The arithmetic theory of local constants for abelian varieties Marco Adamo Seveso Università degli studi di Milano, dipartimento di matematica Federigo Enriques, via Saldini 50, 20133 Milano, Italy [email protected] January 14, 2012 Abstract We present a generalization of the theory of local constant developed by B. Mazur and K. Rubin in order to cover the case of abelian varieties, with emphasis to abelian varieties with real multiplication. Let l be an odd rational prime and let L/K be an abelian l-power extension. Assume that we are given a quadratic extension K/k such that L/k is a dihedral extension and the abelian variety A/k is de…ned over k and polarizable. This theory can be used to relate the rank of the l-Selmer group of A over K to the rank of the l-Selmer group of A over L. Keywords: Abelian varieties, Selmer groups. Subjects: Number theory. MSC classi…cation: 11G05, 11R20 (Primary) 11G10, 11R23, 14G05, 11F80 (Secondary).

Contents 1 Introduction

1

2 Twisting abelian varieties

5

3 The arithmetic theory of local constants

7

A O-polarizations on abelian varieties with real multiplication

1

10

Introduction

In the paper [7] Mazur and Rubin provide an arithmetic theory of local constants to study the parity of the Selmer groups. We brie‡y sketch their ideas and

1

present a generalization of the main result, which applies to abelian varieties with real multiplication. This is needed in [14], where the result is applied in order to relate the growth of the Selmer group to the rationality of Stark-Heegner points. Let K=k be a quadratic extension of a number …eld k and let L=K be a …nite abelian l-power extension, with Galois group , such that L=k is dihedral, where l 6= 2 is a rational prime. Let (A; O; l) = (A; O; l)k be triple where (A; O)k is an abelian variety with multiplication by an order O in a number …eld F (the data being de…ned over k) and l O is a prime ideal of residue characteristic l. Since O is commutative the dual abelian variety At is canonically endowed with multiplication by O by letting : A ! A acts on At through t : At ! At . We make the following assumptions on the triple (A; O; l): it is polarizable, meaning that there exists an O-linear and symmetric isogeny : A ! At of degree prime to l; (O; l) is regular, meaning that l is prime to the conductor of O in OF , the ring of integers of F ; (O; l) is unrami…ed, meaning that the discriminant of F=Q is prime to l. The present paper is organized as follows. Section 2 quickly reviews the theory of twisting abelian varieties and introduces some preliminary material. Section 3 is devoted to review the Mazur-Rubin argument in the more general setting of abelian varieties with multiplication by an order O. The main result is Theorem 3.2, which applies to triples (A; O; l) as above and notably under the assumption [F : Q] = dim A. We note that it is remarked in [7] that their theory generalizes to abelian varieties; Section 3 is understood to provide a suitable reference where unspoken assumptions that are always satis…ed in the case of elliptic curves are made explicit. In particular, when [F : Q] = dim A and one considers a prime l as above, the l-torsion and the l-adic Tate module look essentially like those of an elliptic curve. Finally, the appendix "O-polarizations on abelian varieties with real multiplication " is devoted to eliminate the above polarizability assumption when (A; O) has real multiplication, i.e. F is a totally real …eld and [F : Q] = dim A. Let Ol be the completion of O at the prime ideal l, which is a discrete valuation ring since (O; l) is regular. Then we have M Ol = Ol0 , 0 l jl

thus inducing a decomposition

Sell1 (A=L) =

M

l0 jl

2

Sell01 (A=L) .

We will be interested in the Selmer group Sell1 (A=L) attached to the prime l j l. Note that it is a co…nitely generated module over the discrete valuation ring Ol , so that it makes sense to talk about its corank over Ol . Let Irr ( ) be the set of all the rational irreducible representations of and note that Sell (A=L) and Sell1 (A=L) are Ol [ ]-modules in a natural way. There is a canonical isogeny (map with …nite kernel and cokernel) L

isog

Sell1 (A =K) ! Sell1 (A=L) ,

(1)

where Sell1 (A =K) is the Selmer group of a twist A of A by in the sense of [8] and l is the unique prime ideal of Ol [ ] corresponding to the irreducible representation and dividing l (see the preliminary discussion in Section 2). The action of O [ ] on factors through a quotient O , whose completion O ;l = O ;l at l is again a discrete valuation ring. Denote by F the residue …eld of Ol , which is also the residue …eld of O ;l for every (again we refer to Section 2). The isogeny decomposition (1) yields corankOl (Sell1 (A=L)) =

P

r corankOl;

where we set r := rankOl (Ol; )

Sell1 (A =K) ,

(2)

1.

Hence, in order to get information on corankOl (Sell1 (A=L)) from the knowledge of corankOl (Sell1 (A=K)) we are led to compare corankOl (Sell1 (A=K)) with corankOl; Sell1 (A =K) . It turns out that, assuming that (A; O) has real multiplication, every regular and unrami…ed triple (A; O; l) is polarizable (see Theorem A.11), so that the following generalization of [7, Theorem 7.1, case (a)] is deduced from Theorem 3.2. Let ram (L=K=k) be the set of primes v of K that ramify in L and are inert or rami…ed over k, i.e. v is unrami…ed in L and v c = v for the non-trivial automorphism c of K over k. Denote by Bad (A=K) the set of primes of K that are of bad reduction for A=K. Theorem 1.1 Assume that (A; O; l)k is a regular and unrami…ed triple with real multiplication. If ram (L=K=k) \ Bad (A=K) = and ram (L=K=k) \ fv j lg = , for every dimF Sell (A =K)

dimF (Sell (A=K)) mod 2

and corankOl;

Sell1 (A =K)

corankOl (Sell1 (A=K)) mod 2.

The main application of Theorem 1.1 is the following. De…ne parityl1 (A=K) := corankOl (Sell1 (A=K)) mod 2.

3

Corollary 1.2 Assume that (A; O; l)k is a regular and unrami…ed triple with real multiplication. If ram (L=K=k) \ Bad (A=K) = , ram (L=K=k) \ fv j lg = and parityl1 (A=K) = 1 then corankOl (Sell1 (A=K))

[L : K] .

Proof. This is a consequence of Theorem 1.1, together with (2) and the equality P r = # (see following (5) for this last equality).

Note also that, whenever A [l] (K) = 0, it follows from the subsequent (7), together with the congruence relation appearing in the subsequent Theorem 2.2, that we have: parityl (A=K) := dimF (Sell (A=K))

corankOl (Sell1 (A=K)) mod 2.

Another useful application is the following Corollary, that we state without proof. Corollary 1.3 Under the assumption of the above Corollary, further suppose A [l] (K) = 0. Then, for every n 2 N, Selln (A=L) contains an of rank at least [L : K] and, in particular, dimF (Sell (A=L))

O ln

[ ]-module which is

O ln -free

[L : K] :

The paper [7] deals with the theory of local constants for elliptic curves. As remarked in [7], an extension of the theory of local constants to abelian varieties is expected. The aim of the present work is to provide such a generalization. The results of Section 3 work for arbitrary polarized, regular and unrami…ed triples (A; O; l) = (A; O; l)k . In the setting of elliptic curves the existence of a principal polarization makes the polarizability assumption supe‡uous. As explained, the appendix "O-polarizations on abelian varieties with real multiplication " removes this assumption when A has real multiplication by an order O in a (totally real) number …eld F . We do not prove the existence of a principal polarization, which is false when the class number of F is not one, but rather show the existence of a polarization of degree prime to l. Of course Theorem 3.2 is applicable to other scenarios: for example when (A; O; l) = (A; Z; l) and A is an abelian surface with quaternionic multiplication it is well known that a principal polarization exists. The above corollaries are true for an arbitrary polarized, regular and unrami…ed triple (A; O; l) = (A; O; l)k , but we have choosen to state them without the polarizability assumption when we have real multiplication since this is the statement needed in [14].

4

2

Twisting abelian varieties

Let O = OF be the ring of integers of a number …eld F and let be a …nite abelian l-torsion group. We explain how to get a canonical isogeny decomposition of the group algebra O [ ]. This will be regarded like an algebra with involution using the inversion in . Let IrrF ( ) be the set of isomorphism classes of distinct irreducible representations of over F . Since F [ ] is semisimple there is a canonical decomposition L F[ ]= 2IrrF ( ) F , where F =F are …eld extensions. The projection of F [ ] onto F gives the action of F [ ] on the representation . De…nition 2.1 An ideal I O[ ] is O-‡at. I

O [ ] is said to be saturated whenever the quotient

Tensoring a saturated module I O [ ] over O with F and conversely intersecting an ideal in F [ ] with O [ ] establishes a bijection between the set of the saturated ideals of O [ ] and the ideals of F [ ]. We will denote by I := F \ O [ ] the saturated ideal corresponding to the …eld F . Since 2IrrF ( ) I is an O [ ]-submodule contained in O [ ] with …nite index we get an O [ ]-linear isogeny: L

isog

2IrrF ( )

I ,! O [ ] :

(3)

The aim of the following discussion is to study the O [ ]-module structure of the modules I , which factors through a certainly quotient O contained in the …eld F . From now on we will assume that F is unrami…ed over l 6= 2. The e¤ect of this assumption is that the study of O is essentially reduced to the case O = Z. The following facts can be directly proved or can be deduced from the case O = Z and [8, in particular Lemma 5.4]. The set IrrF ( ) is in bijection with the subgroups 0 such that = 0 is cyclic, i.e. with IrrQ ( ). If 2 IrrF ( ) corresponds to a cyclic quotient = 0 of order lr we have F ' F ( ), where a primitive lr -root of unity. Denote by l the unique prime of Z [ ] dividing l, so that Z [ ] l = le . The ring O [ ] acts on I through the quotient O ' O [ ] and for every prime l0 j l of O there is one only prime l0 of O ' O [ ] over l0 and l0e = O l0 , where e = ' (lr ). We have Q Q lO = l0 jl l0 and l O = l0 jl l0 . (4)

In particular the localization O ;l of O at l is a discrete valuation ring with 1 uniformizers 1 or := and we have O ;l l = O ;l . Indeed F ;l =Fl r is totally rami…ed of degree ' (l ) and O ;l ' Ol [ ]. Under the isomorphism O ' O [ ] we have I ' ( l 1) O , where l is a primitive l-root of unity r 1 1 and in particular O ;l I ' l O . Furthermore sending to induces a 5

well de…ned involution on O ' O [ ] such that the projection of O [ ] onto O ' O [ ] is a morphism of rings with involution. Then the formula [ ; ] : O ;l I 2l

[ ; ] :=

r

1

O ;l I ! O ( )

gives on O ;l I a perfect and (O ; )-Hermetian O -valued pairing. We also note that, if we write I ;Z (resp. I ;O ) for the saturated ideal corresponding to the same irreducible Q-representation (resp. F -representation) it holds the equality I ;O = O Z I ;Z . In particular, setting r := rankOl (Ol; ) as in the introduction, we have P r =# . (5) Recall the extensions L=K=k that was considered in the introduction, as well as the polarizable, regular and unrami…ed triple (A; O; l) = (A; O; l)k . The above discussion applies to := GL=K . We denote by F the common residue …eld of O (resp. Ol ) and O (resp. Ol; ) at l (resp. Ol l) and at l (resp. O l ). The above discussion implies that the twists A := I

;O

O

A=I

;Z

Z

O

O

A=I

;Z

Z

A

do not depend on the ring O over which we are twisting. These twists have been already considered in the literature and we refer to [8] for details. A is again an abelian variety and for every K-algebra X there is a canonical identi…cation (see [8, Theorem 1.4]) (I

;O

O

A) (X) = (I

;O

O

A (X

K

GL=K

L))

.

As it follows from [8, Prop. 4.1], O [ ] O A is identi…ed with the restriction of scalars from K to L of A=K. It then follows that we have (see [7, Prop. 3.1]): Sell (A=L) ' Sell (O [ ]

O

A=K)

and hence the isogeny (3) yields the O [ ]-linear isogeny decomposition (1). The abelian varieties A =K are endowed with multiplication by O acting on I ;O and we can consider the triple (A ; O ; l ). Note that (A ; O ; l ) is again regular and unrami…ed, as it follows from the above discussion, but may fail to be polarizable. The decomposition (4) yields canonical O=Oln -module decompositions (see also [6] for a direct de…nition of the l -adic Selmer groups): L A ln = l0 jl A l0n for n 2 N, (6) M Sell1 (A=K) = Sell01 (A=K) . 0 l jl

As in the setting of classical Selmer groups one can prove that there is an exact sequence: 0!

l 1 O

R

A

l1 (K) ! Sell (A =K) ! Sell1 (A =K) [l ] ! 0 6

(7)

and furthermore dimF

l 1 O

R

A

1

l

!

(K)

= dimF (A [l ] (K)) .

(8)

Let Tl (A) be the Tate module attached to the l-adic representation of the abelian variety A=k over the …eld k. We recall the following fact (see [8, Theorem 2.2]): there is a GK -equivariant isomorphism Tl (A ) = I

;O

O

Tl (A) ,

where GK -acts on the right hand side via g 1 g. The polarizability assumption on (A; O; l) implies that the Ol (1)-valued Weil pairing on Tl (A) is perfect. We only sketch the proof of the following Theorem, which is a straightforward generalization of [7, Theorem A.12 and Prop. A.11], which in turn is an application of the techniques developed in [4] (see also [13, Prop. B.27] for further details). Theorem 2.2 The pairing h ; i induces on Sell1 (A =K) a GK=k -equivariant and skew-(Ol; ; )-Hermetian pairing with kernel Sell1 (A=K)div : Sell1 (A =K)

Sell1 (A =K) !

Fl; Ol;

In particular corankOl;

Sell1 (A =K)

dimF Sell1 (A =K) [l ]

mod 2.

Proof. The …rst step in the proof of the Theorem is to provide a "model" of A over k. This can be done as in [7, Prop. A.9]: choose a lift c of the non-trivial automorphism of the extension K=k to the algebraic closure k=k. For every 2 IrrF ( ) de…ne JO; := (1 + c) IO; , which is a right ideal in O GL=k . The ' left multiplication by 1 + c gives a right O [ ]-module isomorphism I ! J , thus inducing an isomorphism over K: A := I ;O O A ' JO; O A =: A0 . The second step is in producing Ol; (1)-valued perfect, skew-(Ol; ; )-Hermetian and GK -equivariant pairings on Tl (A ), that are Gk -equivariant when making ' the identi…cation Tl (A ) ! Tl A0 . They are obtained setting h ; i := [ ; ] e where e is the Ol (1)-valued Weil pairing on Tl (A ), which is perfect in light of the polarizability assumption on (A; O; l). Finally one applies the Flach construction as explained in [7, Theorem A.12].

3

The arithmetic theory of local constants

Recall our polarized, regular and unrami…ed triple (A; O; l) = (A; O; l)k from the introduction. Since the claim of Theorem 3.2 is invariant under isogenies of degree prime to l we can suppose that O = OF (see Lemma A.10). 7

Lemma 3.1 There are canonical identi…cations of F [GK ]-modules: A [l ] = A [l] : Proof. By [7, Prop. 4.1] there is a canonical identi…cation of Fl [GK ]-modules A [l ] = A [l], which is an identi…cation of O=lO-modules (by the canonicity). The claim follows from the O=Ol-modules decompositions (6). The …rst of the subsequent congruences follows from Theorem 2.2, while the second follows from Lemma 3.1 together with the exact sequence (7): corankOl;

Sell1 (A =K)

corankOl (Sell1 (A=K))

dimF Sell1 (A =K) [l ] dimF (Sell1 (A=K) [l]) dimF Sell (A =K) dimF (Sell (A=K))

(9) mod 2.

The F [GK ]-module identi…cation A [l ] = A [l] allow us to view Sell (A =K) as the sub-F-module of H 1 (K; A [l]) de…ned by the set of local conditions S , S

;v

:= Im

A (Kv ) v 1 ,! H (Kv ; A [l ]) = H 1 (Kv ; A [l]) l A (Kv )

for the local Kummer map v at v. Recall that, as explained in the proof of Theorem 2.2, the O-polarization on A induces on Tl (A ) = Tl (A ) perfect twisted pairings and, up to the canonical identi…cation A [l ] = A [l], they induce the same pairing on A [l]. It follows that, for every , the Selmer structure S is selfdual (for the Weil pairing on A [l]) by the Bloch-Kato generalization of Tate’s local duality applied to Tl (A ) (see [2, Prop. 3.8] and [7, Prop. A.7]). We can now apply [7, Theorem 1.4], whose proof relies on a clever argument of Howard, which implies: dimF Sell (A =K)

P

dimF (Sell (A=K))

v2

dimF

Sv Sv \ S

mod 2, ;v

(10) where denotes the set of primes out of which the Selmer structures S and S coincides, which is …nite. The local constants are de…ned as being v

=

v

(A; O; l; L=K; ) := dimF

Sv Sv \ S

in Z=2Z:

;v

We are going to recover, from the results of [7], an explicit description of the Sv . In order to connect these spaces with those de…ned in [7] when spaces Sv \S ;v working over Z, we will write Sv = Sv;l (resp. S ;v = S ;v;l ) to emphasize the dependence on the chosen prime l j l. Then it is clear that we have Sv;l Sv;l \ S

= ;v;l

L 8

l0 jl

Sv;l0 Sv;l0 \ S

. ;v;l0

S

The space Sv;l \Sv;l;v;l can be explicitly described as follows. For every let L =K be the cyclic subextension of L=K corresponding to the irreducible representation , choose a prime w of L dividing v and set: L0 ;w :=

unique sub…eld such that L L ;w

;w

: L0 ;w = l

if Kv 6= L if Kv = L

;w ;w

The proof of [7, Corollary 5.3] readily generalizes to our setting and gives a canonical identi…cation Sv;l Sv;l \ S

= ;v;l

A (Kv ) A (Kv ) \ NL ;w =L0 ;w A (L

;w )

.

Theorem 3.2 Let ram (L=K=k) be the set of primes of K which ramify in L=K and such that v c = v (if c is the non-trivial automorphism of K=k) and let Goodl (A=K) be the set of primes of K of good reduction for A=K which do not divide l. Then, for every , corankOl; P

Sell1 (A =K)

corankOl; P

Sell1 (A =K)

v2ram(L=K=k) v

corankOl (Sell1 (A=K))

(A; O; l; L=K; ) .

and when [F : Q] = dim A corankOl (Sell1 (A=K))

v2ram(L=K=k) Goodl (A=K) v

(A; O; l; L=K; ) .

Proof. According to (9) and (10) we have to show that v = 0 whenever v 2 = ram (L=K=k) (resp. v 2 = ram (L=K=k) Goodl (A=K) when [F : Q] = dim A). If v 2 = ram (L=K=k) there are two possibilities: v c 6= v or v c = v and v 2 = ram (L=K). In the …rst case it is clear that [7, Lemma 5.1] generalizes: since the entire triples (A; O; l) and (A ; O ; l ) are de…ned over k (see the proof of Theorem 2.2) the automorphism of K=k induces isomorphisms '

(A; O; l)Kv ! (A; O; l)Kvc , '

(A ; O ; l )Kv ! (A ; O ; l )Kvc . '

Therefore, under the isomorphism H 1 (Kv ; A [l]) ! H 1 (Kvc ; A [l]) induced by conjugation, Sv (resp. S ;v ) corresponds to Svc (resp. S ;vc ) and hence vc = v (thus, assuming v c 6= v, we have vc + v = 0). In the second case, since v c = v, by [7, Lemma 6.5 (i)], for every prime w of L the extension L ;w =Kv is (nontrivial) totally rami…ed or v splits completely. Since the …rst possibility has to S be excluded (otherwise L=K should be rami…ed too), the module Sv;l \Sv;l;v;l is trivial by the previous description and hence the direct addend trivial too (so that v = 0). It follows the …rst formula. 9

Sv;l Sv;l \S ;v;l

is

Now assume [F : Q] = dim A and let v 2 ram (L=K=k) be a prime such that v - l and A=K has good reduction at v (more generally assume v c = v, v - l and A=K to have good reduction at v). Since v c = v, by [7, Lemma 6.5 (i)] there are two possibilities: L ;w =Kv is (non-trivial) totally rami…ed for (every) w dividing v or v splits completely in L , but in this latter case we have seen S that Sv;l \Sv;l;v;l is trivial. Thus we may assume L ;w =Kv to be totally rami…ed (non-trivial) and since v - l and A=K has good reduction at v the proof of [7, Theorem 5.6] generalizes. More precisely [7, Lemma 5.4] readily generalizes to the case of an abelian variety A=K with multiplication, giving the identi…cation l 1 l 1 1 O A (K) = O A [l ] (K) ; O O for every local …eld K of residue characteristic coprime with l. It is also true that [7, Lemma 5.5] generalizes to abelian varieties A=K, showing that, whenever in this case L=K is a (non-trivial) totally rami…ed extension and A=K has good reduction, there are equalities: NL=K A (L) = lA (K) if [L : K] = l; A (K) \ lA (L) = lA (K) : These last two equalities applies …rst to L

0 ;w =L ;w

and then to L0 ;w =Kv , giving:

A (Kv ) A (Kv ) = : lA (Kv ) A (Kv ) \ lA L0 ;w L A(Kv ) From the canonical decomposition of lA(K as a module over O=lO = l0 jl Fl0 v) and then using the generalization of [7, Lemma 5.4], we …nd Sv;l Sv;l \ S

=

;v;l

A (Kv ) A (Kv ) \ NL ;w =L0 ;w A (L

Sv;l Sv;l \ S

= ;v;l

By (8) the F-dimension of A [l] (Kv ). Thus we …nd

l

l 1 O

O

1

O

O v

;w )

=

A (Kv ) =

l 1 O

O

A [l1 ] (Kv ) :

A [l1 ] (Kv ) is the same as the F-dimension of dimF A [l] (Kv ) ;

which is the claimed generalization of [7, Theorem 5.6]. Hence to get the second formula it is enough to show that the F-dimension of A [l] (Kv ) is even and this is where the assumption [F : Q] = dim A plays its role. Indeed this last fact follows exactly as in [7, Lemma 6.6], using the F [Gk ]-vector space A [l] (which is 2-dimensional over F since [F : Q] = dim A, by Proposition A.11) and the F (1)-valued Weil pairing on A [l] in a place of the Fl [Gk ]-vector space E [l] and the Fl (1)-valued Weil pairing, which is used in the case of an elliptic curve.

A

O-polarizations on abelian varieties with real multiplication

Let A be an abelian S-scheme with multiplication by an order O in a totally real number …eld F such that dimS A = [F : Q] is constant. The dual abelian 10

S-scheme At =S, that we assume to exists (for example suppose that A=S is projective), is canonically endowed with multiplication by O. Let HomSym A; At := O

: A ! At :

=

t

and

is O-linear

be the set of all symmetric O-linear morphisms (over S) from A=S to the dual abelian scheme At =S (here = t up to the canonical identi…cation A = Att ). Let us be given a prime l O of residue characteristic l invertible in S. For a torsion free …nitely generated Ol -module T , T = T canonically where we write T = HomOl (T; Ol ) to denote the Ol -dual. Using this identi…cation we may de…ne HomAlt Ol (T; T ) := f : T ! T :

=

and

is Ol -linearg ,

called the Ol -module of alternating homomorphisms from T to T . Then Tl (At ) = Tl (A) and we can consider the following commutative diagram (see [3, 1.5]): HomSym (A; At )l ,! HomAlt Ol (T; T ) O (11) \ \ HomO (A; At )l ,! HomOl (T; T ) where we set T := Tl (A). Here Ml = M O Ol is the l-adic completion of M , since the modules involved are …nitely generated. If M is an O-module we write M to denote the constant presheaf M (T =S) := M on the big étale site and we let M be the associated sheaf M (T =S) = M 0 (T ) , where 0 (T ) denotes the set of connected components of T and M 0 (T ) is the O-module of maps from 0 (T ) to M . We also write M and M to denote their restrictions to the étale site. If M is a projective O-module of …nite rank rM , the functor M O A de…ned by the rule (M

O

A) (T =S) := M

O

A (T =S) , T =S arbitrary

is representable by an abelian S-scheme with multiplication by O and dimension dimS M O A = rM dimS A. Indeed this is clear when M = On ; in general we may write On ' M N , so that (M O A) (N O A) ' An holds as functors and then we may apply Yoneda’s Lemma to deduce the existence of an idemponent eM 2 EndS (An ) such that eM An ' M O A. In particular M O A is a sheaf on the big étale site and its restriction to the étale site is a sheaf. On the other hand we may consider the presheaf M pO A de…ned by the rule (M

p O

A) (T =S) := M (T =S)

O

A (T =S) = M

0 (T )

O

A (T =S) ,

where T =S may be an arbitrary S-scheme or an étale S-scheme if we work with the étale site. We write M pO A ! M O A to denote the associated 11

presheaf with the canonical shea…cation morphism. We may therefore consider the canonical morphism p

:M

O

p O

A!M

A!M

A

O

where p is given by the shea…cation morphism M ! M , i.e. the diagonal morphism M (T =S) = M ! M (T =S) = M 0 (T ) . More generally, for an arbitrary sheaf F, we write F O A for the sheaf associated to the presheaf F pO A de…ned by the rule (F pO A) (T =S) := F (T ) pO A (T ). Lemma A.1 The canonical morphism M O A ! M as sheaves on the big étale site or the étale site.

O

A is an isomorphism

Proof. After a base change to an arbitrary S-scheme T , i.e. after restricting the values of our functors to étale T -schemes T 0 =T we may work with the étale site. Su¢ ces to check that the canonical morphism is an isomorphism on the stalks (see [9, II Theorem 2.15] or [15, II Theorem (5:6) iii)]). Let s ! S be a geometric point and let OS;s = lim OU be the strict henselization of OS;s , where the limit !

runs over all étale neightbourhoods of s. Set Ss := Spec (OS;s ) = lim U . If F is a presheaf on the étale site, Fs = F# s where F# is the associated sheaf (see [9, II Remark 2.14 (c)]); in particular (M O A)s = (M pO A)s . There is a canonical morphism Fs = lim F (U ) ! F lim U = F (Ss ) . !

Since F = M

OA

is an S-scheme of …nite type, lim (M !

O

A) (U ) = (M

O

A) lim U

(see [9, II Remarks 2.9 (d)]). Similarly, since we may assume that the U s appearing in lim OU are connected, A is an S-scheme of …nite type, the formation ! of direct limits commutes with tensor products and Ss is connected, lim (M !

p O

A) (U )

= lim (M

O

!

= M

O

A (U )) = M

A lim U = (M

O p O

lim A (U ) !

A) (Ss ) .

Summarizing we have canonical identi…cations (M O A)s = (M O A) (Ss ), induces on the (M O A)s = (M pO A) (Ss ) and the canonical morphism stalks the morphism p (Ss ), which is the identity since Ss is connected. We sketch a proof of the following well known fact. Proposition A.2 Let A be an abelian S-scheme (with O-multiplication) and let 0 6= 2 L be an element of an invertible O-module L. Then L O A is an abelian S-scheme of the same dimension as A, :A!L (a) := 12

A O a

O

is a faithful ‡at morphism, A [ ] := ker

is …nite and ‡at over S and

#A [ ] = #

L 2 , O

where #A [ ] is the order of the …nite S-scheme A [ ]. Furthermore, if # invertible on S, is étale (and hence A [ ] is S-étale).

L O

is

Proof. Consider the exact sequence L ! 0. O We view A as a sheaf for the fppf topology and apply O A. Since L is O-‡at, 1 T orO (L; A) = 0, and, since A is divisible and LO …nite, LO O A = 0. It follows that we have L 1 0 ! T orO ; A ! A ! L O A ! 0. O 0!O!L!

L 1 In particular, since T orO O ; A = A [ ] is …nite, L O A is an abelian S-scheme of the same dimension as A. We also see that is an isogeny, hence a faithful ‡at morphism and an étale morphism when #A [ ] is divisible on S. In order to compute the degree #A [ ] of the …nite S-group scheme A [ ], we …rst assume that ( ; L) = ( ; O), so that 2 O EndS (A) is just the multiplication by map. Since A [ ] is S-‡at, working with each connected component of S, we may …rst assume that S is connected and then compute the degree of A [ ] over S after a base change to the residue …eld k (s) at any s 2 S. Hence we may assume that S = Spec (K) for a …eld K. Setting N ( ) := #A [ ], we see that N is a norm form on F=Q homogeneus of degree 2 dimS A (by [10, III §19 Theorem 2]). It follows from [10, III §19 Lemma] and our assumption 2 2 dimS (A) = [F : Q] that we have N = NF=Q , so that #A [ ] = NF=Q ( ) for 2 O. The claim when L = O follows from the fact that NF=Q ( ) = # OO because O is a lattice in F . We now remark that, if we have given ( i ; Li ) with 2 Li an invertible O-module, i = 1; 2 and if we assume that # L11O is prime to # L22O , the morphism

: L1 L2 ! L1 O L2 (l1 ; l2 ) := l1 2+ 1 2

1; 2 1;

l2

2 induces an isomorphism L11O L22O ' L11 OO L2 O . We apply this remark as follows. 0 We take ( 1 ; L1 ) = ( ; L) and ( 2 ; L2 ) = ; L 1 , where 0 2 L 1 is choosen 1 1 so that # LO is prime to # L0 O . We deduce that # O0 O = # LO L0 O . On the other hand, we may consider the composition

0

and deduce #A

0

#

:A!L

0

O

A!L

= #A [ ] # (L

O

1

A)

O 0

L L 1 # 0 = #A [ ] # (L O O 13

L

O

A=A

. By the case L = O we get O

A)

0

.

L O;A

1 Since T orO

choice of (L O A)

0 0

1 = A [ ] and T orO L O

L

1

L

1

0O

;L

O

A

= (L

O

A)

0

, our

so that # is prime to # 0 O implies that #A [ ] is prime to too. We deduce # LO = #A [ ].

Proposition A.3 If Ol is a discrete valuation ring, the horizontal inclusions appearing in (11) have torsion free cokernel. Proof. It su¢ ces to show that the lower horizontal inclusion and the left vertical inclusion in (11) have torsion free cokernel, since then the torsion freeness of the upper horizontal inclusion follows from the commutativity of (11). Furthermore, to see that the left vertical inclusion has torsion free cokernel it su¢ ces to show that the inclusion of HomSym (A; At ) in HomO (A; At ) has torsion free cokernel. O Suppose that 2 HomO (A; At ) is such that there exists 0 6= n 2 Z such that t n 2 HomSym (A; At ). Then n = (n ) and the right hand side is t nt = t n, O while the left hand side is n. Hence n = t n and, since n is an isogeny, it is an epimorphism in the category of sheaves for the fppf topology on S and the equality = t can be checked at the level of points. That the lower horizontal inclusion has torsion free cokernel follows as in [5, Theorem (12:10), Chapter 12], thanks to our assumption that Ol is a discrete valuation ring. Let L be an invertible O-module. For every : F ,! R we may consider i : L ,! L R and de…ne L := i 1 L R . An oriented invertible Omodule is an invertible O-module L together with the choice of a vector of signs " = (" ) such that " 2 f g. An homomorphism f : (L1 ; "1 ) ! (L2 ; "2 ) of oriented invertibles O-modules is an homomorphism f of O-modules such that " " f L1;1; L2;2; . To an orientation " on L we can associate the set of totally positive elements T " L+ := L L

and indeed, the isomorphism class of the oriented O-module L, is uniquely determined by the couple (L+ ; L). Furthermore, since Aut (L+ ; L) = O+ , the set of totally positive units of O, we see that, to give an oriented O-module (L+ ; L) (up to isomorphism), is the same as to give an element in the narrow class group associated to the order O. Consider the functor of symmetric O-linear morphisms: AT ; AtT L (A) (T =S) := HomSym O and the subfunctor L+ (A) of polarizations (where AT and AtT are the base changes to T ). They are in fact sheaves for the étale topology and in [12, Prop. 1.17], when O = OF is the maximal order in a totally real …eld F , it is proven that L (A) is locally constant with values in invertible O-modules and generated by L+ (A). When S is connected and normal, by [12, Variante 1.18] L (A) is already constant, i.e. L (A) = L (A) for the invertible O-module L (A) := L (A) (T =S) = L (A) (S) where T =S is connected. In general there exists T =S étale such that L (A) = L (AT ) restricted to étale T 0 =T . 14

Let p : L (A) pO A ! At be the morphism given by the rule p ( a) := Sym t t (a), where 2 HomO (AT ; AT ) and a 2 AT (T ) = A (T =S). Since A is a sheaf, it induces a unique : L (A)

O

A ! At .

Note that, when L (A) = L (A) is constant, of Lemma A.1, with the morphism

(12)

is identi…ed, via the isomorphism

: L (A) O A ! At ( a) := (a) 2 At (T =S) , a 2 A (T =S)

(13)

which is an isogeny being a non-zero map between abelian S-schemes of the same dimension (see also [1, proof of (3) ) (1) of Proposition 3.1]). The following proposition is a variant of [1, Proposition 3.1]. Proposition A.4 The following are equivalent, when O = OF : (1) ker

is n-torsion étale sheaf, where

is (12);

(2) for every integer t prime to n there exists T =S étale and of degree prime to t.

2 L+ (A) (T =S)

Proof. In [1, proof of (1) ) (2) of Proposition 3.1] simply replace [1, (3:6)] with the assertion that : (A OL MA ) [t] ,! A_ [t] is an inclusion (notations as in loc. cit.) and note that it is an isomorphism by a comparison of degrees. Hence one recovers [1, (3:6)] and the proof of the implication is the same. [1, proof of (2) ) (3) of Proposition 3.1] (with the obvius modi…cation in the statement) is trivial. The analogous of [1, proof of (3) ) (1) of Proposition 3.1] is the same but we conclude that K has order prime to l for every prime l prime to n. Corollary A.5 Suppose that S is connected and normal or, more generally, that L (A) = L (A) is a constant sheaf and that O = OF . The following are equivalent: (1) ker

is n-torsion étale sheaf, where

is (13);

(2) for every integer t prime to n there exists t.

2 L+ (A) of degree prime to

Proof. We need to prove that, if there exists 0 2 L+ (A) (T =S) of degree prime to t, there exists 2 L+ (A) of the same degree. Since L (A) = L (A), (T )

L (A) (T =S) = L (A) 0 . We take any connected component T0 T of T and we note that the image of 0 in L (A) (T0 =S) = L (A), which is simply the (T ) T0 -component of 0 2 L (A) 0 (that we may assume to be non-zero), is the required polarization.

15

We are now going to show that, under the assumptions of Corollary A.5, condition (1) is always satis…ed if we take n to be the product of the primes that are invertible in S. Indeed we will prove a slightly more general statement without assuming that O = OF . We …rst record the following corollary of Propositions A.2 and A.3. Corollary A.6 If HomSym (A; At ) 6= 0 and Ol = OF;l coincides with the comO pletion at l of the maximal order OF , the upper horizontal inclusion appearing in (11) is an isomorphism A; At HomSym O

l

= HomAlt Ol (T; T ) ' Ol .

Proof. For every Ol -module T let us denote by HomOl ^2Ol T; Ol the Ol module of bilinear forms b : T T ! Ol such that b (x; y) = b (y; x). Under the 2 canonical identi…cation HomOl Ol T; Ol = HomOl (T; T ) the submodule 2 HomOl ^Ol T; Ol corresponds to HomAlt Ol (T; T ). Proposition A.2 implies that Tl (A) is a free rank two module over Ol . It follows that HomOl ^2Ol T; Ol is free of rank one over Ol . Since HomSym (A; At ) is torsion free, the inclusion O Sym (11) shows that the free Ol -module HomO (A; At )l may have rank 0 or rank 1 (over Ol ). Under the assumption HomSym (A; At ) 6= 0 we …nd that it holds O the second possibility and the inclusion is an isomorphism, since it has torsion free cokernel by Proposition A.3. Lemma A.7 Suppose that S is connected and normal or, more generally, that L (A) = L (A) is a constant sheaf. Let nS;O 2 N be the product of those primes l 2 N that are invertible in S or such that Ol = OF;l (i.e. l is coprime with the conductor of O in the maximal order OF ). Then the primes dividing # ker divides nS;O and, in particular, ker is an nS;O -torsion étale sheaf. Proof. As we already remarked in (13) is an isogeny. Let l be a prime that is invertible on S: for every such prime, to prove that is an isogeny of degree coprime with l, it is su¢ cient to show that Tl ( ) is an isomorphism. Whenever Ol = OF;l this is equivalent to checking that Tl ( ) is an isomorphism for every prime l j l. The formation of the l-adic Tate module, commutes with taking tensor product with ‡at O-modules, so that: Tl (L (A)

O

A) = L (A)

O

Tl (A) = L (A)l

Ol

Tl (A) .

For every Ol -module write LAlt (T ) := HomAlt Ol (T; T ) for shortness. Consider the canonical morphism: =

T

: L (A)l

16

Ol

Tl (A) ! T t 7! (t) .

LAlt (Tl (A)) gives the following commuta-

The canonical inclusion L (A)l tive diagram: L (A)l

Ol

Tl (A)

\

LAlt (Tl (A))

Ol

Tl (A)

Tl (

!A

)

Tl (A) k

Tl (A)

!

Tl (A)

Since L (A) 6= 0 and Ol = OF;l , by Corollary A.6 the left vertical inclusion is an isomorphism. In other words the problem is reduced to the analogous problem in the category of Ol -modules, i.e. we must show that, for every free rank 2 Ol -module T , the canonical morphism T is an isomorphism. This is a consequence of the subsequent algebraic Lemma A.8 Lemma A.8 Let T be a …nitely generated free module over a commutative ring R. The canonical morphism T

: HomAlt T !T R (T; T ) f t 7! f (t)

is surjective if rank (T ) 6= 1 with projective kernel. Furthermore we have (with the convention that rank 0 means rank = 0): rank (T ) = rank (T ) , 2

rank (T ) (rank (T ) 1) , 2 rank (T ) (rank (T ) 2) (rank (T ) + 1) rank (ker T ) = . 2 rank Hom ^2 T; R

In particular

T

T =

is an isomorphism if and only if rank (T ) = 2 or 0.

2 Proof. Up to the canonical identi…cation HomAlt R (T; T ) = HomR ^R T; R the morphism T corresponds to the canonical morphism: T

: HomR ^2R T; R T !T b t 7! b (t; ) .

Choose a basis fei : i = 1; :::; ng of T and let f i g be the dual basis of fei g, de…ned by the rule i (ej ) = ij . By viewing the elements of T like column vectors with respect to the basis fei g, the element ei corresponds to the column 1i which is zero at all its entries with the only exception of the i-entry, which is equal to 1, and similarly i corresponds to the row vector 1ti which is equal to zero at all its entries with the only exception of the i-entry, which is equal to 1. Furthermore any element of b 2 Hom ^2 T; R corresponds to a matrix b = (bij ) which has zeros on the diagonal and such that bij = bij for i 6= j. In particular a basis of Hom ^2 T; R is obtained from the elements bi0 j0 2 17

Hom ^2 T; R with i0 > j0 that corresponds to the matrices bi0 j0 with bij = 0 whenever (i; j) 6= (i0 ; j0 ) or (j0 ; i0 ) and bi0 j0 = 1. The assertion on the rank of Hom ^2 T; R T follows. We have 8 if k 6= i0 ; j0 < 0 1tj0 = j0 if k = i0 bi0 j0 (ek ; ) = 1tk bi0 j0 = : 1ti0 = if k = j0 . i0

Now suppose rank (T ) 2, the other cases being trivial, and that we have given 1g: then we have bnh (en ; ) = h . Similarly suppose h with h 2 f1; :::; n that we have given h with h 2 f2; :::; ng: then we have bh1 (e1 ; ) = h. The surjectiveness of T follows and, since T is a free R-module, there is a section of T . Hence ker T appears like a direct addendum of the free module Hom ^2 T; R T and it has to be a projective R-module. Having computed the rank of Hom ^2 T; R T , the assertion on the rank of ker T follows. Lemma A.9 The property of having an O-polarization (over S) of degree coprime with a …xed integer t 2 N is invariant under O-isogenies (over S) of degree prime to t.

Proof. Let : A ! B be an O-isogeny of degree prime to t between two S.-schemes and let B ! B t be the polarization 'L induced by the ample line bundle L, of degree prime to t. By de…nition one easily check the equality ' L = t 'L : the left hand side shows that we get a polarization on A, since the pull-back of an ample line bundle by a …nite morphism is an ample line bundle; the right hand side shows that, this polarization, is of degree coprime with t. Lemma A.10 Let A be an abelian S-scheme with multiplication by an order O in a …eld F and …x an integer t 2 N. There is an O-linear isogeny of degree coprime with t (over S) : A ! B of A into an abelian S-scheme B with multiplication by the maximal order OF . Proof. Choose an invertible ideal I O such that # O I is prime to t and the conductor f (O) of the order O (in the maximal order). Then I is an invertible O-ideal and it is OF -stable. The inclusion O I 1 = HomO (I; O) yields an O-linear isogeny (see [6, 2.4]): A;I

:A!I

1

O

A

which is of degree a power of # O I , hence prime to t, and moreover I has a canonical structure of OF -module, since I 1 is an OF -module. We are now ready to prove the main result of the appendix.

18

1

O

A

Theorem A.11 Let A=S be an abelian S-scheme over a conneced and normal scheme S, with real multiplication by an order O in a totally real number …eld F such that dimS A = [F : Q] and further assume that At =S exists. For every integer t 2 N which is invertible in S there is an O-linear polarization of degree prime to t. In particular, if S = Spec (K) for a …eld K, we may …nd an O-linear polarization of degree prime to t for every t prime to the characteristic of K. Proof. Let nS;O be as in Lemma A.7 and let nS be the product of those primes that are invertible in S. Thanks to Lemmas A.9 and A.10 we may assume O = OF , so that nS;O = nS . By Lemma A.7 ker is an nS;O = nS -torsion étale sheaf. Corollary A.5 gives the claim.

Acknowledgements The author would like to thank Professor Massimo Bertolini for his many advices. It is also a pleasure to thank the anonymous referee for its suggestions, that led to a substantial improvement of the earlier version of the paper. The reference [1], provided by the referee, has been particularly useful for clarifying the results of the appendix.

References [1] F. Andreatta, E. Z. Goren, Geometry of Hilbert modular varieties over totally rami…ed primes, Internat. Math. Res. Notices 33 (2003) 1785-1835. [2] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift, Vol. I, Prog. in Math. 86, Birkhauser, Boston (1990), P. Cartier, et al., eds., pp. 333-400. [3] P. Deligne, G. Pappas, Singularités des espaces de modules de Hilbert, en les caractérisque divisant le discriminant, Compos. Math. 90 (1994) No.1 59-79. [4] M. Flach, A generalisation of the Cassels-Tate pairing, J. Reine Angew. Math. 412 (1990) 113-127. [5] G. van der Geer, Abelian varieties. Manuscript http://sta¤.science.uva.nl/~bmoonen/boek/BookAV.html.

avaiable

at

[6] B. H. Gross and J. Parson, On the local divisibility of Heegner points. Lang Memorial Volume. [7] B. Mazur and K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Ann. of Math. (2) 166 (2007) No. 2 579-612. [8] B. Mazur, K. Rubin and A.Silverberg, Twisting commutative algebraic groups, J. Algebra 314 (2007) No. 1 419-438. 19

[9] J. S. Milne, Etale cohomology, Princeton University Press (1980). [10] D. Mumford, Abelian varieties, Oxford University Press (1970). [11] Jan Nekovár, Selmer complexes, Astérisque 310 (2006). [12] M. Rapoport, Compacti…cations de l’Espace de Modules de HilbertBlumenthal, Compos. Math. 36 (1978) 255-335. [13] M. A. Seveso, Stark-Heegner points and Selmer groups of abelian varieties, PhD Thesis, University of Milan, Federigo Enriques department of mathematics. [14] M. A. Seveso, Congruences and rationality of Stark-Heegner points, J. Number Theory 132 (2012), no. 3, 414-447. [15] G. Tamme, Introduction to étale cohomology, Springer-Verlag (1994).

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