we will show (see Section 4 for the ideal p of EndK (A L), Section 2for the Selmer groups Selp and Selp, and Definition 3.6 for S p) that

rankZp S p (E/K) ≡ dimFpSelp (E/K) ≡ dimFpSelp(A L /K) ≡ d L (mod 2).

(1)

The key step in our proof is the second congruence of (1) We will see

(Proposition 4.1) that E[p] ∼ = AL [p] as G K-modules, and therefore the Selmergroups Selp(E/K) and Selp(A L /K) are both contained in H1(K, E[p]) By

comparing these two subspaces we prove (see Theorem 1.4 and Corollary 4.6)that

dimFpSelp (E/K) − dimFpSelp(A L /K) ≡

v

δ v (mod 2)

summing the local invariants δ v of Definition 4.5 over primes v of K We show how to compute the δ v in terms of norm indices in Section 5 and Section 6,with one important special case postponed to Appendix B

The first congruence of (1) follows easily from the Cassels pairing for E

(see Proposition 2.1) The final congruence of (1) is more subtle, because in

general A L will not have a polarization of degree prime to p, and we deal with this in Appendix A (using the dihedral nature of L/k).

In Section 7 we bring together the results of the previous sections to proveTheorem 7.2, and in Section 8 we discuss some special cases

Generalizations All the results and proofs in this paper hold with E replaced by an abelian variety with a polarization of degree prime to p.

If F/K is not a p-extension, then the proof described above breaks down Namely, if χ is a character whose order is not a prime power, then χ is not

congruent to the trivial character modulo any prime of ¯Q However, by writing

χ as a product of characters of prime-power order, we can apply the methods

of this paper inductively To do this we must use a different prime p at each step, so it is necessary to assume that if A is an abelian variety over K and

is independent of p (This would follow, for example, from the

Shafarevich-Tate conjecture.) To avoid obscuring the main ideas of our arguments, we willinclude those details in a separate paper

The results of this paper can also be applied to study the growth of Selmer

rank in nonabelian Galois extensions of order 2p n with p an odd prime This

will be the subject of a forthcoming paper

field will mean a finite extension of Q in ¯Q If K is a number field then

G K := Gal( ¯Q/K).

1 Variation of Selmer rank

Let K be a number field and p an odd rational prime Let W be a

finite-dimensional Fp-vector space with a continuous action of GKand with a perfect,

skew-symmetric, G K-equivariant self-duality

where μ p is the G K -module of p-th roots of unity in ¯Q.

Theorem 1.1 For every prime v of K, Tate’s local duality gives a perfect symmetric pairing

, v : H1(K v , W ) × H1(K v , W ) −→ H2(K v , μ p) = Fp

Proof See [T1].

un-ramified extension of Kv A Selmer structure F on W is a collection of F

p-subspaces

H F1(K v , W ) ⊂ H1(K v , W )

for every prime v of K, such that H1

F (K v , W ) = H1(Kur

v /K v , W I v) for all but

finitely many v, where I v := G Kur⊂ G K is the inertia group IfF and G are

Selmer structures on W , we define Selmer structures F + G and F ∩ G by

H F+G1 (K v , W ) := H F1(K v , W ) + H G1(K v , W ),

H F∩G1 (Kv , W ) := H F1(Kv , W ) ∩ H1

G (Kv , W ), for every v We say that F ≤ G if H1

is its own orthogonal complement under the Tate pairing of Theorem 1.1

H F1(K, W ) := ker(H1(K, W ) −→v H1(K v , W )/H1

F (K v , W )).

Thus H1

F (K, W ) is the collection of classes whose localizations lie in H F1(K v , W )

for every v If F ≤ G then H1

F (K, W ) ⊂ H1

G (K, W ).

For the basic example of the Selmer groups we will be interested in, where

W is the Galois module of p-torsion on an elliptic curve, see Section 2.

Proposition 1.3 Suppose that F, G are self-dual Selmer structures on

W , and S is a finite set of primes of K such that H F1(Kv , W ) = H G1(Kv , W )

F+G (K, W ) → B Since F and

G are self-dual, Poitou-Tate global duality (see for example [MR1, Th 2.3.4])

shows that the Tate pairings of Theorem 1.1 induce a nondegenerate, ric self-pairing

(1.1)

and C is its own orthogonal complement under this pairing.

Let C F (resp C G) denote the image of ⊕ v∈S H F1(K v , W ) (resp.

their own orthogonal complements under (1.1) In particular we have

dimFp C = dimFp C F = dimFp C G = 12dimFp B.

Since

C ∼ = H F+G1 (K, W )/H F∩G1 (K, W )

C F ∼=⊕ v∈S H1

F (K v , W )/H F∩G1 (K v , W ),

this proves (i)

The proof of (ii) uses an argument of Howard ([Hb, Lemma 1.5.7]) We

by [x, y] := x F , y G , where , is the pairing (1.1) Since the subspaces C,

C F , and C G are all isotropic, for all x, y, ∈ H1

F+G (K, W ) we have

0 =x S , y S = x F + x G , y F + y G = x F , y G + x G , y F = [x, y] + [y, x]

so the pairing (1.2) is skew-symmetric

We see easily that H F1(K, W ) + H G1(K, W ) is in the kernel of the pairing [ , ] Conversely, if x is in the kernel of this pairing, then for every y ∈

G (K, W ), i.e., x ∈ H1

F (K, W ) + H G1(K, W ) Therefore (1.2)

induces a nondegenerate, skew-symmetric, Fp-valued pairing on

H F+G1 (K, W )/(H F1(K, W ) + H G1(K, W )).

Since p is odd, a well-known argument from linear algebra shows that the

dimension of this Fp-vector space must be even This proves (ii)

Theorem 1.4 Suppose that F and G are self-dual Selmer structures on

W , and S is a finite set of primes of K such that H F1(K v , W ) = H G1(K v , W )

Proof We have (modulo 2)

dimFp H F1(K, W ) − dimFp H G1(K, W ) ≡ dimFp H F1(K, W ) + dimFp H G1(K, W )

2 Example: elliptic curves

Let K be a number field If A is an abelian variety over K, and

H1(K, E[α]), sitting in an exact sequence

(2.1)

whereX(A/K) is the Shafarevich-Tate group of A over K If p is a prime we

let Selp ∞ (A/K) be the direct limit of the Selmer groups Sel p n (A/K), and then

induces a perfect G K -equivariant self-duality E[p] × E[p] → μ p Thus we are

in the setting of Section 1

We define a Selmer structure E on E[p] by taking H1

E (K v , E[p]) to be the

image of E(Kv )/pE(Kv) under the Kummer injection

E(K v)/pE(Kv )  → H1(Kv , E[p])

for every v By Lemma 19.3 of [Ca2], H E1(Kv , E[p]) = H1(K vur/K v , E[p]) if

v  p and E has good reduction at v With this definition the Selmer group

H E1(K, E[p]) is the usual p-Selmer group Sel p (E/K) of E as in (2.1).

If C is an abelian group, we let Cdivdenote its maximal divisible subgroup.Proposition 2.1 The Selmer structure E on E[p] defined above is self- dual, and

corankZpSelp∞ (E/K) ≡ dimFp H E1(K, E[p]) − dimFp E(K)[p] (mod 2).

d := dimFp(Selp∞ (E/K)/(Selp ∞ (E/K))div)[p]

= dimFp(X(E/K)[p ∞ ]/(X(E/K)[p ∞])div)[p].

The Cassels pairing [Ca1] shows that d is even Further,

corankZpSelp ∞ (E/K) = dimFpSelp ∞ (E/K)div[p]

= dimFpSelp ∞ (E/K)[p] − d

= rankZE(K) + dimFpX(E/K)[p] − d

by (2.2) with A = E On the other hand, (2.1) shows that

dimF H E1(K, E[p]) = rankZE(K) + dimF E(K)[p] + dimF X(E/K)[p]

so we conclude

corankZpSelp ∞ (E/K) = dimFp H E1(K, E[p]) − dimFp E(K)[p] − d.

This proves the proposition

3 Decomposition of the restriction of scalars

Much of the technical machinery for this section will be drawn from tions 4 and 5 of [MRS]

Sec-Suppose F/K is a finite abelian extension of number fields, G := Gal(F/K), and E is an elliptic curve defined over K We let Res F K E denote the Weil re-

striction of scalars ([W, §1.3]) of E from F to K, an abelian variety over K

with the following properties

Proposition 3.1 (i) For every commutative K-algebra X there is a

canonical isomorphism

(ResF K E)(X) ∼ = E(X ⊗ K F ) functorial in X In particular, (Res F K E)(K) ∼ = E(F ).

(ii) The action of G on the right-hand side of (i) induces a canonical inclusion

Z[G]  → End K(Res F K E).

(iii) For every prime p there is a natural G-equivariant isomorphism,

com-patible with the isomorphism (Res F K E)(K) ∼ = E(F ) of (i),

Selp∞((ResF K E)/K) ∼= Selp∞ (E/F )

where G acts on the left-hand side via the inclusion of (ii).

Proof Assertion (i) is the universal property satisfied by the restriction

of scalars [W], and (ii) is (for example) (4.2) of [MRS] For (iii), Theorem2.2(ii) and Proposition 4.1 of [MRS] give an isomorphism

(ResF K E)[p ∞ ] ∼ = Z[G] ⊗ E[p ∞]

that is G-equivariant (with G acting on Res F K E via the map of (ii) and by

multiplication on Z[G]) and G K -equivariant (with γ ∈ G K acting by γ −1 ⊗ γ

on Z[G] ⊗ E[p ∞]) Then by Shapiro’s lemma (see for example PropositionsIII.6.2, III.5.6(a), and III.5.9 of [Br]), there is a G-equivariant isomorphism

H1(K, (Res F K E)[p ∞])−→ H ∼ 1(F, E[p ∞ ]).

(3.1)

Using (i) with X = Kv, along with the analogue of (3.1) for the local extensions

(F ⊗ K K v )/K v for every prime v of K, one can show that the isomorphism

(3.1) restricts to the isomorphism of (iii)

Definition 3.2 Let Ξ := {cyclic extensions of K in F }, and if L ∈ Ξ

let ρ L be the unique faithful irreducible rational representation of Gal(L/K) Then ρ L ⊗ ¯ Q is the direct sum of all the injective characters Gal(L/K)  → ¯Q×

The correspondence L ↔ ρ Lis a bijection between Ξ and the set of irreducible

rational representations of G Thus the semisimple group ring Q[G]

decom-poses

Q[G] ∼= 

L∈Ξ Q[G] L

(3.2)

where Q[G] L ∼ = ρL is the ρ L -isotypic component of Q[G] As a field, Q[G] L is

isomorphic to the cyclotomic field of [L : K]-th roots of unity.

Let R L be the maximal order of Q[G] L If [L : K] is a power of a prime p, then R L has a unique prime ideal above p, which we denote by p L Also define

as given by Definition 1.1 of [MRS] (see also [Mi,§2]) The abelian variety A L

is defined over K, and its K-isomorphism class is independent of the choice of abelian extension F containing L (see Remark 4.4 of [MRS]) If L = K then

A K = E By Proposition 4.2(i) of [MRS], the inclusion I L  → Z[G] induces an

Let T p (E) denote the Tate module lim ←− E[p n ], and similarly for T p (A L)

The following theorem summarizes the properties of the abelian varieties A L

that we will need

Theorem 3.4 Suppose p is a prime, n ≥ 1, and L/K is a cyclic sion of degree p n Then:

exten-(i) I L= pp L n −1 in R L

(ii) The inclusion Z[G]  → End K(ResF K E) of Proposition 3.1(ii) induces (via

(3.3)) a ring homomorphism Z[G] → End K (A L ) that factors

where the first map is induced by the projection in (3.2).

(iii) Let M be the unique extension of K in L with [L : M ] = p For

ev-ery commutative K-algebra X, the isomorphism of Proposition 3.1(i) stricts (using (3.3)) to an isomorphism, functorial in X,

γ −1 ⊗ γ, and R L -linear, where R L acts on A L via the map of (ii) Proof Assertions (i), (ii), and (iv) are Lemma 5.4(iv), Theorem 5.5(iv),

and Theorem 2.2(iii), respectively, of [MRS] ((iv) is also Proposition 6(b) of[Mi]) Assertion (iii) is Theorem 5.8(ii) of [MRS]

Theorem 3.5 The inclusions A L ⊂ Res F



L ∈Ξ A L −→ Res F

K E.

injects into Z[G] with finite cokernel.

Definition 3.6 Define the Pontrjagin dual Selmer vector spaces

S p(E/K) := Hom(Selp ∞ (E/K), Qp /Z p) ⊗ Q p ,

S p(A L /K) := Hom(Sel p ∞ (A L /K), Q p /Z p) ⊗ Q p

Define S p(E/F ) similarly for every finite extension F of K.

Corollary 3.7 There is a G-equivariant isomorphism

S p(E/F ) ∼= 

L∈Ξ S p(AL /K) where the action of G on the right-hand side is given by Theorem 3.4(ii) Proof We have S p (E/F ) ∼=S p((ResF K E)/K) by (the Pontrjagin dual of)

Proposition 3.1(iii), and S p((ResF K E)/K) ∼=⊕ L∈Ξ S p (A L /K) by Theorem 3.5.

4 The local invariants

Fix an odd prime p and a cyclic extension L/K of degree p n We will

write simply A for the abelian variety AL of Definition 3.3, R for the ring RL

of Definition 3.2, p for the unique prime pL of R above p, and I ⊂ R for the

idealI L of Definition 3.2

Proposition 4.1 There is a canonical G K -isomorphism A[p] −→ E[p] ∼

lies in the maximal ideal of Zp [G] Also, if π and π  are generators of p/p2,

then π/π  ∈ (R/p) × = F×

p , so π p −1 ≡ (π )p −1 (mod pp) It follows that

π p−1 is a canonical generator of pp−1 /p p, so there is a canonical isomorphism

pa(p−1) /p a(p−1)+1 ∼= Fp for every integer a Now using Theorem 3.4(iv) we

have G K-isomorphisms

A[p] ∼= p−1 T p (A)/T p (A) ∼= (pp n−1 −1 /p p n−1)⊗ T p (E) ∼= Fp⊗ T p (E) ∼ = E[p].

inside ResF K E This gives an alternate proof of Proposition 4.1.

Definition 4.3 Recall that in Section 2 we defined a self-dual Selmer

structureE on E[p] We can use the identification of Proposition 4.1 to define

another Selmer structureA on E[p] as follows For every v define H1

A (Kv , E[p])

to be the image of A(K v )/pA(K v) under the composition

A(K v )/pA(K v )  → H1(K v , A[p]) ∼ = H1(K v , E[p])

where the first map is the Kummer injection, and the second map is fromProposition 4.1 The first map depends (only up to multiplication by a unit

in F× p ) on a choice of generator of p/p2, but the image is independent of this

choice With this definition the Selmer group H A1(K, E[p]) is the usual p-Selmer

group Selp(A/K) of A, as in (2.1).

Proposition 4.4 The Selmer structure A is self-dual.

Proof This is Proposition A.7 of Appendix A (It does not follow

im-mediately from Tate’s local duality as in Proposition 2.1, because A has no polarization of degree prime to p, and hence no suitable Weil pairing.)

by

δ v = δ(v, E, L/K) := dimFp (H E1(Kv , E[p])/H E∩A1 (Kv , E[p])) (mod 2).

We will see in Corollary 5.3 below that δv is a purely local invariant, depending

only on K v , E/K v , and L w , where w is a prime of L above v.

Corollary 4.6 Suppose that S is a set of primes of K containing all primes above p, all primes ramified in L/K, and all primes where E has bad

reduction Then

dimFpSelp (E/K) − dimFpSelp(A/K) ≡

v ∈S

δ v (mod 2).

for example [Ca2, Lemma 19.3])

H E1(Kv , E[p]) = H A1(Kv , E[p]) = H1(K vur/K v , E[p]).

Thus the corollary follows from Propositions 2.1 and 4.4 and Theorem 1.4

5 Computing the local invariants

Let p, L/K, A := AL, and p ⊂ R be as in Section 4 Let M be the unique

extension of K in L with [L : M ] = p, and let G := Gal(L/K) (recall that L/K

is cyclic of degree p n) In this section we compare the local Selmer conditions

H E1(K v , E[p]) and H A1(K v , E[p]) for primes v of K, in order to compute the

invariants δv of Definition 4.5

Lemma 5.1 Suppose that c is an automorphism of K, and E is defined

Proof The automorphism c induces isomorphisms

E(K v)−→ E(K ∼ v c ), A(K v)−→ A(K ∼ v c ).

Therefore the isomorphism H1(K v , E[p]) −→ H ∼ 1(K v c , E[p]) induced by c

iden-tifies

H E1(Kv , E[p]) −→ H ∼ 1

E (Kv c , E[p]), H A1(Kv , E[p]) −→ H ∼ 1

A (Kv c , E[p]),

and the lemma follows directly from the definition of δ v

For every prime v of K, let L v := K v ⊗ K L = ⊕ w|v L w , and let G := Gal(L/K) act on L v via its action on L Let M v := K v ⊗ M and let N L/M :

tool for computing δ v

Proposition 5.2 For every prime v of K, the isomorphism

H E1(Kv , E[p]) ∼ = E(Kv)/pE(Kv)

identifies

H E∩A1 (K v , E[p]) ∼ = (E(Kv)∩ N L/M E(L v ))/pE(K v ).

under (3.2) Since σ projects to a p n -th root of unity in R, we see that π is a

generator of p

Note that G and G K v act on E( ¯ K v ⊗L) (as 1⊗G and G K v ⊗1, respectively).

We identify E(L v ), E( ¯ K v ), A(K v ), and A( ¯ K v ) with their images in E( ¯ K v ⊗ L)

under the natural inclusions and Theorem 3.4(iii):

A(K v)⊂ E(L v ) = E(K v ⊗ L) = E( ¯ K v ⊗ L) G Kv ,

E( ¯ K v ) = E( ¯ K v ⊗ K) = E( ¯ K v ⊗ L) G , A( ¯ K v)⊂ E( ¯ K v ⊗ L).

Let ˆπ := (1 ⊗ σ) − 1 on E( ¯ K v ⊗ L), so ˆπ restricts to π on A( ¯ K v) and to

zero on E( ¯ K v ) By Proposition 3.4(iii), A( ¯ K v ) is the kernel of N L/M :=



g ∈Gal(L/M)1⊗ g in E( ¯ K v ⊗ L).

If x ∈ E(K v ), then the image of x in H1(K v , E[p]) is represented by the

cocycle γ → y γ ⊗1 − y where y ∈ E( ¯ K v ) and py = x Similarly, using the identifications above, if α ∈ A(K v ) then the image of α in H1(K v , E[p]) is

represented by the cocycle γ → β γ⊗1 − β where β ∈ A( ¯ K v ) and πβ = α Suppose x ∈ E(K v ), and choose y ∈ E( ¯ K v ) such that py = x Then the image of x in H E1(K v , E[p]) ⊂ H1(K v , E[p]) belongs to H E∩A1 (K v , E[p])

⇐⇒ ∃ β ∈ A( ¯ K v ) : πβ ∈ A(K v ), β γ⊗1 − β = y γ⊗1 − y ∀γ ∈ G K v

⇐⇒ ∃ β ∈ A( ¯ K v ) : β γ ⊗1 − β = y γ ⊗1 − y ∀γ ∈ G K v

where for the second equivalence we use that if γ ∈ G K v and β γ⊗1 − β =

then πβ ∈ A(K v ) Since y ∈ E( ¯ K v ) = E( ¯ K v ⊗ L) G , we have N L/M y = py = x

and the proposition follows

The following corollary gives a purely local formula for δ v, depending only

on E and the local extension L w /K v (where w is a prime of L above v).

Corollary 5.3 Suppose v is a prime of K and w is a prime of L

K v with [L w : L  w ] = p, and otherwise let L  w := L w = K v Let N L w /L 

cyclic, L  w is the completion of M at the prime below w, so we have

E(K v)∩ N L/M E(L v ) = E(Kv)∩ N L w /L 

w E(L w).

This proves the corollary

By local field we mean a finite extension of Q  for some rational prime

Lemma 5.4 If K is a local field with residue characteristic different from p,

particular

Proof There is an isomorphism of topological groups

E( K) ∼ = E( K)[p ∞]⊕ C ⊕ D

with a finite group C of order prime to p and a free Z-module D of finite rank,

where is the residue characteristic of v Since E( K)[p ∞] is finite, the lemma

follows easily

Lemma 5.5 Suppose L/K is a cyclic extension of degree p of local fields

(i) If L/K is unramified and E has good reduction, then N L/K E( L) = E(K).

(ii) If L/K is ramified, ...