The Arithmetic Theory of Local Constants for abelian Varieties

The Arithmetic Theory of Local Constants for Abelian Varieties

MARCOADAMOSEVESO

ABSTRACT- We present a generalization of the theory of local constants 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 defined 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.

1. Introduction

Inthe paper [7] Mazur and Rubinprovide anarithmetic theory of local constants to study the parity of the Selmer groups. We briefly sketch their ideas and 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 inorder to relate the growth of the Selmer group to the rationality of Stark-Heegner points.

LetK=kbe a quadratic extension of a number fieldkand letL=Kbe a finite abelian l-power extension, with Galois group G, such that L=k is dihedral, wherel6ˆ2 is a rational prime. Let…A;O;l† ˆ…A;O;l†_kbe triple where…A;O†_kis anabelianvariety with multiplicationby anorderOina number fieldF(the data being defined overk) an dl Ois a prime ideal of residue characteristicl. SinceOis commutative the dual abelianvarietyA^t is canonically endowed with multiplication byOby lettinga:A!Aacts on A^t through a^t:A^t!A^t. We make the following assumptions on the

(*) Indirizzo dell'A.: UniversitaÁ degli studi di Milano, Dipartimento di Mate- matica Federigo Enriques, via Saldini 50, 20133 Milano, Italy.

E-mail: [email protected]

triple…A;O;l†:

it is polarizable, meaning that there exists anO-linear and symmetric isogeny

l:A!A^t of degree prime tol;

O;… l†is regular, meaning thatlis prime to the conductor ofOinO_F, the ring of integers ofF;

O;… l†is unramified, meaning that the discriminant ofF=Qis prime tol.

The present paper is organized as follows. Section 2 quickly reviews the theory of twisting abelian varieties and introduces some preliminary ma- terial. Section3 is devoted to review the Mazur-Rubinargument inthe more general setting of abelian varieties with multiplication by an orderO.

The mainresult is Theorem 3.2, which applies to triples…A;O;l†as above and notably under the assumption‰F:QŠ ˆdimA. We note that it is re- marked in[7] that their theory generalizes to abelianvarieties; Section3 is understood to provide a suitable reference where unspoken assumptions that are always satisfied inthe case of elliptic curves are made explicit. In particular, when‰F:QŠ ˆdimAand one considers a primelas above, the l-torsionand thel-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 field and

F:Q

‰ Š ˆdimA.

LetO_lbe the completionofOat the prime ideall, which is a discrete valuation ring since…O;l†is regular. Thenwe have

O_lˆM

l⁰jl

O_l⁰;

thus inducing a decomposition

Sel_l¹…A=L† ˆM

l⁰jl

Sel_l⁰¹…A=L†:

We will be interested in the Selmer groupSel_l¹…A=L†attached to the prime ljl. Note that it is a cofinitely generated module over the discrete valuation ringO_l, so that it makes sense to talk about its corank overO_l. Let Irr… †G be the set of all the rational irreducible representations of G and note that Sel_l…A=L†andSel_l¹…A=L†areO_l‰ Š-modules ina natural way. There is aG

canonical isogeny (map with finite kernel and cokernel) M

r

Sel_l¹_r Ar=K

isog!

Sel_l¹…A=L†;

…1†

whereSel_l¹_r Ar=K

is the Selmer group of a twistArofAbyrinthe sense of [8] andl_ris the unique prime ideal ofO_l‰ ŠG corresponding to the irreducible representationrand dividingl(see the preliminary discussion in Section 2).

The actionof O‰ ŠG onrfactors through a quotientO_r, whose completion O_r;lˆ O_r;lratlris again a discrete valuation ring. Denote byFthe residue field ofO_l, which is also the residue field ofO_r;lfor everyr(againwe refer to Section2).

The isogeny decomposition (1) yields corank_Ol…Sel_l¹…A=L†† ˆX

r

r_rcorank_Ol;rSel_l¹_r A_r=K

; where we setrr:ˆrankOl O_l;r

1:

…2†

Hence, in order to get information on corank_Ol…Sel_l¹…A=L†† from the knowledge of corankOl…Sel_l¹…A=K†† we are led to compare

corankOl…Sel_l¹…A=K†† with corankOl;rSel_l¹_r Ar=K .

It turns out that, assuming that …A;O†has real multiplication, every regular and unramified 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. Letram L=K=k… †be the set of primesvofKthat ramify inLand are inert or ramified overk, i.e.vis unramified inLandv^c ˆvfor the non-trivial automorphismcofKoverk. Denote byBad A=K… †the set of primes ofKthat are of bad reductionforA=K.

THEOREM 1.1. Assume that A;… O;l†_k is a regular and unramified triple with real multiplication. If ram L=K=k… † \Bad A=K… † ˆf and ram L=K=k… † \fvjlg ˆf, for everyr

dim_FSel_lr A_r=K

dim_F…Sel_l…A=K††mod 2 and

corank_Ol;rSel_l¹_r A_r=K

corank_Ol…Sel_l¹…A=K††mod 2:

The main application of Theorem 1.1 is the following. Define parity_l¹…A=K†:ˆ corankOl…Sel_l¹…A=K††mod 2:

COROLLARY1.2. Assume that …A;O;l†_k is a regular and unramified triple with real multiplication. If ram L=K=k… † \Bad A=K… † ˆf, ram L=K=k… † \fvjlg ˆfandparity_l¹…A=K† ˆ1then

corank_Ol…Sel_l¹…A=K†† ‰L:KŠ:

PROOF. This is a consequence of Theorem 1.1, together with (2) and the equalityP

r r_rˆ#G (see following (5) for this last equality). p Note also that, whenever A‰ Šl… † ˆK 0, it follows from the subsequent (7), together with the congruence relation appearing in the subsequent Theorem 2.2, that we have:

parity_l…A=K†:ˆdimF…Sel_l…A=K†† corankOl…Sel_l¹…A=K††mod2:

Another useful application is the following Corollary, that we state without proof.

COROLLARY1.3. Under the assumption of the above Corollary, further suppose

A‰ Šl… † ˆK 0:

Then, for every n2N, Sel_lⁿ…A=L† contains an O

lⁿ‰ Š-module which isG O

lⁿ-free of rank at least L‰ :KŠ and, in particular, dim_F…Sel_l…A=L†† ‰L:KŠ:

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 unramified triples…A;O;l† ˆ…A;O;l†_k. Inthe setting of el- liptic curves the existence of a principal polarization makes the polariz- ability assumption supefluous. As explained, the appendix ``O-polarizations onabelianvarieties with real multiplication'' removes this assumption whenAhas real multiplicationby anorderOina (totally real) number field F. We do not prove the existence of a principal polarization, which is false whenthe class number ofFis not one, but rather show the existence of a polarizationof degree prime tol. Of course Theorem 3.2 is applicable to other scenarios: for example when…A;O;l† ˆ…A;Z;l†andAis anabelian surface with quaternionic multiplication it is well known that a principal polarizationexists. The above corollaries are true for anarbitrary polar-

ized, regular and unramified triple …A;O;l† ˆ…A;O;l†_k, but we have choosento state them without the polarizability assumptionwhenwe have real multiplication since this is the statement needed in [14].

2. Twisting abelian varieties

LetO ˆ OF be the ring of integers of a number fieldFand letG be a finite abelianl-torsion group. We explain how to get a canonical isogeny decompositionof the group algebra O‰ Š. This will be regarded like anG algebra with involution using the inversion inG.

Let IrrF… †G be the set of isomorphism classes of distinct irreducible representations ofG overF. SinceF‰ ŠG is semisimple there is a canonical decomposition

F‰ Š ˆG M

r2IrrF… †G

F_r;

whereF_r=Fare field extensions. The projection ofF‰ ŠG ontoF_rgives the actionofF‰ ŠG onthe representationr.

DEFINITION2.1. AnidealI O‰ ŠG is said to be saturated whenever the quotientO‰ ŠG

I isO-flat.

Tensoring a saturated moduleI O‰ ŠG overOwithFand conversely intersecting an ideal inF‰ ŠG withO‰ ŠG establishes a bijectionbetweenthe set of the saturated ideals ofO‰ ŠG and the ideals ofF‰ Š. We will denote byG Ir:ˆFr\ O‰ ŠG the saturated ideal corresponding to the fieldFr.

Since _{r2IrrF… †G} Ir is an O‰ Š-submodule contained inG O‰ ŠG with finite index we get anO‰ Š-linear isogeny:G

M

r2IrrF… †G

Ir^isog,!O‰ Š:G …3†

The aim of the following discussion is to study theO‰ Š-module structureG of the modulesIr, which factors through a certainly quotientOrcontained in the fieldF_r. From now on we will assume thatFis unramified overl6ˆ2. The effect of this assumptionis that the study ofO_ris essentially reduced to the caseO ˆZ. The following facts canbe directly proved or canbe deduced from the caseO ˆZand [8, in particular Lemma 5.4]. The set IrrF… †G is in bijectionwith the subgroups G⁰G such that G=G⁰ is cyclic, i.e. with

IrrQ… †. IfG r2IrrF… †G corresponds to a cyclic quotientG=G⁰of orderl^rwe haveFr'F… †, wherez za primitivel^r-root of unity. Denote bylrthe unique prime ofZ‰ Šz dividingl, so thatZ‰ Šlz ˆl^e_r. The ringO‰ ŠG acts onIrthrough the quotientO_r' O‰ Šz and for every primel⁰jlofOthere is one only prime l⁰_rofO_r' O‰ Šz overl⁰andl^0e_r ˆ O_rl⁰, whereeˆW… †. We havel^r

lO ˆY

l⁰jl

l⁰andlrOrˆY

l⁰jl

l⁰_r: …4†

Inparticular the localizationOr;lofOratlis a discrete valuationring with uniformizersz 1 orp:ˆz z ¹and we haveO_r;llrˆ O_r;lp. IndeedF_r;l=F_l is totally ramified of degreeW… †l^r andO_r;l' O_l‰ Š. Under the isomorphismz O_r' O‰ Šz we haveI_r'…z_l 1†O_r, wherez_lis a primitivel-root of unity and inparticularO_r;lI_r'p^{lr 1}O_r. Furthermore sendingztoz ¹induces a well defined involution ion Or' O‰ Šz such that the projectionof O‰ ŠG onto Or' O‰ Šz is a morphism of rings with involution. Then the formula

;

‰ Š_r:O_r;lI_r O_r;lI_r! O_r a;b

‰ Š_r:ˆp ^{2lr 1}ai b… † gives onO_r;lIra perfect and Or;i

-HermetianOr-valued pairing. We also note that, if we writeI_r;Z(resp.I_r;O) for the saturated ideal corresponding to the same irreducibleQ-representation (resp.F-representation) it holds the equalityI_r;O ˆ O _ZI_r;Z. Inparticular, settingr_r:ˆrank_Ol O_l;r

as inthe introduction, we have

X

r

rrˆ#G:

…5†

Recall the extensionsL=K=kthat was considered in the introduction, as well as the polarizable, regular and unramified triple…A;O;l† ˆ…A;O;l†_k. The above discussionapplies toG:ˆG_L=K. We denote byFthe common residue field ofO(resp.O_l ) an dO_r (resp.O_l;r) atl(resp.O_ll) an d atl_r (resp.O_rl_r). The above discussionimplies that the twists

Ar:ˆ Ir;OOAˆ Ir;ZZO OAˆ Ir;ZZA

do not depend on the ringOover which we are twisting. These twists have been already considered in the literature and we refer to [8] for details.Ar

is againanabelianvariety and for everyK-algebraXthere is a canonical identification (see [8, Theorem 1.4])

Ir;OOA X

… † ˆ Ir;OOA X… KL†_GL=K :

As it follows from [8, Prop. 4.1], O‰ Š G OA is identified with the re- strictionof scalars fromKtoLofA=K. It thenfollows that we have (see [7, Prop. 3.1]):

Sel_l…A=L† 'Sel_l…O‰ Š G _OA=K†

and hence the isogeny (3) yields theO‰ ŠG -linear isogeny decomposition (1).

The abelianvarieties Ar=K are endowed with multiplication by Or

acting on Ir;O and we can consider the triple Ar;Or;lr

. Note that A_r;O_r;l_r

is again regular and unramified, as it follows from the above discussion, but may fail to be polarizable. The decomposition (4) yields canonicalO=Olⁿ-module decompositions (see also [6] for a direct definition of thelr-adic Selmer groups):

Arh ilⁿ_r

ˆM

l⁰jl

Arh il⁰ⁿ_r

forn2N;

Sel_l¹_r…A=K† ˆM

l⁰jl

Sel_l⁰¹_r …A=K†:

…6†

As inthe setting of classical Selmer groups one canprove that there is anexact sequence:

0!l_r¹

O_rRArh il¹_r K

… † !Sel_lr Ar=K

!Sel_l¹_r Ar=K

lr !0 …7†

and furthermore dimF l_r¹

O_rRArh il¹_r K … †

!

ˆdimF Ar lr … †K : …8†

LetT_l… †A be the Tate module attached to thel-adic representation of the abelianvarietyA=kover the fieldk. We recall the following fact (see [8, Theorem 2.2]): there is aG_K-equivariant isomorphism

T_l A_r

ˆ I_r;OOT_l… †;A

where GK-acts onthe right hand side via g ¹g. The polarizability as- sumptionon…A;O;l†implies that theO_l… †-valued Weil pairing on1 T_l… †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 inturnis anapplicationof the techniques developed in[4] (see also [13, Prop. B.27] for further details).

THEOREM 2.2. The pairing h ; i_r induces on Sell¹_r Ar=K a GK=k-equivariant and skew- Ol;r;i

-Hermetian pairing with kernel Sell¹_r…A=K†_div:

Sell¹_r Ar=K

Sell¹_r Ar=K

!Fl;r

Ol;r

In particular

corankOl;rSel_l¹_r Ar=K

dimF Sel_l¹_r Ar=K lr

mod 2:

PROOF. The first step inthe proof of the Theorem is to provide a

``model'' ofAroverk. This canbe done as in[7, Prop. A.9]: choose a liftcof the non-trivial automorphism of the extensionK=kto the algebraic closure k=k. For everyr2Irr_F… †G defineJO;r:ˆ…1‡c†IO;r, which is a right ideal in OG_L=k

. The left multiplicationby 1‡c gives a right O‰ Š-moduleG isomorphism I_r!^' J_r, thus inducing an isomorphism over K: A_r:ˆ

I_r;OOA' J_O;rOAˆ:A⁰_r. The second step is in producingO_l;r… †-val-1 ued perfect, skew- O_l;r;i

-Hermetianand GK-equivariant pairings on T_l Ar

, that are G_k-equivariant when making the identification T_l Ar

!^' T_l A⁰_r

. They are obtained settingh ; i_r:ˆ‰ ; Š_rewhereeis the O_l… †-valued Weil pairing on1 T_l A_r

, which is perfect inlight of the polarizability assumptionon…A;O;l†. Finally one applies the Flach con- structionas explained in[7, Theorem A.12].

3. The arithmetic theory of local constants

Recall our polarized, regular and unramified triple …A;O;l† ˆ A;O;l

… †_k from the introduction. Since the claim of Theorem 3.2 is invariant under isogenies of degree prime to lwe cansuppose that O ˆ O_F (see Lemma A.10).

LEMMA3.1. There are canonical identifications ofF‰G_KŠ-modules:

Ar lr ˆA‰ Š:l

PROOF. By [7, Prop. 4.1] there is a canonical identification of F_l‰G_KŠ-modules A_r l_r ˆA l‰ Š, which is anidentificationof O=lO-mod- ules (by the canonicity). The claim follows from the O=Ol-modules

decompositions (6). p

The first of the subsequent congruences follows from Theorem 2.2, while the second follows from Lemma 3.1 together with the exact sequence (7):

corankOl;rSel_l¹_r Ar=K

corankOl…Sel_l¹…A=K†† dim_F Sel_l¹_r A_r=K

l_r

dim_F…Sel_l¹…A=K†‰ Šl† dimFSel_lr Ar=K

dimF…Sel_l…A=K†† mod 2:

…9†

The F‰G_KŠ-module identification Ar lr ˆA‰ Šl allow us to view Sel_lr A_r=K

as the sub-F-module ofH¹…K;A‰ Šl†defined by the set of local conditionsS_r,

Sr;v:ˆIm Ar… †Kv

l_rA_r… †K_v ,!^dv H¹ Kv;Ar lr ˆH¹…Kv;A‰ Šl†

for the local Kummer mapd_vatv. Recall that, as explained in the proof of Theorem 2.2, theO-polarizationonAinduces onT_l A_r

ˆT_lr A_r perfect twisted pairings and, up to the canonical identificationAr lr ˆA‰ Š, theyl induce the same pairing on A‰ Š. It follows that, for everyl r, the Selmer structure Sr is selfdual (for the Weil pairing on A‰ Š) by the Bloch-Katol generalization of Tate's local duality applied toT_l Ar

(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:

…10†dimFSel_lr Ar=K

dimF…Sel_l…A=K†† X

v2S

dimF S_v Sv\Sr;v

mod 2;

whereSdenotes the set of primes out of which the Selmer structuresSand Srcoincides, which is finite. The local constants are defined as being

d_vˆd_v…A;O;l;L=K;r†:ˆdim_F S_v Sv\ Sr;v

inZ=2Z:

We are going to recover, from the results of [7], an explicit description of the spaces S_v

Sv\ Sr;v. In order to connect these spaces with those defined in[7] whenworking overZ, we will writeSvˆ S_v;l (resp.Sr;vˆ S_r;v;lr) to emphasize the dependence on the chosen primeljl. Thenit is clear that we have

S_v;l

S_v;l\ S_r;v;lrˆM

l⁰jl

S_v;l⁰ S_v;l⁰\ S_r;v;l⁰_r:

The space S_v;l

S_v;l\ S_r;v;lrcanbe explicitly described as follows. For everyr let Lr=K be the cyclic subextension of L=K corresponding to the irre- ducible representationr, choose a primewofL_r dividingvand set:

L⁰_r;w:ˆ unique subfield such that hL_r;w:L⁰_r;wi

ˆl if K_v6ˆL_r;w

Lr;w if KvˆLr;w

(

The proof of [7, Corollary 5.3] readily generalizes to our setting and gives a canonical identification

S_v;l Sv;l\ Sr;v;lr

ˆ A K… †_v

A K… † \_v N_Lr;w=L⁰_r;wA L_r;w:

THEOREM3.2. Let ram L=K=k… †be the set of primes of K which ramify in L=K and such that v^c ˆv (if c is the non-trivial automorphism of K=k) and let Good_l…A=K†be the set of primes of K of good reduction for A=K which do not divide l . Then, for everyr,

corankOl;rSel_l¹_r Ar=K

corankOl…Sel_l¹…A=K††

X

v2ram L=K=k… †

d_v…A;O;l;L=K;r†:

and when F‰ :QŠ ˆdimA

corankOl;rSel_l¹_r Ar=K

corankOl…Sel_l¹…A=K††

X

v2ram L=K=k… † Goodl…A=K†

d_v…A;O;l;L=K;r†:

PROOF. According to (9) and (10) we have to show that dvˆ0 whenever v2=ram L=K=k… † (resp. v2=ram L=K=k… † Good_l…A=K† when

F:Q

‰ Š ˆdimA).

If v2=ram L=K=k… † there are two possibilities: v^c 6ˆv or v^cˆv and v2=ram L=K… †. Inthe first case it is clear that [7, Lemma 5.1] gen- eralizes: since the entire triples …A;O;l† and Ar;Or;lr

are defined overk(see the proof of Theorem 2.2) the automorphism ofK=kinduces isomorphisms

A;O;l

… †_Kv!^' …A;O;l†_Kvc; A_r;O_r;l_r

Kv!^' A_r;O_r;l_r

Kvc:

Therefore, under the isomorphism H¹…Kv;A‰ Šl† !^' H¹…Kv^c;A‰ Šl† induced by conjugation,Sv (resp. Sr;v) corresponds to Sv^c (resp.Sr;v^c) and hence dv^c ˆdv(thus, assumingv^c 6ˆv, we havedv^c‡dvˆ0). Inthe second case, sincev^cˆv, by [7, Lemma 6.5… †], for every primei wofL_rthe extension L_r;w=K_v is (non-trivial) totally ramified or v splits completely. Since the first possibility has to be excluded (otherwise L=K should be ramified too), the module S_v;l

Sv;l\ Sr;v;lr

is trivial by the previous descriptionand hence the direct addend S_v;l

S_v;l\ S_r;v;lr is trivial too (so that dvˆ0). It follows the first formula.

Now assume ‰F:QŠ ˆdimA and let v2ram L=K=k… † be a prime such that v4jl and A=K has good reductionat v (more generally as- sume v^cˆv, v4jland A=K to have good reductionat v). Since v^cˆv, by [7, Lemma 6.5 … †] there are two possibilities:i Lr;w=Kv is (non-tri- vial) totally ramified for (every) wdividing v or v splits completely in Lr, but inthis latter case we have seenthat S_v;l

S_v;l\ S_r;v;lr is trivial. Thus we may assume Lr;w=Kv to be totally ramified (non-trivial) and since v4jl and A=K has good reductionat v the proof of [7, Theorem 5.6]

gen eralizes. More precisely [7, Lemma 5.4] readily generalizes to the case of anabelianvariety A=K with multiplication, giving the identi- fication

l ¹

O OA… † ˆK l ¹

O OA‰ Š Kl¹… †;

for every local fieldKof residue characteristic coprime withl. It is also true that [7, Lemma 5.5] generalizes to abelian varieties A=K, showing that, whenever in this caseL=Kis a (non-trivial) totally ramified extension and A=Khas good reduction, there are equalities:

N_L=KA… † ˆL lA… †K if ‰L:KŠ ˆl;

A… † \K lA… † ˆL lA… †:K

These last two equalities applies first to Lr;w=L⁰_r;w and then to L⁰_r;w=Kv, giving:

Sv;l

S_v;l\ S_r;v;lrˆ A K… †v

A K… † \v N_Lr;w=L⁰_r;wA Lr;wˆ A K… †v

A K… † \_v lA L⁰_r;wˆA K… †v

lA K… †_v :

From the canonical decomposition of A K… †_v

lA K… †v as a module over O=lO ˆ L

l⁰jlF_l⁰ and then using the generalization of [7, Lemma 5.4], we find S_v;l

S_v;l\ S_r;v;lr ˆl ¹

O OA K… † ˆv l ¹

O OA‰ Šl¹… †:Kv

By (8) theF-dimension ofl ¹

O _OA‰ Šl¹… †K_v is the same as theF-dimen- sionofA‰ Šl… †. Thus we fin dK_v

dvdimFA‰ Šl… †;Kv

which is the claimed generalization of [7, Theorem 5.6]. Hence to get the second formula it is enough to show that theF-dimension ofA‰ Šl… †Kv is even and this is where the assumption‰F:QŠ ˆdimAplays its role. Indeed this last fact follows exactly as in[7, Lemma 6.6], using theF‰ Š-vector spaceG_k A‰ Šl (which is 2-dimensional overFsince‰F:QŠ ˆdimA, by Proposition A.11) and theF… †-valued Weil pairing on1 A‰ Šl ina place of theF_l‰ Š-vectorG_k spaceE l‰ Šand theFl(1)-valued Weil pairing, which is used in the case of an

elliptic curve. p

A O-polarizations on abelian varieties with real multiplication Let Abe anabelianS-scheme with multiplicationby anorderOina totally real number fieldFsuch that dim_SAˆ‰F:QŠis constant. The dual abelianS-schemeA^t=S, that we assume to exists (for example suppose that A=Sis projective), is canonically endowed with multiplication byO.

Let

Hom^Sym_O A;A^t

:ˆl:A!A^t:lˆl^t andlisO linear

be the set of all symmetricO-linear morphisms (overS) fromA=Sto the dual abelianscheme A^t=S (here lˆl^t up to the canonical identification AˆA^tt). Let us be givena primel Oof residue characteristiclinvertible inS. For a torsion free finitely generatedO_l-moduleT,TˆTcanonically where we write TˆHom_Ol…T;O_l† to denote the O_l-dual. Using this identification we may define

Hom^Alt_Ol…T;T†:ˆfl:T!T:lˆ l andl isO_l linearg;

called the Ol-module of alternating homomorphisms fromT toT. Then T_l… † ˆA^t T_l… †A and we can consider the following commutative diagram (see [3, 1.5]):

Hom^Sym_O …A;A^t†_l ,! Hom^Alt_Ol…T;T†

\ \

Hom_O…A;A^t†_l ,! Hom_Ol…T;T† …11†

where we setT:ˆT_l… †. HereA M_lˆM_OO_lis thel-adic completionof M, since the modules involved are finitely generated.

If M is an O-module we write M to denote the constant presheaf M T=S… †:ˆM onthe big eÂtale site and we letMbe the associated sheaf M T=S… † ˆM^{p0… †T}, wherep0… †T denotes the set of connected components of TandM^{p0… †T} is theO-module of maps fromp₀… †T toM. We also writeM andMto denote their restrictions to the eÂtale site.

IfM is a projective O-module of finite rankrM, the functorMOA defined by the rule

MOA

… †…T=S†:ˆMOA T=S… †;T=Sarbitrary

is representable by an abelianS-scheme with multiplicationbyOand di- mension dim_SM_OAˆr_Mdim_SA. Indeed this is clear whenMˆ Oⁿ; in general we may write Oⁿ'MN, so that …MOA† …NOA† 'Aⁿ holds as functors and then we may apply Yoneda's Lemma to deduce the existence of an idemponente_M2End_S… †Aⁿ such thate_MAⁿ'M_OA. In particularM_OAis a sheaf onthe big eÂtale site and its restriction to the eÂtale site is a sheaf.

On the other hand we may consider the presheafM^p_OAdefined by the rule

M^p_OA T=S

… †:ˆM T=S… † OA T=S… † ˆM^{p0… †T} OA T=S… †;

whereT=Smay be anarbitraryS-scheme or aneÂtaleS-scheme if we work with the eÂtale site. We writeM^p_OA!MOAto denote the associated presheaf with the canonical sheafication morphism. We may therefore consider the canonical morphism

D:MOA!^Dp M^p_OA!MOA

whereD^pis givenby the sheaficationmorphismM!M, i.e. the diagonal morphism M T=S… † ˆM!M T=S… † ˆM^{p0… †T}. More generally, for an ar- bitrary sheafF, we writeFOAfor the sheaf associated to the presheaf F^p_OAdefined by the rule F^p_OA

T=S

… †:ˆF… † T ^p_OA T… †.

LEMMA A.1. The canonical morphism MOA!MOA is an iso- morphism as sheaves on the big eÂtale site or the eÂtale site.

PROOF. After a base change to an arbitraryS-schemeT, i.e. after re- stricting the values of our functors to eÂtaleT-schemesT⁰=Twe may work with the eÂtale site. Suffices to check that the canonical morphism is an isomorphism onthe stalks (see [9, II Theorem 2.15] or [15, II Theorem (5.6) iii)]). Lets!Sbe a geometric point and letO_S;sˆlim

! OU be the strict henselization ofO_S;s, where the limit runs over all eÂtale neightbourhoods of s. Set S_s:ˆSpec O_S;s

ˆlimU. If F is a presheaf onthe eÂtale site, F_sˆ F^#

swhereF^#is the associated sheaf (see [9, II Remark 2.14 (c)]); in particular…MOA†_sˆ M^p_OA

s. There is a canonical morphism F_sˆlim

! F… † !U FlimU

ˆF… †:S_s Since FˆM_OA is an S-scheme of finite type, lim

!…M_OA†… † ˆU M_OA

… †

limU

(see [9, II Remarks 2.9 (d)]). Similarly, since we may assume that theUs appearing in lim

! O_Uare connected,Ais anS-scheme of finite type, the formation of direct limits commutes with tensor products andS_sis connected,

lim! M^p_OA U … † ˆlim

!…M_OA U… †† ˆM_Olim

! A U… †

ˆMOAlimU

ˆ M^p_OA S_s … †:

Summarizing we have canonical identifications…MOA†_sˆ…MOA†… †,S_s M_OA

… †_sˆ M^p_OA S_s

… †and the canonical morphismDinduces on the stalks the morphismD^p… †, which is the identity sinceS_s S_sis connected. p

We sketch a proof of the following well known fact.

PROPOSITIONA.2. LetAbe an abelianS-scheme (withO-multiplica- tion) and let06ˆl2Lbe an element of an invertibleO-moduleL. Then L_OAis an abelianS-scheme of the same dimension asA,

l:A!L_OA l… †a :ˆl_Oa

is a faithful flat morphism,A‰ Šl :ˆkerlis finite and flat overSand

#A‰ Š ˆl #L² lO;

where#A‰ Šl is the order of the finite S-scheme A‰ Š. Furthermore, ifl # L is invertible on S,lis eÂtale (and hence A‰ Šl is S-eÂtale). lO

PROOF. Consider the exact sequence 0! O !^l L! L

lO!0:

We view Aas a sheaf for the fppf topology and apply OA. Sin ceL is O-flat, Tor¹_O…L;A† ˆ0, and, since A is divisible and L

lO finite, L

lOOAˆ0. It follows that we have 0!Tor_O¹ L

lO;A

!A!^l LOA!0:

Inparticular, sinceTor¹_O L lO;A

ˆA‰ Šl is finite,LOAis anabelianS- scheme of the same dimension asA. We also see thatlis an isogeny, hence a faithful flat morphism and an eÂtale morphism when#A‰ Šl is divisible onS.

Inorder to compute the degree#A‰ Šl of the finiteS-group schemeA‰ Š, wel first assume that …l;L† ˆ…l;O†, so that l2 O End_S… †A is just the multiplicationbylmap. SinceA‰ Šl isS-flat, working with each connected component ofS, we may first assume thatSis connected and then compute the degree ofA‰ Šl overSafter a base change to the residue fieldk s… †at any s2S. Hence we may assume that SˆSpec… †K for a field K. Setting N… †l :ˆ#A‰ Š, we see thatl Nis a norm form onF=Qhomogeneus of degree 2dim_SA(by [10, III §19 Theorem 2]). It follows from [10, III § 19 Lemma]

and our assumption dim_S… † ˆA ‰F:QŠ that we have NˆN_F=Q² , so that

#A‰ Š ˆl N²_F=Q… †l forl2 O. The claim whenLˆ Ofollows from the fact thatNF=Q… † ˆl # O

lObecauseOis a lattice inF. We now remark that, if we have given…l_i;L_i†with l2L_i aninvertible O-module,iˆ1;2 an d if we assume that# L₁

l1Ois prime to# L₂

l2O, the morphism m_l1;l2 :L1L2!L1OL2

m_l1;l2…l₁;l₂†:ˆl₁l₂‡l₁l₂

induces an isomorphism L1

l₁O L2

l₂O' L1OL2

l_1Ol₂O. We apply this remark

as follows. We take …l1;L1† ˆ…l;L† and …l2;L2† ˆ l⁰;L ¹

, where l⁰2L ¹ is choosenso that # L

lO is prime to #L ¹

l⁰O. We deduce that

# O

ll⁰Oˆ# L lOL ¹

l⁰O. Onthe other hand, we may consider the composition l⁰l:A!^l L_OA!^l0 L ¹_OL_OAˆA

and deduce#A ll⁰ ˆ#A‰ Š l #…L_OA† l⁰ . By the caseLˆ Owe get

# L

lO#L ¹

l⁰Oˆ#A‰ Š l #…L_OA† l⁰ :

Since Tor¹_O L lO;A

ˆA‰ Šl and Tor_O¹ L ¹

l⁰O;L_OA

!

ˆ…L_OA† l⁰ , our choice of l⁰ so that # L

lO is prime to #L ¹

l⁰O implies that #A‰ Šl is prime to …LOA† l⁰ too. We deduce# L

lOˆ#A‰ Š.l

PROPOSITION A.3. If Ol is a discrete valuation ring, the horizontal inclusions appearing in(11)have torsion free cokernel.

PROOF. It suffices 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 com- mutativity of (11). Furthermore, to see that the left vertical inclusion has torsion free cokernel it suffices to show that the inclusion ofHom^Sym_O …A;A^t† inHom_O…A;A^t†has torsionfree cokernel. Suppose thatl2Hom_O…A;A^t† is such that there exists 06ˆn2Zsuch that nl2Hom^Sym_O …A;A^t†. Then nlˆ… †nl ^tand the right hand side isl^tn^t ˆl^tn, while the left hand side is ln. Hencelnˆl^tnand, sincenis anisogeny, it is anepimorphism inthe category of sheaves for the fppf topology onSand the equalitylˆl^tcanbe checked at the level of points. That the lower horizontal inclusion has tor- sion free cokernel follows as in [5, Theorem (12.10), Chapter 12], thanks to our assumptionthatO_lis a discrete valuationring. p LetLbe aninvertibleO-module. For everys:F,!Rwe may consider i_s:L,!L_sR and define L_s :ˆi_s¹ L_sR

. An oriented invertible O- module is aninvertibleO-moduleLtogether with the choice of a vector of

signseˆ… †es such thates2 f g. Anhomomorphismf :…L1;e1† !…L2;e2† of oriented invertibles O-modules is anhomomorphism f of O-modules such thatf L^e_1;s^1;s

L^e_2;s^2;s. To anorientationeonLwe canassociate the set of totally positive elements

L^‡:ˆ\

s

L^e_s^s L

and indeed, the isomorphism class of the orientedO-moduleL, is uniquely determined by the couple L^‡;L

. Furthermore, sinceAut L^‡;L

ˆ O_‡, the set of totally positive units ofO, we see that, to give anorientedO- module L^‡;L

(up to isomorphism), is the same as to give anelement inthe narrow class group associated to the orderO.

Consider the functor of symmetricO-linear morphisms:

L… †A…T=S†:ˆHom^Sym_O AT;A^t_T

and the subfunctor L^‡… †A of polarizations (where AT and A^t_T are the base changes toT). They are infact sheaves for the eÂtale topology and in [12, Prop. 1.17], whenO ˆ O_Fis the maximal order ina totally real field F, it is proven thatL… †A is locally constant with values in invertibleO- modules and generated byL^‡… †. WhenA Sis connected and normal, by [12, Variante 1.18] L… †A is already constant, i.e. L… † ˆA L… †A for the invertible O-module L… †A :ˆL… †A…T=S† ˆL… †A … †S where T=S is con- nected. In general there exists T=S eÂtale such that L… † ˆA L… †A_T re- stricted to eÂtale T⁰=T.

Let m^p:L… † A ^p_OA!A^t be the morphism givenby the rule m^p…la†:ˆl… †, wherea l2Hom^Sym_O AT;A^t_T

anda2AT… † ˆT A T=S… †.

SinceA^t is a sheaf, it induces a unique

m:L… † A OA!A^t: …12†

Note that, when L… † ˆA L… †A is constant, m is identified, via the iso- morphismDof Lemma A.1 , with the morphism

m:L… † A _OA!A^t

m l… a†:ˆl… † 2a A^t…T=S†;a2A T=S… † …13†

which is an isogeny being a non-zero map between abelianS-schemes of the same dimension (see also [1, proof of (3))(1) of Proposition3.1]).

The following proposition is a variant of [1, Proposition 3.1].

PROPOSITIONA.4. The following are equivalent, whenO ˆ OF: (1) kermis n-torsion eÂtale sheaf, wheremis(12);

(2) for every integer t prime to n there exists T=S eÂtale and l2L^‡… †A …T=S†of degree prime to t.

PROOF. In[1, proof of (1) )(2) of Proposition3.1] simply replace [1, (3.6)] with the assertionthat F:…A_OLM_A†‰ Št ,!A^_‰ Št is anin- clusion (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 implicationis the same. [1, proof of (2) ) (3) of Proposition3.1]

(with the obvius modificationinthe statement) is trivial. The analogous of [1, proof of (3) ) (1) of Proposition3.1] is the same but we conclude that K has order prime to l for every prime l prime to n.

COROLLARY A.5. Suppose thatS is connected and normal or, more generally, that L… † ˆA L… †A is a constant sheaf and that O ˆ O_F. The following are equivalent:

(1) kermis n-torsion eÂtale sheaf, wheremis(13);

(2) for every integer t prime to n there exists l2L^‡… †A of degree prime to t.

PROOF. We need to prove that, if there existsl⁰2L^‡… †A…T=S†of de- gree prime to t, there exists l2L^‡… †A of the same degree. Since L… † ˆA L… †,A L… †A …T=S† ˆL… †A^{p0… †T}. We take any connected component T₀TofTand we note that the imagelofl⁰inL… †A …T₀=S† ˆL… †, whichA is simply theT₀-component ofl⁰2L… †A ^{p0… †T} (that we may assume to be non-

zero), is the required polarization. p

We are now going to show that, under the assumptions of Corollary A.5, condition (1) is always satisfied if we takento be the product of the primes that are invertible in S. Indeed we will prove a slightly more general statement without assuming thatO ˆ O_F. We first record the following corollary of Propositions A.2 and A.3.

COROLLARYA.6. IfHom^Sym_O …A; A^t† 6ˆ0 andO_lˆ O_F;l coincides with the completion at l of the maximal order O_F, the upper horizontal inclusion appearing in(11)is an isomorphism

Hom^Sym_O A; A^t

lˆHom^Alt_Ol…T; T† ' O_l:

PROOF. For everyOl-moduleTlet us denote byHomOl ^²_Ol T;Ol the O_l-module of bilinear forms b:TT! O_l such that b x;… y† ˆ b y;… x†.

Under the canonical identification HomOl ²_OlT;O_l

ˆHomOl…T;T† the submoduleHomOl ^²_OlT;O_l

corresponds toHom^Alt_Ol…T;T†. PropositionA.2 implies that T_l… †A is a free rank two module over O_l. It follows that HomOl ^²_OlT;O_l

is free of rank one overO_l. SinceHom^Sym_O …A;A^t†is torsion free, the inclusion (11) shows that the freeO_l-moduleHom^Sym_O …A;A^t†_lmay have rank 0 or rank 1 (overO_l). Under the assumptionHom^Sym_O …A;A^t† 6ˆ0 we find that it holds the second possibility and the inclusion is an iso- morphism, since it has torsion free cokernel by Proposition A.3. p LEMMA A.7. Suppose that S is connected and normal or, more gen- erally, thatL… † ˆA L… †A is a constant sheaf. Let nS;O2Nbe the product of those primes l2Nthat are invertible in S or such thatO_lˆ O_F;l(i.e. l is coprime with the conductor of O in the maximal order O_F). Then the primes dividing#kermdivides n_S;O and, in particular,kermis an n_S;O- torsion eÂtale sheaf.

PROOF. As we already remarked min(13) is anisogeny. Let lbe a prime that is invertible onS: for every such prime, to prove thatmis an isogeny of degree coprime withl, it is sufficien t to show thatT_l… †m is an isomorphism. Whenever O_lˆ O_F;l this is equivalent to checking that T_l… †m is anisomorphism for every primeljl. The formationof thel-adic Tate module, commutes with taking tensor product with flatO-modules, so that:

T_l…L… † A _OA† ˆL… † A _OT_l… † ˆA L… †A _lOlT_l… †:A

For every O_l-module write L^Alt… †T :ˆHom^Alt_Ol…T;T† for shortness.

Consider the canonical morphism:

mˆm_T:L… †A _lOlT_l… † !A T lt7!l… †:t

The canonical inclusion L… †A_lL^Alt…T_l… †A † gives the following com- mutative diagram:

L… †A _lOlT_l… †A ^Tl!^†mA T_l… †A

\ k

L^Alt…Tl… †A † OlTl… †A ^m!^{Tl… †A} Tl… †A

Since L… † 6ˆA 0 an d Ol ˆ OF;l, by Corollary A.6 the left vertical inclusionis anisomorphism. Inother words the problem is reduced to the analogous problem in the category of O_l-modules, i.e. we must show that, for every free rank 2 O_l-module T, the canonical morphism m_T is an isomorphism. This is a consequence of the subsequent alge-

braic Lemma A.8 p

LEMMAA.8. Let T be a finitely generated free module over a commu- tative ring R. The canonical morphism

m_T:Hom^Alt_R …T;T† T!T f t7!f t… †

is surjective ifrank… † 6ˆT 1with projective kernel. Furthermore we have (with the convention thatrank0meansrankˆ0):

rank… † ˆT rank… †;T rank Hom ^²T;R

T

ˆrank… †T ²…rank… †T 1†

2 ;

rank ker… m_T† ˆrank… †T …rank… †T 2†…rank… † ‡T 1†

2 :

In particularm_Tis an isomorphism if and only ifrank… † ˆT 2or0.

PROOF. Up to the canonical identification Hom^Alt_R …T;T† ˆ HomR ^²_RT;R

the morphism m_T corresponds to the canonical mor- phism:

m_T:Hom_R ^²_RT;R

T!T bt7!b t;… †:

Choose a basisfe_i:iˆ1;. . .;ngofTand letf gd_i be the dual basis off g,e_i defined by the ruled_i e_j

ˆd_ij. By viewing the elements ofTlike column vectors with respect to the basisf g, the elemente_i e_i corresponds to the column1_iwhich is zero at all its entries with the only exception of thei- entry, which is equal to 1, and similarlydicorresponds to the row vector 1^t_i which is equal to zero at all its entries with the only exception of thei-entry, which is equal to 1. Furthermore any element ofb2Hom ^²T;R

corre- sponds to a matrixbˆ b_ij

which has zeros on the diagonal and such that b_ijˆ b_ijfori6ˆj. Inparticular a basis ofHom ^²T;R

is obtained from the elements bi0j02Hom ^²T;R

with i0 >j0 that corresponds to the

matrices bi0j0 with bijˆ0 whenever … † 6ˆi;j …i0;j0†or…j0;i0†andbi0j0ˆ1.

The assertiononthe rank ofHom ^²T;R

Tfollows. We have

bi0j0…ek; † ˆ1^t_kbi0j0 ˆ

0 if k6ˆi0;j0

1^t_j0ˆd_j0 if kˆi₀ 1^t_i0ˆ d_i0 if kˆj₀: 8>

><

>>

:

Now suppose rank… † T 2, the other cases being trivial, and that we have given d_h with h2f1; :::;n 1g: thenwe haveb_nh…e_n; † ˆd_h . Similarly suppose that we have given d_h with h2f2; :::;ng: thenwe have b_h1…e₁; † ˆ d_h. The surjectiveness ofm_Tfollows and, sinceTis a freeR- module, there is a sectionof m_T. Hence kerm_T appears like a direct ad- dendum of the free moduleHom ^²T;R

Tand it has to be a projective R-module. Having computed the rank ofHom ^²T;R

T, the assertion

onthe rank of kerm_Tfollows. p

LEMMAA.9. The property of having anO-polarization (over S) of de- gree coprime with a fixed integer t2N is invariant underO-isogenies (over S) of degree prime to t.

PROOF. Letl:A!Bbe anO-isogeny of degree prime totbetween twoS-schemes and letB!B^tbe the polarizationW_Linduced by the ample line bundle L, of degree prime to t. By definition one easily check the equalityW_lLˆl^tW_Ll: the left hand side shows that we get a polarization on A, since the pull-back of an ample line bundle by a finite morphism is an ample line bundle; the right hand side shows that, this polarization, is of

degree coprime witht. p

LEMMAA.10. Let A be an abelian S -scheme with multiplication by an orderOin a field F and fix an integer t2N. There is anO-linear isogeny of degree coprime with t (over S)l:A!B of A into an abelian S-scheme B with multiplication by the maximal orderO_F.

PROOF. Choose aninvertible idealI Osuch that#O

Iis prime tot and the conductorf… †O of the orderO(inthe maximal order). ThenIis an invertible O-ideal and it is O_F-stable. The inclusion O I ¹ ˆ Hom_O…I;O†yields anO-linear isogeny (see [6, 2.4]):

l_A;I:A!I ¹OA

which is of degree a power of#O

I, hence prime tot, and moreoverI ¹OA has a canonical structure ofO_F-module, sinceI ¹is anO_F-module. p

We are now ready to prove the main result of the appendix.

THEOREMA.11. Let A=S be an abelian S-scheme over a conneced and normal scheme S, with real multiplication by an orderOin a totally real number field F such thatdim_SAˆ‰F:QŠand further assume that A^t=S exists. For every integer t2Nwhich is invertible in S there is anO-linear polarization of degree prime to t.

In particular, if SˆSpec… †K for a field K, we may find anO-linear polarization of degree prime to t for every t prime to the characteristic of K.

PROOF. Letn_S;Obe as inLemma A.7 and letn_Sbe the product of those primes that are invertible inS. Thanks to Lemmas A.9 and A.10 we may assumeO ˆ OF, so thatn_S;Oˆn_S. By Lemma A.7 kermis ann_S;Oˆn_S- torsioneÂtale sheaf. Corollary A.5 gives the claim. p Acknowledgments. 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 versionof the paper. The reference [1], provided by the referee, has beenparticularly useful for clarifying the results of the appendix.

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Manoscritto pervenuto in redazione il 6 agosto 2010.