Arithmetic local constants for abelian varieties with extra endomorphisms
aa r X i v : . [ m a t h . N T ] F e b ARITHMETIC LOCAL CONSTANTS FOR ABELIAN VARIETIESWITH EXTRA ENDOMORPHISMS
SUNIL CHETTY
Abstract.
This work generalizes the theory of arithmetic local constants,introduced by Mazur and Rubin, to better address abelian varieties with alarger endomorphism ring than Z . We then study the growth of the p ∞ -Selmer rank of our abelian variety, and we address the problem of extendingthe results of Mazur and Rubin to dihedral towers k ⊂ K ⊂ F in which [ F : K ] is not a p -power extension. Introduction
In [9], Mazur and Rubin introduce a theory of arithmetic local constants for anelliptic curve E in terms of Selmer structures associated to E . With this theorythey study, for an odd prime p , the growth in Z p -corank of the p ∞ -Selmer groupSel p ∞ ( E/K ) (see §5) over a dihedral extension of number fields. To be precise,an extension F/k is dihedral if k ⊂ K ⊂ F is a tower of number fields with K/k quadratic,
F/k
Galois,
F/K p -power abelian, and a lift of the non-trivial element c ∈ Gal ( K/k ) acts on each σ ∈ Gal ( F/K ) as cσc − = σ − . They prove (undermild assumptions, see [9, §7]) that the growth in the Z p -corank of Sel p ∞ ( E/ ) over F/K must be at least [ F : K ] .Here, we consider a more general context for the theory of local constants. Inparticular, we replace the elliptic curve E/k with a pair ( X/k, λ ) of an abelianvariety X/k and a polarization λ : X → X ∨ on X of degree prime to p , where X ∨ is the dual abelian variety. We consider the ring of integers O of a numberfield K , and assume O ⊂
End K ( X ) is contained in the ring of endomorphisms of X defined over K . The case O = Z and K = Q is that of Mazur and Rubin in[9]. Recent work of Seveso [15] addresses similar questions for abelian varieties withreal multiplication.The condition that X has a polarization degree prime to p implies that many ofthe constructions of [9] generalize verbatim , with E replaced by X . The goal inthe present work is, in particular, to generalize Theorem 6.4 of [9] in the case thatthe endomorphism ring of X is strictly larger than Z .As a motivating example, consider p an odd rational prime, X = E an ellipticcurve defined over Q with complex multiplication by the ring of integers O of aquadratic imaginary field K in which p does not split, and set K = K . The Z p -corank of Sel p ∞ ( E/K ) would be even, so E would not satisfy the hypotheses ofTheorem 7.2 of [9] and hence one does not obtain a lower bound for the Z p -corank Date : February 9, 2021.2010
Mathematics Subject Classification.
Primary 11G05, 11G10; Secondary 11G07, 11G15.
Key words and phrases. elliptic curve, abelian variety, Selmer rank, complex multiplication. see subsection “Generalizations” in [9, §1] of Sel p ∞ ( E/F ) . One needs to consider Sel p ∞ ( E/F ) as a module over O ⊗ Z p inorder to obtain any useful generalization of the main tool (Theorem 6.4 of [9]) inthe proof of Theorem 7.2 of [9].1.1. Notation and Assumptions.
Before continuing, we introduce some nota-tion and assumptions that will be used until §6.1, where we will ease the restrictionson
F/K .Fix an odd rational prime p . The tower k ⊂ K ⊂ F is as above, with K/k quadratic,
F/K an abelian p -extension, and F/k dihedral. Also,
X/k and
O ⊂
End K ( X ) are as above, and we denote the cohomology groups H i ( Gal ( ¯
K/K ) , X ( ¯ K )) by H i ( K, X ) . Define a set S F of primes v of K by S F := { v | p , or v ramifies in F/K , or where
X/K has bad reduction } , and define S L similarly for intermediate fields K ⊂ L ⊂ F . For a cyclic extension L/K contained in F , define A L to be the twist of X , in the sense of [10], associatedto L/K (see §3 below).We assume that our prime p is unramified in O ⊂
End K ( X ) and we denote K p and O p for the local field and ring, respectively, at a prime p of O above p .For each prime v of K we fix an extension of v to ¯ K , which in turn fixes anembedding of ¯ K into an algebraic closure of K v and a decomposition subgroup G K v = Gal ( ¯ K v /K v ) ⊂ G K .We fix a polarization λ : X → X ∨ on X of degree prime to p , thus fixingan isogeny λ ∈ Hom ( X, X ∨ ) which has an inverse in λ − ∈ Hom ( X ∨ , X ) ⊗ Q .Associated to λ is the Rosati involution on End ( X ) ⊗ Q , given by α α † := λ − ◦ α ∨ ◦ λ, where α ∨ is the dual of α . This in particular satisfies,e ℓ,λ ( αa, a ′ ) = e ℓ,λ (cid:0) a, α † a ′ (cid:1) , where e ℓ,λ ( · , · ) = e ℓ ( · , λ ( · )) is the Weil pairing and a, a ′ ∈ T ℓ ( X ) ⊗ Q (see [11, §16-17]).We assume that the non-trivial element c ∈ Gal ( K/k ) acts as the Rosati involu-tion on O ⊂
End K ( X ) ⊗ Q , and that O is taken to itself by the Rosati involution,i.e. O c = O † = O . Remark 1.1.
Suppose X = E is an elliptic curve defined over k with complexmultiplication by O ⊂ K and O ⊂
End K ( E ) . We know that the Rosati involutionis the automorphism of O ⊗ Q = K of order 2. If K * k , then k K = K and so theaction of the Rosati involution and c ∈ Gal ( K/k ) on O must coincide.1.2. Main Results.
With the above discussion in mind, the goal in the followingis to keep track of the extra endomorphisms of the variety
X/k . Effectively thisamounts to extending the base ring (from Z p to O ⊗ Z p ) for the p ∞ -Selmer module,and as such the main results address this base extension.In §2 we address the important properties, for our purposes, of torsion O -modules, noting Proposition 2.8 for those modules equipped with a certain biliearform. In §3 we extend the results of [9] regarding Selmer structures and duality,and in §4 we apply those results to obtain information about the O /p O -rank ofthe relevant modules (as in §2 of [9]). This, in particular, motivates a generalizeddefinition (in §6) of the arithmetic local constant δ v , and combining §2-§4 in §5leads to our main result, Theorem 6.2. OCAL CONSTANTS FOR ABELIAN VARIETIES 3
As an application, in §6.1 we are able to address another generalization men-tioned in the introduction of [9]. In particular, we will consider dihedral towers k ⊂ K ⊂ F where [ F : K ] is not a prime power. For example, suppose [ F : K ] isdivisible by two distinct odd primes p, q and L/K is a cyclic extension containedin F . Then we have a p -power extension M/K and a q -power extension M ′ /K in L (one of these may be trivial) such that M ∩ M ′ = K and L = M M ′ . Wecan apply Theorem 6.2 for X , A M , and the ( p -power) dihedral extension M/k andthen separately for A M , A L , and a ( q -power) dihedral extension M ′ /k . AssumingConjecture 6.6, we can combine this information to compare X and A L .In addition to applications to growth in p -Selmer rank, it would be interesting tocompare the individual δ v to a quotient of the local root numbers for the L -functionassociated to X , as in [3]. We leave this question to future work.2. Torsion O -modules In this section we consider various O -modules, and so we prove some generalresults before applying them to our specific situation. Our abelian variety X andthe associated cohomology groups H i ( K, X ) are the basic examples of O -modulesto keep in mind.As O /p O may not be an integral domain, one does not have a natural definitionof the O /p O -rank of an O /p O -module via its fraction field (since there would beno such field). However, since p O = Q i p i with p i = p j when i = j , one has O /p O ∼ = ⊕ i ( O / p i ) induced by the natural O → ⊕ i ( O / p i ) maps. Thus, O /p O is a direct sum of fields O / p i , and each of these is a finite extension of F p . Definition 2.1.
Set R = O /p O and R i = O / p i , so R ∼ = ⊕ mi =1 R i . For any R -module M of finite type, define the R -rank of M to berank R M := ( . . . , dim R i M ⊗ R R i , . . . ) ∈ Z m . We say a = ( a , . . . , a m ) ∈ Z m is even if a i is even for each i .A first, and most important, property of this definition of R -rank is that itbehaves as one expects with respect to short exact sequences. We will exploitthis property frequently. The proof of this and the subsequent Lemma are left asexercises for the reader. Proposition 2.2. If → M → M → M → is a short exact sequence of R -modules then rank R M = rank R M + rank R M . Lemma 2.3. If M is an O /p O -module of finite type (i.e. M is p -torsion as an O -module) then M ⊗ R ( O / p ) ∼ = M [ p ] . For any R -module M , we denote M † for the R -module which has the same under-lying set as M , but with R -action given by rm := r † m . Also, for any abelian group Γ , we denote Hom ( M, Γ) :=
Hom Z ( M, Γ) for the R -module of group homomor-phisms from M to Γ , with the R -action on Hom ( M, Γ) given by ( rf )( x ) = f ( rx ) . Alternatively, one has O /p O = O ⊗ Z ( Z /p Z ) and that O is a torsion-free, hence flat, Z -module(see [7, §XVI.3]), which yields the same decomposition. CHETTY
Lemma 2.4.
Suppose M is an R -module and c : M ∼ −→ M is an isomorphismof groups with c ( rm ) = r † c ( m ) . Then M ∼ = M † as R -modules and in particularrank R M = rank R M † . Proof.
The isomoprhism c induces an R -isomorphism, since c ( rm ) = r † c ( m ) . (cid:3) Lemma 2.5. rank R R t = rank R Hom ( R † t , F p ) † , for each t ∈ { , . . . , m } .Proof. By Definition 2.1,rank R R t = ( . . . , dim R j R t ⊗ R R j , . . . )= (0 , . . . , dim R t R t , . . . , , rank R Hom ( R t , F p ) † = ( . . . , dim R j Hom ( R t , F p ) † ⊗ R R j , . . . ) . Since Hom ( R, F p ) † is an O /p O -module, we can use Lemma 2.3 to obtain(2.1) dim R j Hom ( R t , F p ) † ⊗ R R j = dim R j Hom ( R t , F p ) † [ p j ] , and we claim that(2.2) dim R j Hom ( R t , F p ) † [ p j ] = ( when R t = R † j when R t = R † j ) Consider f ∈ Hom ( R t , F p ) † [ p j ] , with R t = R † j . If f ( r † x ) = 0 for all x ∈ R t and all r † ∈ p † j then f is the zero map, since there exists some r † ∈ p † j such that r † p t andhence r † R t = R t . When R t = R † j , we have r † x = 0 for all x ∈ R t , and so f ( r † x ) = 0 is satisfied for every f ∈ Hom ( R † j , F p ) † [ p j ] , and this set has R j -dimension 1.Now consider R t = R s . Then viewing R t ⊗ R R s either as R t [ p s ] or R s [ p t ] shows that R t ⊗ R R s is trivial, and hence has rank 0. When R t = R s , we have R t ⊗ R R t = R t . From this and (2.2), we obtain dim R R t = dim R Hom ( R † t , F p ) † . (cid:3) Remark 2.6.
Alternatively, one can prove Lemma 2.5 as follows. Define a perfectpairing ( , ) : R t × R † t → F p via ( x, y ) Tr R t / F p ( xy † ) . This pairing satisfies ( rx, y ) = ( x, r † y ) and hence gives an R t -module isomoprhism R t ∼ = Hom ( R † t , F p ) † . Corollary 2.7. If M is an R -module of finite type, thenrank R M = rank R Hom ( M † , F p ) † . Proof.
This follows from the Lemma and M ∼ = ⊕ t R n t t . (cid:3) The next proposition is analogous to a well-known theorem for alternating pair-ings on vector spaces. Specifically, if k is a field with char ( k ) = 2 and there is anon-degenerate, skew-symmetric pairing on a finite dimensional k -vector space V ,then dim k V is even (see [7, §XV.8] or [12, §9.5]). Proposition 2.8.
Suppose A is a commutative ring, char ( A ) = 2 , and A ∼ = ⊕ nj =1 A j , where each A j is a local ring with principal maximal ideal m j . Let M , N be A -modules with M finite and [ , ] : M × M → N be a non-degenerate, skew-symmetric pairing which satisfies [ sx, y ] = [ x, sy ] for all x, y ∈ M and s ∈ A . Thenthere exist A -submodules M ′ , M ′′ with M ′ ∼ = M ′′ and M ∼ = M ′ ⊕ M ′′ .Proof. Let M j = M ⊗ A j . We first note that A ∼ = ⊕ j A j implies M ∼ = ⊕ M j . Since M is finite, we see that x ∈ M j implies x ∈ M j [ m tj ] for some t . For i = j , x ∈ M j OCAL CONSTANTS FOR ABELIAN VARIETIES 5 and y ∈ M i , we have that [ x, y ] = 0 . Indeed, there is some α ∈ m j with αx = 0 which acts as a unit on M i . Thus, there is some y ′ ∈ M i with αy ′ = y and so αx, y ′ ] = [ x, αy ′ ] = [ x, y ] . Now, suppose x ∈ M j is of maximal order, i.e. that x ∈ M j [ m tj ] but x M j [ m t − j ] and that t is maximal. Let π be a generator of m j in A j . Since π t − x = 0 thereis some y ∈ M j such that [ π t − x, y ] = 0 . We then have = [ π t − x, y ] = [ x, π t − y ] and so π t − y = 0 . In particular, this implies that y M j [ m t − j ] and y ∈ M j [ m tj ] ,since x was chosen to be of maximal order. Moreover, we have that span A j { x } ∼ = span A j { y } . We also note that if w = ax for some a ∈ A then [ x, w ] = [ x, ax ] = [ ax, x ] = [ w, x ] and so [ x, w ] = 0 .Set U := span A j { x, y } . We claim that U ∩ U ⊥ = { } . Let z ∈ U ∩ U ⊥ with z = ax + by for some a, b ∈ A j , and suppose that a = 0 . Since A j is a local ring,we have π t ∤ a , so a | π t − . So, we can find a ′ ∈ A j such that aa ′ = π t − . Now, for w = a ′ y we have [ z, w ] = [ ax + by, a ′ y ] = [ ax, a ′ y ]= [( aa ′ ) x, y ] = [ π t − x, y ] = 0 , contradicting z ∈ U ⊥ . In the same way we can see that if π t ∤ b then we can find w ∈ U such that [ z, w ] = 0 .We are now left with the case that π t | a and π t | b . Since x was chosen to be ofmaximal order, this forces z = 0 and it follows that U ∩ U ⊥ = { } . Also, the aboveargument shows that U ∼ = A j x ⊕ A j y . The finiteness of M (and hence M j ) thenimplies that we can decompose M j as M j = U ⊕ U ⊥ and by induction we obtainthe claim. (cid:3) Remark 2.9.
Recall that R = O /p O , R j = O / p j , and set S = O ⊗ Z p and S j = O p j . We have decompositions R ∼ = ⊕ j R j and S ∼ = ⊕ j S j . For R L := R L ⊗ Z p ,where R L is as in §3 of [9] (see also §3 below), we again have a decomposition O ⊗ R L ∼ = ⊕ j ( O p j ⊗ R L ) . In what follows, these rings will play the role of A in theabove proposition.3. Selmer Structures and Tate Duality
As our goal is to establish a theorem analogous to Theorem 6.4 of [9], we needto generalize the results of [9] regarding the pairing of Tate’s local duality in orderto yield information about the Selmer structures of Definition 3.3 as O -modules.Using Definition 3.3 of [9] (see also Definition 1.1 of [10]), we have the I -twist A of X exactly as in the elliptic curve case X = E . Specifically, for a cyclicextension L/K contained in F , let ρ L denote the unique faithful irreducible rationalrepresentation of Gal ( L/K ) . Define the I L -twist of X to be A L := I L ⊗ X , where I L := Q [ Gal ( F/K )] L ∩ Z [ Gal ( F/K )] and Q [ Gal ( F/K )] L is the sum of all (left) ideals of Q [ Gal ( F/K )] isomorphic to ρ L . We define the ring R L (mentioned in Remark 2.9) as the maximal order of Q [ Gal ( F/K )] L , and when [ L : K ] = p m we have that R L ∼ = Z [ µ p m ] has a uniqueprime above p . CHETTY
Remark 3.1.
By definition (in [10]), when I L is a Z -module, the twist A L = I L ⊗ X is a Z -module. However, we may regard it as an O -module, simply by letting O acton I L ⊗ X via its action on X . The resulting module coincides with the O -module I ′ L ⊗ X obtained by twisting X with the O -module I ′ L := K [ Gal ( F/K )] L ∩ O [ Gal ( F/K )] . Proposition 3.2.
For ˆ p the unique prime above p in I L , there is a canonicalGal ( ¯ K/K ) -isomorphism A L [ˆ p ] ∼ = X [ p ] .Proof. This is exactly as in Proposition 4.1 of [9] (also Remark 4.2 in [9]), whereour ˆ p is their p = p L . (cid:3) We are concerned with the following Selmer structures, analogous to those of §2and §4 of [9].
Definition 3.3.
Define a Selmer structure X on X [ p ] as the collection of O -modules H X ( K v , X [ p ]) , defined to be, for each v , the image of X ( K v ) /pX ( K v ) ֒ → H ( K v , X [ p ]) . Fix a generator π of ˆ p , with ˆ p as in Proposition 3.2. Define a Selmer structure A on X [ p ] by setting, for each v , H A ( K v , X [ p ]) to be the image of A L ( K v ) /πA L ( K v ) ֒ → H ( K v , A L [ˆ p ]) ∼ = H ( K v , X [ p ]) . We note that the image in H ( K v , X [ p ]) is independent of the choice of our gener-ator. As in [9, §1], define H X + A ( K v , X [ p ]) := H X ( K v , X [ p ]) + H A ( K v , X [ p ]) H X ∩A ( K v , X [ p ]) := H X ( K v , X [ p ]) ∩ H A ( K v , X [ p ]) . Definition 3.4.
We say that a Selmer structure F on X [ p ] is self-dual if for everyprime v , H F ( K v , [ X [ p ]) is its own orthogonal complement under the pairing ofTate’s local duality:(3.1) h , i v : H ( K v , X [ p ]) × H ( K v , X [ p ]) → H ( K v , µ p ) = F p . We note that in Definition 3.4, we are making use of our assumption that X hasa polarization of degree prime to p in order to have (3.1) as a self -pairing. Definition 3.5.
Given a Selmer structure F on X [ p ] , define the Selmer group tobe H F ( K, X [ p ]) := ker H ( K, X [ p ]) → Y v H ( K v , X [ p ]) /H F ( K v , X [ p ]) ! . Thus, H F ( K, X [ p ]) is the set of classes whose localizations are in H F ( K v , X [ p ]) , orin other words the classes satisfying the local conditions defined by F . Proposition 3.6.
The Selmer structures X and A on X [ p ] are self-dual.Proof. The Tate pairing is the same as that in [9], and Tate local duality holds fora general abelian variety (see [9, §1.4]). This shows that X is self-dual. For A , theproof is exactly Proposition A.7 of Appendix A of [9], noting that we need onlyregard A L as a Z -module here. (cid:3) OCAL CONSTANTS FOR ABELIAN VARIETIES 7
The pairing (3.1) is not O -linear, but understanding the interplay of the pairingand the map induced by c on the local cohomology groups H ( K v , X [ p ]) providesinformation (see Lemma 4.4 below) about the R -rank of certain Selmer groups.Now, we fix a lift of the nontrivial element c ∈ Gal ( K/k ) to Gal ( ¯ K/k ) , whichwe also denote c . As c ∈ G k with c ( K ) = K , we have that c : K v ∼ −→ K v c .The maps c : G K → G K : s c − sc and c : M → M : a c ( a ) , for any G k -module M , are compatible in the sense of [14, §VII.5], and hence induce c ∗ : H ∗ ( K, M ) → H ∗ ( K, M ) on cohomology. Similarly, from c : G K v → G K vc weobtain c ∗ : H ∗ ( G K vc , M ) → H ∗ ( G K v , M ) . Lemma 3.7.
For G k -module M , the map c ∗ : H ( G K , M ) → H ( G K , M ) inducedby the lift c ∈ G k of c is independent of the choice of lift.Proof. The claim follows from a special case of Proposition 3 of §VII.5 of [14]. (cid:3)
Lemma 3.8.
Let M and N be two G k -modules and ϕ : M → N a G k -equivariantmap. Then for the map ϕ ∗ : H ∗ ( K, M ) → H ∗ ( K, N ) induced by ϕ , ϕ ∗ ◦ c ∗ = c ∗ ◦ ϕ ∗ : H ∗ ( K v c , M ) → H ∗ ( K v , N ) . Proof.
Let G = Gal ( ¯ K v /K v ) and G ′ = Gal ( ¯ K v c /K v c ) . We prove the claim oncochains. For each i ≥ , let P i := Z [ G i +1 ] , be the free module generated byelements ( g , . . . , g i ) ∈ G i +1 , with a G -action by s. ( g , . . . , g i ) = ( s.g , . . . , s.g i ) . These form the standard resolution for Z (see [14, §VII.3] or [1, §I.5]).Suppose f ∈ Hom G ′ ( P ′ i , M ) . Then c ∗ ( f )( g , . . . , g i ) = c ( f ( c − g c, . . . , c − g i c )) ϕ ∗ ( f )( g , . . . , g i ) = ϕ ( f ( g , . . . , g i )) , and it follows that ( ϕ ∗ ◦ c ∗ )( f )( g , . . . , g i ) = ( c ∗ ◦ ϕ ∗ )( f )( g , . . . , g i ) , using the G k -equivariance of ϕ . (cid:3) Let W = X [ p ] . Denote e ∗ : H ∗ ( K, W ⊗ W ) → H ∗ ( K, µ p ) for the map inducedby the Weil pairing e p,λ on W . We will also use e ∗ for the maps induced by e p,λ on G K v -cohomology and G K vc -cohomology, and context will make the notation clear.We know that e p,λ is Gal ( ¯ K/k ) -equivariant (see [16, §III.8] or [11, §12]). By Lemma3.8, we see that e ∗ ◦ c ∗ = c ∗ ◦ e ∗ : H ∗ ( K v c , W ⊗ W ) → H ∗ ( K v , µ p ) . Proposition 3.9.
Suppose S is a finite set of primes v of K such that v ∈ S ifand only if v c ∈ S . For any a , b ∈ ⊕ v ∈ S H ( K v , W ) , let h a, b i := P v ∈ S h a v , b v i v .Then h a, c ∗ ( b ) i = h c ∗ ( a ) , b i . Proof.
Recall that h , i v is defined via the composition (cf. [13, §1.4]) H ( K v , W ) ⊗ H ( K v , W ) ∪ (cid:15) (cid:15) H ( K v , W ⊗ W ) e ∗ / / H ( K v , µ p ) inv v / / µ p . CHETTY
The cup product ∪ is functorial, so the commutative diagram H ( K v c , W ) c ∗ (cid:15) (cid:15) ⊗ H ( K v c , W ) c ∗ (cid:15) (cid:15) ∪ / / H ( K v c , W ⊗ W ) c ∗ (cid:15) (cid:15) H ( K v , W ) ⊗ H ( K v , W ) ∪ / / H ( K v , W ⊗ W ) implies a ∪ c ∗ ( b ) = c ∗ c ∗ ( a ) ∪ c ∗ ( b ) = c ∗ ( c ∗ ( a ) ∪ b ) . Also we can see that, for all i ≥ , H i ( K, W ) ∼ c ∗ / / res vc (cid:15) (cid:15) H i ( K, W ) res v (cid:15) (cid:15) H i ( K v c , W ) ∼ c ∗ / / H i ( K v , W ) . commutes by recalling that on cochains res v ( f ) is restriction of the map f . UsingLemma 3.8 and the property inv v ◦ c ∗ = inv v c (see [14, §§XI.1-XI.2], particularlyProposition 1) of the local invariant map, we see h a, c ∗ ( b ) i = h c ∗ ( a ) , b i . (cid:3) The next proposition shows how the R -action on our cohomology groups interactswith the pairing (3.1). Proposition 3.10.
For any a , b ∈ H ( K v , X [ p ]) and r ∈ R , h ra, b i v = h a, r † b i v .Proof. Let W = X [ p ] as above, and let x, y ∈ W and r ∈ O . The claim is aconsequence of the identity e p,λ ( rx, y ) = e p,λ (cid:0) x, r † y (cid:1) . As e p,λ is bilinear, it canbe viewed as a map on W ⊗ Z W , and the above property becomes e p,λ ( rx ⊗ y ) = e p,λ (cid:0) x ⊗ r † y (cid:1) . Now, for a, b ∈ H ( K v , W ) we have r and r † acting by ( ra )( g ) = r.a ( g ) and ( r † b )( g ) = r † .b ( g ) . Thus, keeping in mind that
O ⊂
End K ( X ) , it followsthat e ∗ (( ra ) ∪ b )( g, h ) = e p,λ ((( ra ) ∪ b )( g, h ))= e p,λ (cid:0) ( a ∪ ( r † b ))( g, h ) (cid:1) = e ∗ p,λ (( a ∪ ( r † b ))( g, h ) , and so h ra, b i v = inv v ◦ e ∗ p,λ (( ra ) ∪ b )= inv v ◦ e ∗ p,λ ( a ∪ ( r † b )) = h a, r † b i v . (cid:3) Corollary 3.11.
The orthogonal complement of H ( K v , X [ p ])[ p ] under (3.1) is ⊕ q = p † H ( K v , X [ p ])[ q ] .Proof. Set M = H ( K v , X [ p ]) . Let a ∈ M [ p ] , b ∈ M , and r ∈ p . Then h , b i v = h ra, b i v = h a, r † b i v , so r † M ⊂ M [ p ] ⊥ and in turn p † M ⊂ M [ p ] ⊥ . Since M = ⊕ q | p M [ q ] , we see that p † M = ⊕ q = p † M [ q ] ⊂ M [ p ] ⊥ , and non-degeneracy finishes the claim. (cid:3) OCAL CONSTANTS FOR ABELIAN VARIETIES 9 O /p O -rank Recall S L is a finite set of primes of K containing those which divide p or areramified in L/K or where X does not have good reduction. In this section we fixa cyclic extension L/K contained in F . Lemma 4.1.
For v S L , the Selmer structures X and A on X [ p ] coincide.Proof. This is Corollary 4.6 of [9], which uses Lemma 19.3 of [2]. Specifically, both X and A are self-dual (cf §3) and when v S L then both T p ( X ) and T p ( A L ) areunramified at v . Thus, H X ( K v , X [ p ]) = H A ( K v , X [ p ]) = H ( K urv /K v , X [ p ]) . (cid:3) Let R = O /p O and R i = O / p i be as in the previous section. We now generalizethe main results of §1 of [9] regarding self-dual Selmer structures. Later, determin-ing the difference in the ( O ⊗ Z p ) -corank of the p ∞ -Selmer groups associated to X and A will be reduced to determining the difference in the R -corank of the p -Selmergroups, and Theorem 4.5 below describes the latter. We phrase the result specifi-cally in terms of the Selmer structures X and A , as we make use of the assumptionon c introduced in the beginning of §1 to prove Lemma 4.3. Remark 4.2.
The following is an example of an application of Lemma 2.4. Set W = X [ p ] and B = M v ∈ S L ( H X + A ( K v , W ) /H X ∩A ( K v , W )) . We check that v ∈ S L if and only if v c ∈ S L . Since c ∈ Gal ( K/k ) , we have v | p implies v c | p . Also, if w witnesses that v is ramified in L/K then w c witnesses that v c is ramified in L/K . Lastly, since X is defined over k , X has good reduction at v if and only if X has good reduction at v c .The automorphism c induces an isomorphism X ( K v ) ∼ −→ X ( K v c ) and in turn H ( K v , W ) ∼ −→ H ( K v c , W ) . This restricts to a group isomorphism H X ( K v , W ) ∼ −→ H X ( K v c , W ) . We have analogous isomorphisms for H A ( K v , W ) . As B is a direct sum taken overall v ∈ S L , we know that H X + A ( K v , W ) and H X + A ( K v c , W ) occur symmetricallyin B . Thus, B = L v ∈ S L ( H X + A ( K v , W ) /H X ∩A ( K v , W )) ∼ = L v c ∈ S L ( H X + A ( K v c , W ) /H X ∩A ( K v c , W )) = B and so c : B ∼ −→ B . Lemma 2.4 then gives rank R B = rank R B † .Recall the definition of a Selmer group, e.g. H X ( K, X [ p ]) , in Definition 3.5. Thefollowing Lemmas generalize Proposition 1.3 of [9]. Lemma 4.3.
Since X and A are self-dual,rank R H X + A ( K, X [ p ]) /H X ∩A ( K, X [ p ])= P v ∈ S rank R ( H X ( K v , X [ p ]) /H X ∩A ( K v , X [ p ])) . Proof.
We follow the ideas of Proposition 1.3 of [9], noting the adjustments neededto address R -rank. Let W and B be as in Remark 4.2. The Tate pairing restrictsto H X + A ( K v , W ) for each v , and since X and A are self-dual we obtain a pairing h , i : B × B → F p .Defining C X (resp. C A ) to be the projection of ⊕ v H X ( K v , W ) (resp. ⊕ v H A ( K v , W ) )in B , the local self-duality of X (resp. A ) implies that C X (resp. C A ) is its ownorthogonal complement under h , i . Using these orthogonality relations, we willshow(4.1) rank R C = rank R C X = rank R C A = 12 rank R B. First we note B = C X ⊕ C A , and since C ⊥X = C X and C ⊥A = C A , the pairing h , i restricts to a non-degenerate pairing on C X × C A . From this we obtain in the usualway (see [7, §I.9] or [7, §XIII.5]) an R -isomorphism C X → Hom ( C A , F p ) † whichimplies rank R C X = rank R Hom ( C A , F p ) † = rank R C †A , using Corollary 2.7 for the right-hand equality. Then by Lemma 2.4, as in Remark4.2, we see rank R C X = rank R C †A = rank R C A . Thus, we have the middle and right-hand equalities of (4.1).Similarly, from B × B → F p and C = C ⊥ , we obtain C × ( B/C ) → F p whichgives rank R C = rank R Hom ( B/C, F p ) † = rank R ( B/C ) † , and in turn, again by Lemma 2.4, we have rank R C = rank R B/C . Now using theexact sequence (of R -modules) → C → B → B/C → and Proposition 2.2, we haverank R B = rank R C + rank R ( B/C ) = 2 rank R C, and hence the left-hand equality of (4.1). The result now follows from C ∼ = H X + A ( K v , W ) /H X ∩A ( K v , W ) and C X ∼ = ⊕ v H X ( K v , W ) /H X ∩A ( K v , W ) . (cid:3) Lemma 4.4.
With the same assumptions and notation of Lemma 4.3,rank R H X + A ( K, W ) ≡ rank R ( H X ( K, W ) + H A ( K, W )) ( mod . Proof.
Again, we follow Proposition 1.3 of [9]. For u ∈ H X + A ( K, W ) , write u s ∈ C for the localization of u , and u x , u a for the projections of u s to C X , C A , respectively.Using the symmetry of h , i , the pairing [ , ] : H X + A ( K, W ) × H X + A ( K, W ) → F p : [ u, w ] := h u x , w a i is skew-symmetric. Also, exactly as in [9], the kernel of [ , ] is exactly H X ( K, W ) + H A ( K, W ) , and so [ , ] induces an F p -valued, non-degenerate, skew-symmetricpairing on H := H X + A ( K, W ) / ( H X ( K, W ) + H A ( K, W )) . Since [ , ] is defined in terms of P v ∈ S L h , i v , we use Propositions 3.10 and 3.9,respectively, to see that [ u, rw ] = [ r † u, w ] and [ u, c ∗ ( w )] = [ c ∗ ( u ) , w ] . OCAL CONSTANTS FOR ABELIAN VARIETIES 11
Define [ , ] ′ on H by [ u, w ] ′ := [ u, c ∗ ( w )] . The non-degeneracy and skew-symmetryof [ , ] imply that [ , ] ′ is non-degenerate and skew-symmetric also. In addition, thetwo properties above imply that [ ru, w ] ′ = [ u, rw ] ′ and with this pairing Proposition2.8 (with A = R ) shows that rank R H is even. (cid:3) Theorem 4.5.
Since X and A are self-dual,rank R H X ( K, X [ p ]) − rank R H A ( K, X [ p ]) ≡ P v ∈ S rank R ( H X ( K v , X [ p ]) /H X ∩A ( K v , X [ p ])) ( mod . Proof.
Applying Lemma 4.1, the claim follows from the congruencesrank R H X ( K, X [ p ]) − rank R H A ( K, X [ p ]) ≡ rank R H X ( K, X [ p ]) + rank R H A ( K, X [ p ]) ≡ rank R ( H X ( K, X [ p ]) + H A ( K, X [ p ])) + rank R H X ∩A ( K, X [ p ]) ≡ rank R H X + A ( K, X [ p ]) − rank R H X ∩A ( K, X [ p ]) ≡ P v ∈ S rank R ( H X ( K v , X [ p ]) / dim H X ∩A ( K v , X [ p ])) ( mod . The last two steps follow from Lemmas 4.3 and 4.4. (cid:3)
Remark 4.6.
The summands in the right-hand side of Theorem 4.5 motivateDefinition 6.1 below of the arithmetic local constants δ v .5. p -Selmer corank The p -Selmer group H X ( K, X [ p ]) = Sel p ( X/K ) sits in the exact sequence (seefor example [16, §X.4])(5.1) → X ( K ) ⊗ Z /p m Z → Sel p m ( X/K ) → X ( X/K )[ p m ] → and passing to the limit Sel p ∞ ( X/K ) sits in(5.2) → X ( K ) ⊗ Q p / Z p → Sel p ∞ ( X/K ) → X ( X/K )[ p ∞ ] → . We have similar sequences for H A ( K, X [ p ]) = Sel ˆ p ( A L /K ) and for the associateddirect limit Sel p ∞ ( A L /K ) .We next generalize Proposition 2.1 of [9], but in order to do so we need to definea notion of corank over the ring O ⊗ Z p (particularly in the case that it is not anintegral domain). Again, we have a decomposition O ⊗ Z p ∼ = ⊕ i O p i . Definition 5.1.
Let S := O ⊗ Z p and S i := O p i . For an S -module M , define the S -corank of M to becorank S M := ( . . . , corank S i M ⊗ O p i , . . . ) . Proposition 5.2. corank S Sel p ∞ ( X/K ) ≡ rank R Sel p ( X/K ) − rank R X ( K )[ p ] ( mod . Proof.
We follow the strategy of Proposition 2.1 of [9]. Let d := rank R ( Sel p ∞ ( X/K ) / Sel p ∞ ( X/K ) div )[ p ]= rank R ( X ( X/K )[ p ∞ ] / X ( X/K )[ p ∞ ] div )[ p ] . We havecorank S Sel p ∞ ( X/K ) = ( . . . , corank S i Sel p ∞ ( X/K ) ⊗ S i , . . . )= ( . . . , rank R i Sel p ∞ ( X/K ) div [ p ] ⊗ R i , . . . )= ( . . . , rank R i Sel p ∞ ( X/K )[ p ] ⊗ R i , . . . ) − d = rank R Sel p ∞ ( X/K )[ p ] − d, with the first and last equalities by definition, and the others as in [9]. From (5.2)we obtain another sequence → ( X ( K ) ⊗ Q p / Z p )[ p ] → Sel p ∞ ( X/K )[ p ] → X ( X/K )[ p ] → and then applying Proposition 2.2 we haverank R Sel p ∞ ( X/K )[ p ] = rank R ( X ( K ) ⊗ Q p / Z p )[ p ] + rank R X ( X/K )[ p ] . From (5.1) and Proposition 2.2 we obtainrank R Sel p ( X/K ) = rank R ( X ( K ) /pX ( K )) + rank R X ( X/K )[ p ] . Combining these, we see thatcorank S Sel p ∞ ( X/K ) − rank R Sel p ( X/K )= rank R Sel p ∞ ( X/K )[ p ] − d − rank R Sel p ( X/K )= rank R ( X ( K ) ⊗ Q p / Z p )[ p ] − rank R ( X ( K ) /pX ( K )) − d = − rank R X ( K )[ p ] − d. Here we have cancelled the X ( X/K )[ p ] terms in the second equality, and the lastequality follows from the exact sequence → X ( K )[ p ] → X ( K ) ⊗ Z /p Z → ( X ( K ) ⊗ Q p / Z p )[ p ] → , defined by considering each term as an O -module and decomposing each term asin [12, §11.2], and applying [7, §XVI.2].It remains to see that d is even, which will show that the above equality impliesthe desired congruence ( mod . We prove d is even below in Proposition 5.8. (cid:3) First, we recall some definitions and results of Appendix A of [9]. For a cyclicextension
L/K of degree p n in F we define R L := R L ⊗ Z p , where R L is as in §3,and consider R L as a G K -module by letting G K act trivially. Let ζ be a primitive p n root of unity and denote ι for the involution of R L induced by ζ ζ − , andsimilarly for R L . Let π := ζ − ζ − , which is a generator of the unique prime ˆ p of R L above p and of the maximal ideal P of R L .For W an R L -module and B a Z p -module, a pairing h , i : W × W → B is ι - adjoint if for each r ∈ R L and x, y ∈ W , h rx, y i = h x, r ι y i . Also, a pairing h , i : W × W → R L ⊗ Z p B is R L - semilinear if for each r ∈ R L and x, y ∈ W h rx, y i = r h x, y i = h x, r ι y i , and is skew-Hermitian if it is R L -semilinear and h y, x i = −h x, y i ι ⊗ .Mazur and Rubin construct a map τ : R L → Z p such that composition with τ ⊗ R L ⊗ Z p B → B gives a bijection (Lemma A.3 and Proposition A.4 of [9]) be-tween the set of R L -semilinear pairings W × W → R L ⊗ Z p B and the set of ι -adjointpairings W × W → B . Also, if h , i R L corresponds to h , i Z p then h , i R L is perfect(resp. G K -equivariant) if and only if h , i Z p is perfect (resp. G K -equivariant). OCAL CONSTANTS FOR ABELIAN VARIETIES 13
Definition 5.3 (Definition A.5 of [9]) . Let p n = [ L : K ] . Define two pairings: f : I L × I L → R L by f ( α, β ) := π − p n − αβ ι , and h , i R L := f ⊗ e p,λ on T p ( A L ) = I L ⊗ T p ( X ) by(5.3) h α ⊗ x, β ⊗ y i := ( π − p n − αβ ι ) ⊗ e p,λ ( x, y ) ∈ R L ⊗ Z p Z p (1) . In Theorem A.12 of [9], Mazur and Rubin use the pairing (5.3) and arguments ofFlach [5] to obtain a perfect, skew-Hermitian, Gal ( K/k ) -equivariant pairing [ , ] R L on X ( A L /K ) / div := X ( A L /K ) / X ( A L /K ) div , taking values in D p := R L ⊗ Z p Q p / Z p . Using Flach’s arguments, we can also obtainthe classical Cassels-Tate pairing on X ( X/K ) / div from the Weil pairing on X [ p ] .We first show that these pairings satisfy [ sx, y ] = [ x, s † y ] , for each s ∈ O . Proposition 5.4.
Suppose
Y /k is an abelian variety with an action of O and B = Q p / Z p or B = D p . If h , i : T p ( Y ) × T p ( Y ) → B induces (via Flach’sconstruction) [ , ] on X ( Y /K ) / div and h sx, y i = h x, s † y i for all s ∈ O , then [ sx, y ] = [ x, s † y ] for all s ∈ O .Proof. We recall the construction of [ , ] from p.116 of [5]. Let V p ( Y ) = T p ( Y ) ⊗ Q .From x, x ′ ∈ Sel p ∞ ( Y /K ) , we obtain cocylces α, α ′ ∈ Z ( K, Y [ p ∞ ]) . From theexact diagram C ( K, V p ( Y )) / / d (cid:15) (cid:15) C ( K, Y [ p ∞ ]) d (cid:15) (cid:15) / / C ( K, T p ( Y )) / / C ( K, V p ( Y )) / / C ( K, Y [ p ∞ ]) we see that α and α ′ can be lifted to β, β ′ ∈ C ( K, V p ( Y )) , and we have dβ, dβ ′ ∈ C ( K, T p ( Y )) . The pairing h , i induces a cup-product ∪ C i ( K, V p ( Y )) × C j ( K, V p ( Y )) ∪ −→ C i + j ( K, B ) . Since H ( K, B ) = 0 , there is some ǫ ∈ C ( K, B ) such that dβ ∪ β ′ = dǫ . Since α ′ represents x ∈ Sel p ∞ ( Y /K ) , res v ( α ′ ) is the image of some cocycle β ′ v ∈ Z ( K v , V p ( Y )) .Define γ v := res v ( β ) ∪ β ′ v − res v ( ǫ ) ∈ C ( K v , B ) , and then [ x, x ′ ] := P v inv v ( γ v ) . Just as in Proposition 3.10, the cup-product ∪ satisfies an O -adjoint property,so d ( sβ ) ∪ β ′ = s ( dβ ) ∪ β ′ = dβ ∪ s † β ′ , giving the same ǫ for both pairs ( sx, x ′ ) and ( x, s † x ′ ) . Also,res v ( sβ ) ∪ β ′ v = s ( res v ( β )) ∪ β ′ v = res v ( β ) ∪ s † β ′ v . Thus the pairs ( sx, x ′ ) and ( x, s † x ′ ) define the same γ v , for each v , and so [ sx, x ′ ] =[ x, s † x ′ ] . (cid:3) Corollary 5.5. If [ , ] is obtained from e p,λ or h , i R L , then [ sx, y ] = [ x, s † y ] forall s ∈ O . Proof.
We have already seen that e p,λ ( sx, y ) = e p,λ (cid:0) x, s † y (cid:1) . By definition, the O -action on I L ⊗ T p ( X ) is s ( α ⊗ x ) = α ⊗ ( sx ) . Therefore, h s ( α ⊗ x ) , β ⊗ y i = h α ⊗ ( sx ) , β ⊗ y i = ( π − p n − αβ ι ) ⊗ e p,λ ( sx, y )= ( π − p n − αβ ι ) ⊗ e p,λ (cid:0) x, s † y (cid:1) = h α ⊗ x, β ⊗ ( s † y ) i = h α ⊗ x, s † ( β ⊗ y ) i , and Proposition 5.4 gives the claim. (cid:3) Proposition 5.6.
Let [ , ] denote the Cassels-Tate pairing X ( X/K ) / div × X ( X/K ) / div → Q p / Z p . Then [ c ∗ ( x ) , x ′ ] = [ x, c ∗ ( x ′ )] .Proof. Recall that e p,λ is G k -equivariant. We keep the notation in the proof ofProposition 5.4. Specifically, let B = Q p / Z p and let x, x ′ ∈ Sel p ∞ ( X/K ) . Just asin Proposition 3.8 the G k -equivariance of e p,λ implies, for any cochains ω , ω ′ ,(5.4) c ∗ ( c ∗ ( ω ) ∪ ω ′ ) = ω ∪ c ∗ ( ω ′ ) . Let the pair c ∗ ( β ) , β ′ (resp. β, c ∗ ( β ′ ) ) define ǫ ∈ C ( K, B ) and γ v ∈ C ( K v , B ) (resp. ǫ ′ , γ ′ v ) as in Proposition 5.4. Property (5.4) then implies that c ∗ ( ǫ ) = ǫ ′ .From c ∗ ◦ res v = res v c ◦ c ∗ , we obtain γ ′ v = res v ( β ) ∪ c ∗ ( β ′ v c ) − res v ( c ∗ ( ǫ ))= c ∗ ( res v c ( c ∗ ( β )) ∪ β ′ v c − res v c ( ǫ ))= c ∗ ( γ v c ) , and so P v inv v ( γ ′ v ) = P v inv v ◦ c ∗ ( γ v c ) = P v inv v c ( γ v c ) . Thus, we conclude that [ x, c ∗ ( x ′ )] = [ c ∗ ( x ) , x ′ ] . (cid:3) Remark 5.7.
The proposition also follows from Theorem A.12 of [9]. In partic-ular, Mazur and Rubin show that the G k -equivariance of e p,λ implies Gal ( K/k ) -equivariance of [ , ] , and Gal ( K/k ) acts trivially on Q p / Z p .The following proposition shows that d = rank R X ( X/K ) / div [ p ] is even. Theo-rem 1 of [5] shows that X ( X/K ) / div is finite, and in particular it is a finite p -group.Thus, for some t ≥ X ( X/K ) / div = X ( X/K ) / div [ p t ] = ⊕ i X ( X/K ) / div [ p ti ] . Proposition 5.8. d = rank R ( X ( X/K )[ p ∞ ] / X ( X/K )[ p ∞ ] div )[ p ] is even.Proof. From Corollary 5.5 and Proposition 5.6 the pairing [ , ] on X ( X/K ) / div satisfies [ sx, x ′ ] = [ x, s † x ′ ] and [ c ∗ ( x ) , x ′ ] = [ x, c ∗ ( x ′ )] for all s ∈ O and x, x ′ ∈ X ( X/K ) / div . Define [ , ] ′ by [ x, y ] ′ := [ x, c ∗ ( y )] as in Lemma 4.4, obtaining a non-degenerate, skew-symmetric, Z p -bilinear pairing on X ( X/K ) / div with [ sx, y ] ′ =[ x, sy ] ′ for all s ∈ O and x, y ∈ X ( X/K ) / div . Since X ( X/K ) / div is finite, Propo-sition 2.8 (with A = O ⊗ Z p ) then shows that d is even. (cid:3) We now provide the analogous statement to Proposition 5.2 for A L . Previously,we noted that the twist A L is defined over K , but in fact it is essential that A L have a model over k in order to apply Theorem A.12 of [9]. Again, the resultsof Appendix A of [9] (Definition A.8 and on, or alternatively [10, §6]) allow us toconsider A L defined over k . Combining Propositions 5.2 and 5.9 in Theorem 6.2below proves a generalization of Theorem 6.4 of [9]. Recall R L = R L ⊗ Z p . OCAL CONSTANTS FOR ABELIAN VARIETIES 15
Proposition 5.9. corank
O⊗R L Sel p ∞ ( A L /K ) ≡ rank R Sel ˆ p ( A L /K ) − rank R X ( K )[ p ] ( mod . Proof.
The proof is the same as Proposition 5.2, using Proposition 3.2 to identify A L ( K )[ˆ p ] with E ( K )[ p ] , and seeing that d = rank R X ( A L /K ) / div [ˆ p ] is even asfollows. Theorem 1 of [5] shows M = X ( A L /K ) / div is an O ⊗ R L -module of finitecardinality. Since O ⊗ R L = O ⊗ ( Z p ⊗ R L ) = ( O ⊗ Z p ) ⊗ R L , we have M = ⊕ j ( M ⊗ O ′ ˆ p j ) , where O ′ ˆ p j = O p j ⊗ R L . As noted above, TheoremA.12 of [9] produces a perfect, skew-Hermitian, Gal ( K/k ) -equivariant pairing [ , ] .Defining [ x, y ] ′ = [ x, c ∗ ( y )] as before gives a non-degenerate, skew-symmetric, R L -bilinear pairing with [ sx, y ] ′ = [ x, sy ] ′ for all s ∈ O . We can therefore applyProposition 2.8 (with A = O ⊗ R L ) to see d is even. (cid:3) Main Results
We are now in a position to define and make use of the arithmetic local constantsfor our abelian variety X . Recall R = O /p O , where O ⊂
End K ( X ) . Also, recallthat for each cyclic L/K , we have a twist A L of X and rings R L (see §3) and R L = R L ⊗ Z p . Definition 6.1.
As in Definition 4.5 of [9], for each cyclic
L/K contained in F , wedefine the arithmetic local constant δ v := δ ( v, X, L/K ) by δ v := rank R ( H X ( K v , X [ p ]) /H X ∩A ( K v , X [ p ])) ( mod . Theorem 6.2.
For S L as in §1.1,corank O⊗ Z p Sel p ∞ ( X/K ) − corank O⊗R L Sel p ∞ ( A L /K ) ≡ X v ∈ S L δ v ( mod . Proof.
First, Lemma 4.1 and Theorem 4.5 giverank R Sel p ( X/K ) − rank R Sel ˆ p ( A L /K ) ≡ X v ∈ S L δ v ( mod . The claim then follows from this, Proposition 5.2 and Proposition 5.9. (cid:3)
Corollary 5.3 of [9] shows that in the elliptic curve case, δ v can be computed viaa completely local formulae, and the same arguments apply in our more generalsetting. For v a prime of K and w a prime of L above v , if L w = K v , let L ′ w bethe unique subfield of L w containing K v with [ L w : L ′ w ] = p , and otherwise let L ′ w := L w = K v . Proposition 5.2 of [9] provides an O -module isomorphism(6.1) H X ∩A ( K v , X [ p ]) ∼ = ( X ( K v ) ∩ N L w /L ′ w X ( L w )) /pX ( K v ) . Proposition 6.3 (Corollary 5.3 of [9]) . For every prime v of K , (6.1) implies δ v ≡ rank R X ( K v ) / ( X ( K v ) ∩ N L w /L ′ w X ( L w )) ( mod . Corollary 6.4.
Let S cL be the set of primes v of K such that v ramifies in L/K and v c = v . Thencorank O⊗ Z p Sel p ∞ ( X/K ) − corank O⊗R L Sel p ∞ ( A L /K ) ≡ X v ∈ S cL δ v ( mod . Proof.
The arguments are as in the proof of Theorem 7.1 of [9]. If v S cL then v c = v or v is unramified in L/K . If v c = v then Lemma 5.1 of [9] shows that δ v + δ v c ≡ . If v c = v and v is unramified then, Lemma 6.5 of [9] shows that v splits completely in L/K and hence N L w /L ′ w is surjective. Using Proposition 6.3,we see that δ v ≡ . (cid:3) The following is a first example of a class of abelian varieties for which Proposi-tion 6.2 can be used to produce a lower bound for the growth in p -Selmer ( O ⊗ Z p )-rank. Corollary 6.5.
Suppose that for every v ∈ S cF , we have v | p and X has goodordinary, non-anomalous reduction at v . If corank O⊗ Z p Sel p ∞ ( X/K ) is odd thencorank O⊗ Z p Sel p ∞ ( X/F ) ≥ ([ F : K ] , . . . , [ F : K ]) . Proof.
Suppose
L/K is a cyclic extension contained in F . Theorem 6.2 and Corol-lary 6.4 show that we need only see that δ v = 0 for all v ∈ S cF . Since v ∈ S cF , wehave v is totally ramified in L w /K v by Lemma 6.5 of [9].The assumptions that v | p and that X has good ordinary, non-anomalous reduc-tion at v allow us to apply the arguments of Appendix B of [9] to see δ v = 0 . Thekey ingredients therein are, firstly, the diagram on page 239 of [8], which applies toabelian varieties of any dimension. Secondly, non-anomalous reduction guaranteesthe relevant norm maps are surjective.Now, for each cyclic L in F , we havecorank O⊗ Z p Sel p ∞ ( X/K ) ≡ corank O⊗R L Sel p ∞ ( A L /K ) ( mod , and by our hypotheses, the left-hand side is odd. As in Theorem 7.1 of [9], thePontrjagin dual S p ( X/F ) of Sel p ∞ ( X/F ) (see for example [9, §3]) decomposes as S p ( X/F ) ∼ = ⊕ L S p ( A L /K ) , with each S p ( A L /K ) a K [ Gal ( F/K )] L ⊗ Q p -module (see §3 and Remark 3.1), and wehave just seen each has odd dimension. From K [ Gal ( F/K )] ∼ = ⊕ L K [ Gal ( F/K )] L ,we see that S p ( X/F ) contains a submodule isomorphic to K p [ Gal ( F/K )] ∼ = ⊕ L ( K [ Gal ( F/K )] L ⊗ Q p ) , and the claim follows. (cid:3) Composite Dihedral Extensions.
We now consider an abelian extension
F/K of odd degree [ F : K ] = m , and a cyclic extension L/K inside F . To easenotation, we fix some ordering of the primes in [ L : K ] = Q i p e i i , where e i > foreach i . For such L/K in F and each i , there exists a p i -power subextension M i /K such that L/M i is of degree prime to p i .By Proposition 5.10 of [10], if M and M ′ are cyclic extensions of K inside L with [ M : K ] and [ M ′ : K ] coprime and L = M M ′ , then the twist A L of X withrespect to L/K may also be realized as a twist of A M , i.e. A L ∼ = ( A M ) M ′ . Thus,if we want to compare A L and X , it suffices to compare X with A M , and also A M with ( A M ) M ′ . As in the paragraph preceeding Proposition 5.9, we consider A M and ( A M ) M ′ as defined over k .In order to inductively apply Theorem 6.2 (see Theorem 6.9 below), we assumethe following conjecture. Conjecture 6.6.
Suppose p is a prime, Y /L is an abelian variety, B ⊂ End L ( Y ) is an integral domain, and q and q ′ are primes of B above p . Then OCAL CONSTANTS FOR ABELIAN VARIETIES 17 (1) corank B ⊗ Z p Sel p ∞ ( Y /L ) is independent of p ,(2) corank B q Sel p ∞ ( Y /L ) ⊗ B q = corank B q ′ Sel p ∞ ( Y /L ) ⊗ B q ′ , Remark 6.7.
Both parts of the conjecture follow from the Shafarevich-Tate Con-jecture. Indeed, when X ( Y /L ) < ∞ , (5.2) impliescorank B ⊗ Z p Sel p ∞ ( Y /L ) = rank B ⊗ Z p ( Y ( L ) ⊗ Q p / Z p ) = ( . . . , rank B Y ( L ) , . . . ) . Each entry in the tuple is identical, giving (2), and independent of p , giving (1).For the remainder, we let F/K be as at the beginning of §6.1 with
F/k dihedral,
X/k and
O ⊂
End K ( X ) as in the previous sections (see §1.1), and assume thateach prime dividing [ F : K ] is unramified in O . For Theorem 6.9 below, we also fixa cyclic extension L/K in F .For each M/K in L , let R M denote the maximal order in Q [ Gal ( F/K )] M (as in§3 for M = L ) and O M = O ⊗ R M . Recall c is the non-trivial element of Gal ( K/k ) .Let (as in Corollary 6.4) S cM := { primes v of K : v c = v and v ramifies in M/K } . Set M = K and for each i > set M i ⊂ L to be a p i -extension of K such that p i ∤ [ L : M i ] .Using Conjecture 6.6 (2), for any p , the tuple defining corank B ⊗ Z p Sel p ∞ ( Y /L ) may be thought of as a single value, so we define r p ( Y /L, B ) ∈ Z by r p ( Y /L, B ) := corank B q Sel p ∞ ( Y /L ) ⊗ B q , where q is some prime of B above p . In turn, one may interpret the right-hand sideof Theorem 6.2 as a single value, so we define δ ( X, L/K ) ∈ Z / Z as δ ( X, L/K ) := the first component of X v ∈ S L δ ( v, X, L/K ) ! ( mod . Remark 6.8.
We emphasize that the sum of the local constants δ ( v, X, L/K ) , forfixed X and L/K , has constant parity across components, by Conjecture 6.6 (2)and Theorem 6.2. It would be interesting to determine under what conditions onecan prove that the individual δ ( v, X, L/K ) have constant parity across components. Theorem 6.9.
Assume Conjecture 6.6. For K = M , M , · · · , L as above and p aprime dividing [ L : K ] , r p ( A L /K, O L ) − r p ( X/K, O ) ≡ X i ≥ δ ( A M i − , M i /K ) ( mod . Proof.
Without loss of generality we may assume p = p . We proceed by inductionon the number j of primes dividing [ L : K ] , and the case j = 1 is that of Theorem6.2. Suppose now that j > , and let M = M and let M ′ correspond to thecompositum of the M i for < i ≤ j . Recall from the discussion above thatProposition 5.10 of [10] shows A L ∼ = ( A M ) M ′ . Arguments of Howe [6, §2] showthat A M has a polarization degree of p , in particular prime to [ L : M ] , and so we We note that O L ⊗ Z p ∼ = O ⊗ R L , with the latter as in Theorem 6.2. The new notation ismore convenient for dealing with more than one prime. can apply Theorem 6.2 in L/M with A M playing the role of X . For p ′ any primedividing [ L : M ] , by induction we have r p ′ ( A L , O L ) − r p ′ ( A M , O M ) ≡ X i ≥ δ ( A M i − , M i /M ) ( mod . Using Conjecture 6.6 (1), we have r p ( Y /K, B ) ≡ r p ′ ( Y /K, B ) ( mod , for Y = X , A M , A L , and B = O , O M , O L , respectively, and hence r p ( A L /K, O L ) − r p ( X/K, O ) ≡ r p ′ ( A L /K, O L ) − r p ′ ( A M /K, O M )+ r p ( A M /K, O M ) − r p ( X/K, O ) ≡ P i ≥ δ ( A M i − , M i /M )+ δ ( X, M/K ) ≡ P i ≥ δ ( A M i − , M i /K ) ( mod . We are able to restrict the primes v in the preceeding sums to those in S cM i just asin Corollary 6.4. (cid:3) As in Corollary 6.5, the following is a first example of a setting in which Theorem6.9 can be used to provide a lower bound for growth in the rank of E (i.e. when X = E is an elliptic curve). Corollary 6.10.
Let
E/k be an elliptic curve, K k , and assume X ( E/F ) < ∞ . For each cyclic L/K let M L,i ⊂ L be as in the paragraphs preceeding Theorem6.9. Suppose that for every prime v of K ,(1) if v = v c then v is unramified in M L,i /K for every L and each i ≥ ,(2) if v = v c and v ramifies in M L, /K then v ∤ p and E has good reductionat v .Let m be the number of primes v satisfying (2). If rank O E ( K ) + m is odd, thenrank O E ( F ) ≥ [ F : K ] . Proof.
Fix a cyclic extension
L/K inside F , and set M i = M L,i . From X ( E/F ) < ∞ we have (e.g.) rank O E ( K ) = r p ( E/K, O ) and Conjecture 6.6, so we are in thesituation of Theorem 6.9. As in Corollary 6.4, if v is unramified or v = v c then δ ( v, A m i − , M i /K ) ≡ or δ ( v, A m i − , M i /K ) + δ ( v c , A m i − , M i /K ) ≡ , respectively, for every i ≥ . For v = v c , condition (1) gives δ ( v, A m i − , M i /K ) ≡ ,for every i ≥ . Thus δ ( E, M i /K ) ≡ for i ≥ . By Theorem 2.8 of [4], condition(2) along with K k gives δ ( v, E, M /K ) ≡ (1 , , and so δ ( E, M /K ) ≡ m .Using Theorem 6.9, we combine the calculations to see that r p ( A L /K, O L ) ≡ r p ( E/K, O ) + m ( mod . By assumption, this is forces r p ( A L /K, O L ) to be odd and hence at least 1. Theclaim then follows just as in Corollary 6.5. (cid:3) OCAL CONSTANTS FOR ABELIAN VARIETIES 19
Acknowledgements.
This material is based upon work supported by the NationalScience Foundation under grant DMS-0457481. The author would like to thank KarlRubin for his many helpful conversations on this material, and thank Karl Rubinand Jan Nekovář for comments on initial drafts of this paper.
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