Semi-abelian Schemes and Heights of Cycles in Moduli Spaces of abelian Varieties

Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

JEAN-BENOIÃTBOST(*) - GERARDFREIXAS IMONTPLET(**)

Questo articolo eÁ dedicato a Francesco Baldassarri con i nostri migliori auguri

ABSTRACT- We study extension properties of Barsotti-Tate groups and we establish diophantine inequalities involving heights of cycles with respect to logarith- mically singular hermitian line bundles on arithmetic varieties. We apply these results to bound heights of cycles on moduli spaces of abelian varieties, induced by quotients of abelian varieties by levels of Barsotti-Tate subgroups, over function fields over number fields. To achieve this aim, we combine our results with an effective version of Rumely's theorem on integral points on possibly open arithmetic surfaces and with Faltings' theorems on heights of abelian varieties in isogeny classes.

1. Introduction

Letkbe a field andAag-dimensional abelian variety overk. For any prime number l different from the characteristic of k, the l-adic Tate module is defined as

T_lAˆlim

n

A[lⁿ](k);

where A[n] denotes then-torsion subgroup scheme ofAand kan alge- braic closure of k. It is well known thatT_lAis a freeZ_l-module of rank 2g. The action of the absolute Galois groupG_k:ˆGal(k=k) ofkon thek-

(*) Indirizzo dell'A.: DeÂpartement de MatheÂmatiques, Universite Paris-Sud 11, BaÃtiment 425, 91405 Orsay cedex, France.

E-mail: jean-benoit.bost@math.u-psud.fr

(**) Indirizzo dell'A.: CNRS-Institut de MatheÂmatiques de Jussieu, 4 Place Jussieu, 75005 Paris, France.

E-mail: freixas@math.jussieu.fr

valued torsion points of A endows TlA with the structure of a Galois module. Besides, the endomorphism ring Endk(A) (consisting of the k- endomorphisms of the abelian variety) is a torsion free Z-algebra of fi- nite type. The Q-algebra End_k(A)_ZQ is semi-simple (and conse- quently, for anylas above, theQ_l-algebra End_k(A)_ZQ_l also), and the action of End_k(A) onT_lAis compatible with the structure ofG_k-module onTlA.

THEOREM1.1 (Tate's conjecture for abelian varieties). Assume that k is finitely generated over its prime field. The following assertions hold:

a. T_lA_ZlQ_lis a semi-simpleQ_l[G_k]-module;

b. the natural map

Endk(A)ZZl !EndZl[Gk](TlA) is an isomorphism.

Theorem 1.1 was proven by Tate himself whenkis a finite field [32], and by Zarhin whenkis a function field over a finite field [34, 35] (see also [22], notably Chapter XII). Faltings established it in the number field case in his famous article [12] (see also [13], [15], [30]). His result was extended to the case of a fieldkfinitely generated overQin [15, Chap. VI] and [30, Exp. IX].

In all cases, the proof is based on the following proposition:

PROPOSITION1.2 (Tate). LetAbe an abelian variety over a fieldk. Let lbe a prime number different from the characteristic ofk. Suppose thatA satisfies the following property:

T(A=k): for any l-divisible groupfGng_n0with GnA[lⁿ], infinitely many of the abelian varieties A=G_nover k are isomorphic to each other.

Then

i. for every Ql[Gk]-submodule W of TlAZlQl, there exists u in Endk(A)ZQlsuch that u(TlAZlQl)ˆW;

ii. Theorem1.1holds for the abelian variety A.

Proofs of T(A=k) have to deal with the difficulty of establishing fi- niteness results for isomorphism classes of abelian varieties not endowed with a polarization. A crucial contribution of Zarhin's [35] has been to in- troduce the so-called ``Zarhin trick'', that allowed him to get rid of the (lack of) polarization problems, and to establish the following result:

PROPOSITION1.3 (Zarhin). Letkbe a field, andlbe a prime number different from the characteristic ofk.

Assume that, for every abelian variety A over k and any principal polarizationlover A (defined over k¹), the following property is satisfied:

Z(A;l=k): for any l-divisible group fG_ng_n0 with G_n a subgroup of A[lⁿ]lagrangian with respect to the Weil pairing attached tol, the quo- tients A=Gnbelong to a finite family of k-isomorphism classes.

Then PropertyT(A=k)and Theorem1.1hold for any abelian variety A over k.

We refer to [32] and to [15, Chap. IV, Sec. 2] for discussions of the reduction of the Tate conjecture to assertionsT(A=k) andZ(A;l=k).

The method of Faltings for establishingZ(A;l=k) whenkis a number field combines results of Tate and Raynaud on Barsotti-Tate groups, and some ideas from Arakelov geometry. If indeedK denotes a number field andO_Kits ring of integers, Faltings attaches some canonical heighth_F(A) to (the NeÂron model overOKof) an abelian varietyAoverK. The Faltings height is a counterpart of Zarhin's geometric height used in [34], and a key point in Faltings proof is a finiteness property aÁ la Northcott for this height [12, Sec. 3].

When k is finitely generated over Q, the proof of Z(A;l=k) and of Theorem 1.1 in [15, Chap. VI] and [30, Exp. IX] uses Ð besides Faltings original arguments in the number field case Ð some results of Deligne or Faltings concerning abelian schemes over varieties on fields of characteristic zero ([8], [11]) that are established by Hodge theoretic methods.

It is possible to avoid the use of Hodge theory in the proof ofZ(A;l=k) whenkis finitely generated over some fieldk0by using Zarhin's results on bounded families of abelian schemes over (open dense subschemes) of a normal projective schemeXoverk₀(herek₀ˆQ;andk₀(X)ˆk) in their full strength version established by Moret-Bailly [22, Chap. XII] to show that, with the notation used inZ(A;l=k) above, the quotientsA=Gnarise in some bounded family. By considering the restrictions of these quotients to suitable many closed points of X, and by using the validity of Z(A;l=k) when k is a number field, one concludes to the finiteness of their iso- morphism classes.

(¹) Namely, such that the attached isogenyA!A^_fromAto its dual abelian variety is defined overk.

In this note we sketch a variant to these diverse approaches to Z(A;l=k), in the situation where k is a function field of transcendence degree 1 overQ. Our approach also avoids Hodge theoretic arguments, but relies on higher-dimensional Arakelov geometry. It involves bounds on heights of cycles in relevant moduli spaces, and consequently provides some kind of ``quantitative control'' in the finiteness property asserted by Z(A;l=k) orT(A=k), in the spirit of [30, Exp. VII, 4.4].

Actually our derivation ofZ(A;l=k) provides us with an excuse to deal with the following subjects, which constitute the core of the article:

i. extension properties of Barsotti-Tate subgroups of abelian sche- mes with semi-stable reduction, over two dimensional regular bases (section 2);

ii. heights of cycles on arithmetic varieties, with respect to logarith- mically singular hermitian line bundles (section 3). These heights satisfy a Liouville type inequality (Theorem 3.2) and Northcott's finiteness property (Corollary 3.4);

iii. a generalization to logarithmically singular hermitian line bundles of a theorem of Zhang for heights of generic sequences of points in arithmetic varieties (Theorem 3.5);

iv. a theorem of Rumely [29], and a refinement by Autissier [2], on the existence of integral points on open arithmetic surfaces. This is combined with the previous points and Faltings' theorem [12] on heights of quotients of an abelian scheme by the levels of a Barsotti- Tate subgroup (section 4).

Most of our constructions might be extended to function fields of ar- bitrary positive transcendence degree overQ. For the sake of simplicity, we have considered the case of transcendence degree 1 only. Observe also that, by using the work of Zarhin and Moret-Bailly as discussed above, the validity of Z(A;l=k) when k is a function field of transcendence degree larger than 1 overQmay be derived from its validity when k has tran- scendence degree 1, combined with purely geometric arguments.

To our knowledge, the considerations ini-iii, although not unexpected, do not appear in the literature. The use made of them inivshows that they may be of interest for diophantine geometry. More than the application to the Tate conjecture over function fields, we believe that the interest of this text lies in the tools we develop. We expect them to be useful in other contexts. Finally, let us mention the related work of H. Ikoma [19], that also avoids the use of complex Hodge theory by using instead the theory of heights developed by Moriwaki [24].

2. Barsotti-Tate subgroups of semi-abelian schemes

2.1. Let Sdenote a regular, integral and noetherian scheme of Krull dimension 2. We denote its generic point byhand suppose its character- istic to be 0. In the sequel, we fix a primepand we deal with Barsotti-Tate subgroups in the p-torsion of abelian schemes. Only basic notions con- cerning Barsotti-Tate groups will be required. To shorten the exposition, we refer the reader to Tate's article [33] and Illusie's expose [30, Exp. VI]

for an introduction.

PROPOSITION 2.1. Let A be a semi-abelian scheme over S with abelian generic fiber A, and fGng_n0 a Barsotti-Tate subgroup of A[p¹]. Assume that, for every n0; there exists a finite flat group subscheme P_nA such that G_nP_n;h. Then there exists a big open subsetUofS(namely an open subsetUsuch thatcodim(SnU)2) and an integerm0 such that:

i. Gm extends to a finite flat group subschemeGmA[p^m]_jU; ii. the Barsotti-Tate group defined by HnˆGm‡n=Gm extends to a

Barsotti-Tate groupfHng_n0over U;

iii. letB ˆA_jU=G_m.² ThenH_n is a group subscheme of B[pⁿ]for all n0.

REMARK2.2. i. Letm0 be such that the assertion of the proposi- tion holds. We claim that for anym⁰m, the analogous assertion obtained by replacingmbym⁰is also true. For this, it is enough to show thatG_m‡n extends to a finite flat group subscheme ofAjU, for every n0. Define Gm‡n to be the kernel of the isogeny

A_jU !B !B=H_n:

It is a quasi-finite flat subgroup ofA_jU containingG_mand such that the quotientG_m‡n=G_mis isomorphic toH_n. BecauseG_m andH_nare finite overS, we infer thatG_m‡nis finite as well.

ii. After possibly replacingU by a smaller big open subset, one can suppose that all the groups Gn extend to finite flat subgroups of A_jU.

(²) The quotient is well defined because S is regular and A is generically abelian. This ensures the existence of relatively ample line bundleA=S[27, Cor. XI 1.16], and hence the quotient by a finite flat subgroup is defined [SGA III, 5.4.1].

Indeed, we already saw that this is true for allnm. On the other hand, thefinitely manyremaining groupsGn,nm, extend to finite flat sub- groups ofA_jU⁰, for some other big open subset U⁰. For this, recall that G_nP_n;h and P_n is a finite flat subgroup of A. Thus, replacing U by U\U⁰, we obtain a big open subset for which the claim holds.

The purity lemma [14, Chap. V, Lemma 6.2] will be needed in the course of the proof of Proposition 2.1. We quote it for the convenience of the reader.

LEMMA2.3. Let R be a regular local ring of Krull dimension2. Write SˆSpecR, h for the generic point of S, s for the closed point of S, UˆSn fsgand j:U,!S for the open immersion. The functor jdefines an equivalence between the categories of locally free sheaves of finite rank over U and over S. Moreover it is compatible with tensor product.

REMARK 2.4. It is important to bring the reader's attention to the following subtlety: the functorj does not preserve exact sequences. Ac- tually, de Jong-Oort show in [18, Sec. 6] that it is possible to find a regular local schemeSof dimension 2 and generic characteristic 0, finite flat group schemes G, Hover S and a morphism W:G!H which is a closed im- mersion over UˆSn fsg but not over S. In particular, the schematic closure ofG_hinHis not flat overS.

PROOF OFPROPOSITION2.1. By Remark 2.2 and by the noetherianity hypothesis, we may assume that SˆSpecR is affine. The proof is then divided into several steps.

STEP1. We observe that the group schemeG_n can be extended to a finite flat group schemeGnoverS, and that there exists a big open subset Un ofSsuch thatG_njUnis a subgroup scheme of A_jUn. This is easily ob- tained from finite flat closures along codimension 1 closed subsets and Lemma 2.3.

STEP2. The next point is an adaptation of Tate's trick ([33], proof of Proposition 12 and Lemma 2.5 below) to construct a candidate to the de- sired Barsotti-Tate group from suitable quotients of the group schemesGn. Letk;n0 be integers. The intersectionV ˆU_k\U_k‡nis a big open subset ofS. We have inclusions

G_kjV G_k‡njV A_jV

so there is a well-defined quotientGk‡njV=GkjV. This is a finite flat group scheme over V. By Lemma 2.3 this group scheme uniquely extends to a finite flat group scheme over S, to be denoted H^k‡n_k . If nowV denotes another small enough big open subset, then we have a map between well defined quotients

G_k‡n‡1jV

G_k‡1jV !^p G_k‡njV G_kjV

induced by multiplication byp, which is an isomorphism over the generic point. By the purity lemma this morphism extends in a unique way to a morphism of group schemes

H^k‡n‡1_k‡1 !^~p H^k‡n_k :

Remark that~pis surjective. Indeedp~is clearly finite and is dominant too, because the involved groups are flat over S with reduced generic fiber (recall that the characteristic ofhis 0, so Cartier's theorem applies to the generic fibers, showing they are reduced). Furthermore, p~ is an iso- morphism over the generic fiber. The conditions of Lemma 2.5 below (Ta- te's trick) are fulfilled: it follows that there exists an integerm1 such that the morphism

H^k‡2_k‡1 !^p~ H^k‡1_k

is an isomorphism forkm. Therefore, for everyn0, then-th compo- sition

H^m‡n‡1_m‡n ^~p!ⁿ H^m‡1_m is an isomorphism as well.

Define the finite flat group scheme overS H_nˆH^m‡n_m :

The natural inclusions among the groups H_n restricted on suitable big open subsets extend to a compatible system of morphisms

i^n‡k_n :H_n !H_n‡k:

Let us fix an integern0. For some big open subsetVofS(depending on n) we have a chain of inclusions

G₁j_V. . .G_m‡n‡1j_VAj_V :

The following diagram commutes:

The morphism pis the natural projection. It follows from the diagram that the kernel of multiplication by pⁿ on H_n‡1jV is the same as the kernel of p, namely H_njV. Furthermore we have a closed immersion induced bypⁿ

m_V :H_n‡1jV

H_njV ,!H_1jV:

We actually see that m_V is an isomorphism. Indeed, m_V is an isomor- phism on the generic point h, so m_V is dominant. Let W be a big open subset such that i^n‡1_n :H_njW!H_n‡1jW is a closed immersion. Up to shrinking V we may assume that V W. Then the isomorphism m_V extends to an analogous isomorphismm_W, in a unique way, by the purity Lemma 2.3.

STEP 3. We show that Gm and fHng_n0 are the desired group schemes, after restriction to a suitable big open subset ofS.

LetUSbe a big open subset such thatG_mjU A_jU. To simplify the notations, let us replace Sby U. There is a well defined quotient semi- abelian schemeB ˆA=G_moverS. By definition, for everyn, over some big open subsetVwe haveH_njV P_m‡njV=G_mjV. This inclusion uniquely extends to a morphism

H_n ^h!ⁿ P_m‡n Gm B;

by the purity Lemma 2.3. Moreover, the relationh_nˆh_n‡1i^n‡1_n holds. In particular, ifh_nis a closed immersion over a big open subsetW_n, theni^n‡1_n is a closed immersion overW_nas well.

Let us now choose a big open subsetWnsuch thath1andhnare closed immersions when restricted toWn. We claim thath_n‡1jWn is a closed im- mersion as well. Indeed we have a commutative square

It follows that the arrow~h_n‡1is a closed immersion. We thus have a well defined quotient semi-abelian scheme

CnˆB_jWn=H_njWn Imh~n‡1

and a natural isogeny

q:B_jWn !C_n:

By construction and uniqueness of extensions over S, the kernel of the isogenyqis a finite flat group subscheme ofB_jWnisomorphic toH_n‡1jWn, and the morphismhn‡1is a closed immersion overWn.

Let us finally fix a big open subsetUofSsuch thath1 is a closed im- mersion overU. Then the preceding argument shows thath_n is a closed immersion for alln. Furthermore we have isomorphisms

m_U :H_n‡1jU

H_njU ! H_1jU

induced by multiplication by pⁿ. We conclude that the family fH_njUg_n constitutes a Barsotti-Tate group contained in B_jU. This completes the

proof. p

LEMMA2.5 (Tate's trick). Let S be a normal, integral and noetherian affine scheme SˆSpecR, of generic characteristic 0. Let fX_kg_k1 be a family of finite flat schemes over S, with reduced generic fibers, endowed with surjective morphismsp_k:X_k‡1!X_k, that become isomorphisms over the generic point of S. Then there exists an integer m1such thatp_kis an isomorphism for every km.

PROOF. First of all, we may assume that the schemes X_k are non- empty. Indeed, in the empty case the conclusion trivially holds. LetD_k6ˆ0 be the affine R-algebra ofX_k. IfF is the fraction field ofR, all theF-al- gebrasDkRFcan be identified to a fixed finiteF-algebraD, for instance D1RF. This is so becausepkis an isomorphism over the generic pointhfor

allk1. Observe thatDis a separableF-algebra, because it is reduced by assumption andFhas characteristic 0. The mapspkare surjective and the schemesX_kare flat, so theD_kform an increasing sequence of rings. TheR- algebraD_kcontainsRas a subring, sinceX_kis flat overSand non-empty.

By definitionFD_k ˆD. The finiteness condition says thatD_k is integral overR. Hence the ring

Gˆ[

k1

Dk

is an integral extension ofRwithFGˆD. Therefore,Gis contained in the integral closure ofRin the separableF-algebraD. Since Ris noetherian and integrally closed, it follows thatG is a noetherianR-submodule ofD.

Consequently, the increasing sequenceDk of subrings ofG stabilizes. Let m0 be an integer such thatD_k ˆD_k‡1 for allkm. Thenp_kis an iso-

morphism forkm. p

REMARK 2.6. i. The assumption that the schemes X_k are reduced cannot be dropped in the preceding statement. Otherwise, one can con- struct a counterexample. For instance, let us consider a strictly increasing infinite sequence of freeR-modules of finite constant rankr,

E₁₄E₂₄ . . .: Define finiteR-algebrasD_k by

D_k ˆRE_k; with x²ˆ0 forx2E_k:

The schemesX_kˆSpecD_k are not reduced but satisfy all the other hy- potheses of the lemma. By construction, the conclusion of the lemma is not satisfied: the sequencefD_kg_kdoes not stabilize.

ii. As we already recalled in the proof of Proposition 2.1, when the schemesX_k are finite flat group schemes over S, they are automatically reduced over the generic fiber. This is the case since the generic char- acteristic ofSis 0, and due to Cartier's theorem: a finite group scheme over a field of characteristic 0 is always reduced. Therefore, in this situation Lemma 2.5 applies.

2.2. We now discuss an application of Proposition 2.1 to the following particular setting. Let Sbe a local, henselian and integral scheme of di- mension 2, whose closed and generic points we denoteyandhrespectively.

LetUˆSn fygbe the unique strict big open subset ofS. Consider a semi- abelian schemeA!Swith abelian generic fiberA. We can decompose

the quasi-finite flat groupsA[pⁿ] overSas A[pⁿ]ˆA[pgⁿ][L_n;

whereA[gpⁿ] is the maximal finite subgroup andLnhas an empty special fiber [22, Chap. IV, Sec. 1]. These are open and closed flat subschemes of A[pⁿ]. Moreover this construction is functorial in S, with respect to morphisms of local henselian schemes.

Suppose in addition thatS is noetherian, complete (hence excellent), and regular (hence normal), of generic characteristic 0, and that a Barsotti- Tate subgroupfGng_n ofA[p¹] is given. The groupsGn extend to quasi- finite flat closed subgroups G_nA[pⁿ]_jU over U, with inclusions G_nG_n‡1. We will be concerned with the pull-back ofG_nto the generic points of morphismsf :T!S, whereTis atrait.³In other words, ifmis the generic point ofTandf(m)2U, then we can define the pull-backGn;m

ofGntom. This extends to a quasi-finite flat subgroup scheme of the base changed semi-abelian schemeA_T. Let it beG_n;T. Therefore we can take the finite part

Gen;TˆA[pgⁿ]_T\Gn;T:

We are interested in the behavior of the familyfGe_n;Tg_n.

PROPOSITION 2.7. There exist a finite sequence of blow-ups Se!S (centered over the closed point), a big open subsetV ofSeand a positive integer n11 such that, for every morphism f:T !S, where T is a trait, which factors through V and sends the generic point m2T to U, the quotients Gen‡n1;T=Gen1;T define a Barsotti-Tate subgroup of A_T=Ge_n1;T. The conclusion remains true if one replaces n₁ by a larger integer.

PROOF. We first define the finite part ofGnoverU. It is the finite flat closed subgroup ofA[pgⁿ]_jU defined by

Ge_nˆG_n\A[gpⁿ]_jU:

This family may not constitute a Barsotti-Tate group. However there is a

(³) We define a trait (in a non-standard manner) as a 1-dimensional, local, henselian, regular and noetherian scheme.

sequence of closed immersions Ge_n‡1;h

Gen;h ,!^p Ge_n;h

Gen 1;h ,!^p . . . ,!^p Ge1;h:

BecauseGe1;his a noetherian scheme, this sequence stabilizes at some point.

Therefore, for some integern₀, the groupsfQ_n:ˆGe_n‡n0;h=Ge_n0;hg_nform a Barsotti-Tate group. Observe these groups extend to closed subgroups of the quotient semi-abelian schemeAjU=Gen0, but they may not extend over S. By a variant of Gabber's lemma [18, Thm 2.1, Rmk. 2.2], there exists a sequence of blow-ups Se!S centered over the closed point y such that A_jU=Ge_n0extends to a semi-abelian schemeB !S.e⁴We can and (for later purposes) we will assume that we have effected at least one blow-up. Re- mark that the baseeSis still regular, integral and noetherian of dimension 2 and generic characteristic 0. By a theorem of Raynaud [27, p.130], [14, Chap. I, Prop. 2.7], the natural commutative diagram of isogenies

extends overS.e⁵The kernel of the extended isogenyA_~S!B is a quasi- finite flat closed subgroup ofA[pgⁿ⁰]~S, hence finite. It is an extension ofGen0

over Se and we use the same notation for it. Thus, we can write B ˆAS~=Ge_n0. Observe now that

QnˆGe_n‡n0;h

Gen0;h A[pg^n‡n0]~S

Gen0

;

and the last quotient is a finite flat subgroup ofB. Hence Proposition 2.1 applies to the semi-abelian scheme B and the Barsotti-Tate subgroup fQ_ng_n ofB[p¹]_h. Namely, there exists a big open subsetV ofSeand an integerm0 such thatQmextends to a finite flat subgroupQmofBjV, the quotientsQn‡m=Qmextend to finite flat subgroupsHnofB_jV=Q_mand the sequencefHng_ndefines a Barsotti-Tate group.

(⁴) In this version of Gabber's lemma, it is only required that S be quasi- compact, excellent, reduced and normal. This is satsified under our assumptions.

(⁵) For Raynaud's theorem we only need normality ofeS.

Let nowf :T!S be as in the statement, withV given by the pre- ceding construction. Because the generic pointmis sent intoU, we can pull-back Gen over U to the generic point. By the functoriality of the finite part construction with respect to morphisms of local henselian schemes, this coincides with the previously constructed Ge_n;m. Further- more, because f factors through V, the preceding discussion for the quotients Qn‡m=Qm shows that, putting n1ˆn0‡m, the family fGen‡n1;T=Gen1;Tg defines a Barsotti-Tate subgroup of AT=Gen1;T.⁶ We stress the important fact that the index n₁ does not depend on the morphismf.

For the last claim of the proposition, we refer to Remark 2.2. p REMARK2.8. It follows from the proof that we may suppose that the sequence of blow-upseS!Sis non-trivial: namely, it consists of at least one blow-up.

CONSTRUCTION2.9. Let us place ourselves under the assumptions of Proposition 2.7. We consider a particular class of morphisms f :T!S,e obtained as follows. Recall we assumed thateSis obtained after at least one blow-up. Therefore we have a decomposition

where Bly(S) is the blow-up ofSalong y. The image in Bly(S) of the ex- ceptional divisor ofpis a finite closed subschemeF overy. The big open subset Bly(S)nFof Bly(S) naturally embeds intoeS. We may form

W ˆV\(Bl_y(S)nF):

(2:1)

This is a big open subset of Bly(S). A morphism f :T!S that factors throughWBly(S) also factors throughV S.e

2.3. We now apply the considerations in 2.2 to semi-abelian schemes over arithmetic surfaces.

(⁶) Compare with the discussion in [15, pp. 134±136].

Fig. 1. The data in Theorem 2.10.

LetKbe a number field,O_Kits ring of integers,SˆSpecO_K. L etX be a proper arithmetic surface overS, namely a regular, flat and proper scheme overS, with geometrically irreducible generic fiber of dimension 1.

Suppose given a semi-abelian scheme A!U, where U is a big open subset ofX with generic pointh, namely the complement of a finite set fP_ig_i2Iof closed points inX. We assume thatA_his abelian. LetDUbe the divisor of bad reduction of A. We decompose DˆH‡V into hor- izontal and vertical components. Define V ˆUnD. L et fGng_n be a Barsotti-Tate subgroup ofA[p¹]_jV . For every non-zero primep of OK

such that VpˆV\Xp6ˆ ;, we choose a closed point yp2VpnH. This provides a finite collection of closed points Y ˆ fy_pg_p2P. Introduce S_pˆSpecOd_X;yp. We still denote y_p its closed point. The Barsotti-Tate group fG_ng_n can be base-changed into a Barsotti-Tate group over the generic point of Sp. Let us denote by Wp the big open subscheme of

Blyp(Sp) provided by Construction 2.9 applied to these data (see equation (2.1)). We have morphisms

f_p:Blyp(Sp) !Bl_Y(U)

which lift the natural morphisms Sp!U. L et Fpˆf_p(Blyp(Sp)nWp), which is a finite closed subset of Bl_Y(U). By construction,Y\Hˆ ;andH can be seen as a divisor in Bl_Y(U). WriteVe_pfor the strict transform ofV_p. We then define the open subset of Bl_Y(U)

W ˆBlY(U)n H[ [

p2P

(Fp[Vep) : (2:2)

There is a natural inclusion of open subsetsV W . An important feature of this construction is that the structure map Bl_Y(X)!S is surjective when restricted toW .

Let nowf :SpecO_L!W be an integral point ofW (in other words,L is a finite extension of K, O_L its ring of integers, andf a morphism of schemes over SpecOK)⁷. In particular it provides an integral point of UnH. For every primeqofO_Llying abovep2P,f induces a morphism f_q:SpecOd_L;q!S_p which factors through W_p. We also obtain by base change a semi-abelian schemeA_qoverOd_L;qand extended quasi-finite flat subgroupsG_n;q. Take the finite partsGe_n;q. By Proposition 2.7, there exists an integer n1, which depends onY butnotonf, such that the quotients fGen‡n1;q=Gen1;qg_ndefine a Barsotti-Tate group. For any other primeqnot lying aboveP, the analogous conclusion, with the same indexn1, is trivially true. Indeedqis sent byf intoV andfG_ng_nis assumed to be a Barsotti- Tate subgroup of the abelian schemeA_jV .

We summarize the preceding observations into the following theorem.

THEOREM2.10. LetX !SpecOKbe an arithmetic surface andAa semi-abelian scheme over a big open subset U of X, generically abe- lian. Let D be the divisor of bad reduction in U and decompose DˆH‡V in horizontal and vertical components. PutV ˆUnD, and choose a closed point yp2V\XpnH for every p in P, the subset of SpecOKn f0g defined by thepsuch that V\Xp6ˆ ;:

Then, given a Barsotti-Tate subgroup fGng_n ofA_jV , there exists an integer n₁0such that for every integral point f :SpecO_L!Ufactor-

(⁷) Because the structure morphism W !Sis surjective, Rumely's theorem ([29], [23]) will ensure that there exist infinitely many such points (for varying finite extensionsLofK).

ing throughW defined by(2.2), we have:

i. the image of f is disjoint with the horizontal component H;in particular, the pull-back fA is a semi-abelian scheme over OL, generically abelian.

ii. For every prime q of O_L lying above one of the primes p in P, denote byA_qthe base change ofA toSpecOd_L;qinduced by f . Let G_n;q be the quasi-finite flat subgroup of A_q extending the base change ofGn to the generic point ofSpecOd_L;q, andGen;qits finite part. Then the family fGe_n‡n1;q=Ge_n1g_n defines a Barsotti-Tate subgroup of Aq=Gen1.

3. Some diophantine inequalities for heights

In this section we discuss some generalities and diophantine inequal- ities for heights of cycles in arithmetic varieties, with respect to hermitian line bundles whose metrics are of some logarithmic singular type. This kind of heights will appear in later considerations, when dealing with minimal compactifications of moduli spaces of abelian varieties. The ar- ithmetic intersection formalism leading to these heights has been devel- oped by Burgos-Kramer-KuÈhn [6]. Their finiteness property has been explored by the second author of the present article in [16]. A diophantine inequality, generalizing results by Zhang [36] in the spirit of Faltings' lemma [12, Lemma 3] on heights defined by means of metrics with loga- rithmic singularities, has been established in some unpublished work by the first author [4]. Here we provide the precise statements and some details.

3.1. We introduce the heights and formulate the finiteness property we will need. LetX andY be arithmetic varieties over the ring of integers OKof a number fieldK, namely, projective and flat schemes over SpecOK. We make the following geometric assumptions:

the generic fiberY_K is smooth overK;

there is a surjective morphismW:Y !X the restriction of which to the generic fiber induces an isomorphism Y_KnW_K¹(F)! X_KnF, whereFis a Zariski closed subset ofX_K andDˆW_K¹(F)_red is a di- visor such thatD(C) has normal crossings.

In particular,UˆXKnFis contained in the regular locusX^reg_K and U(C):ˆ a

s:K,!C

Us(C)

is a complex quasi-projective manifold. LetL be a line bundle onX and k k a smooth hermitian metric on L_jU(C), invariant under the action of complex conjugation. We denote byL ˆ(L;k k) this singular hermitian line bundle in the sense of Arakelov geometry.

DEFINITION3.1. We say thatL is apre-log-loghermitian line bundle, with singularities alongF, with respect toW, if the pull-back singular her- mitian line bundleWL is a pre-log-log hermitian line bundle onY, with singularities alongD[6, Sec. 7], [16, Sec. 3]. Similarly, one defines the notion ofcontinuousorsmoothhermitian line bundle onX.

Under the pre-log-log assumption, according to [6, Sec. 7], there is a well defined height functionh_WL on algebraic cycles onY whose flat ir- reducible components intersect Dproperly (or equivalently, are not con- tained inD) overY_K. We refer toloc. cit.for details, as well as to [16, Sec.

6]. We observe that the constructions of height functions in these refer- ences extend to non-necessarily regular arithmetic varieties, with smooth generic fibers. We refer to [5, Rmk ii, p. 941] in the case of smooth her- mitian metrics and to [7, Sec. 4.3] for the general case.

The theory of heights with respect to semi-positive continuous hermi- tian line bundles is developed in [21, Sec. 5]. We will only need the case of an ample line bundleL onX endowed with a continuous metrick k₁with semi-positive first Chern current (after pull-back toY). Then,WL1is an integrable hermitian line bundle onY, in the sense of [21, Def. 4.7.1], and the height functionh_WL1is defined. The integrability property requires an easy proof, that we give in Lemma 3.7 below.

Let us denote byZp;U(X) the free abelian group ofp-dimensional cy- cles on X, whose flat irreducible components intersect the open subset UˆX_KnF of X_K. We similarly define Z_p;W ¹_(U)(Y). To a pre-log-log (resp. smooth, ample continuous semi-positive) hermitian line bundle L, we attach a height function

h_L :M

p Zp;U(X) !R characterized by the following properties:

h_L isZ-linear;

h_WL(z)ˆh_L(Wz), where W is the usual push-forward morphism of cycles

W:M

p

Z_p;W1(U)(Y) !M

p

Z_p;U(X):

Ifz2Z_p;U(X) is a cycle such that deg_LKz_K 6ˆ0, its normalized height is defined by

eh_L(z)ˆ h_L(z) [K:Q] deg_LKz_K:

Some properties of these height functions, such as compatibility with change of ground ring, are straightforward consequences of the corre- sponding properties forh_WL.

The next statement is adapted from [16, Thm. 1.3], and may be seen as a Liouville type inequality for the approximation ofFby cycles onY.

THEOREM3.2. LetX,Y,W:Y !X be as above, andL a pre-log-log hermitian line bundle onX with singularities along F, with respect toW.

Assume thatL is ample overX and that the first Chern formc1(L)is semi-positive on U(C). Then, for any smooth hermitian metrick k₀onL with semi-positive first Chern form (after pull-back toY), there exist real constantsa;b;g>0such that, for every effective cycle z2Zp;U(X), such thatdeg_LKzK6ˆ0we have:

eh_L0(z)‡g>0 and

eh_L(z) eh_L0(z)

a‡blog (eh_L0(z)‡g):

REMARK 3.3. Since the line bundle L is ample, there exist smooth metrics onL with semi-positive first Chern form. For example, one may consider a very ample power L^N and use the restriction of the Fubini- Study metric on the sheafO(1) of the projective spaceP(H⁰(X;L^N)). For this particular metrick k₀, a multiple of the heighth_L0is the restriction to cycles on X of the usual Faltings height onP(H⁰(X;L^N)). Any other height for a pre-log-log metric (and in particular for a smooth metric) onL then differs fromh_L0by an archimedean factor, expressed by integrals of so-called Bott-Chern secondary forms.

PROOF OFTHEOREM3.2. The proof follows the lines of [16, Sec. 6.2], with several simplifications. The idea is to adapt the argument due to

Faltings, for heights of points, to the higher dimensional situation [12, Lemma 3], to reduce to an application of Jensen's inequality. We sketch the proof and refer toloc. cit.for more details. First of all, by assumptionL is ample. After possibly replacing it by some positive power, we find global sectionss₁;. . .;s_m2H⁰(X;L) such that

F(C)ˆ(divs1)(C)\. . .\(divsm)(C):

One may suppose that the cyclezis flat, integral and properly intersects divs₁overC. By the definition of the height, one is reduced to bound in- tegrals of the form

Z

W ¹(z(C))

jW(f)jc₁(WL)ⁱc₁(WL₀)^j; i‡jˆdimz_K:

Here we wrotek k ˆe ^f=2k k₀, and we note that the integral has to be computed on a suitable resolution of singularities ofW ¹(z(C)). Recall that zKmeets properly divs1, that the metrick khas logarithmic singularities, and that the first Chern forms are semi-positive. After a rescaling of the smooth metric, we may thus replace jW(f)j by log logkWs₁k₀². This re- scaling can be made independently ofz. (Observe that, by comparison toloc.

cit., there is an important simplification in this step. Namely the positivity assumption made on the first Chern forms allows to avoid the decomposi- tion result in Theorem 4.3.⁸) Now, one proceeds to decrease the exponenti.

For this, we first write

c1(WL)ˆc1(WL0)‡dd^cf

onW ¹(U(C)). One is reduced to bound integrals of the form Z

W ¹(z(C))

log logkWs1k₀²dd^cf c1(WL)^{i 1}c1(WL0)^j:

In this situation, one can apply Stokes theorem. Taking into account the following relation of differential forms outsideF(C)

dd^clog logks1k₀²ˆ i 2p

@logks1k₀²^@logks1k₀² (logks1k₀²)²

c1(L0) logks1k₀²;

(⁸) With the notations of [16, Thm. 1.3], this explains that we can takerˆ1.

the positivity of the Chern differential forms, and elementary bounds be- tween functions with logarithmic singularities, one reduces to bound in- tegrals of the form

i 2p

Z

W ¹(z(C))

@ 1

(logkWs₁k₀²)¹⁼²

!

^@logkWs1k₀²c1(WL)^{i 1}c1(WL0)^j:

Such integrals are treated inloc. cit., Lemma 5.5. After reducing the ex- ponentito 0, the main term to bound is an integral of the form

Z

W¹(z(C))

log logkWs₁k₀²c₁(WL₀)^d:

By Jensen's inequality, this is bounded in terms of

log Z

W ¹(z(C))

logkWs1k₀¹ c1(WL0)^d [K:Q] deg_LKz_K 0

B@

1 CA:

Finally, by the definition of the height and the existence of a universal lower bound for normalized heights with respect to ample smooth hermitian line bundles, this last term is bounded by log (eh_L0(z)‡g), where the constantg only depends on the smooth metrick k₀. This completes the proof. p COROLLARY3.4. LetL be an ample pre-log-log hermitian line bundle onX with semi-positive first Chern form. Then, given a real numberA >0, there exist finitely many effective cyclesz2Z_p;U(X)withh_L(z)Aand deg_LKz_K A.

PROOF. After possibly replacingL by a high power, we may assume thatL is very ample. Then, using a projective embedding and the Fubini- Study metric on projective space, we can construct a smooth hermitian metric on L, say k k₀, with positive first Chern form and, more im- portantly, such that the heighth_L

0is the restriction of some usual Faltings height for the ambient projective space. Then the finiteness property is an immediate consequence of the comparison theorem above and the finite- ness property forh_L0proven by Bost-Gillet-Soule in [5, Thm. 3.2.5]. p 3.2. We now proceed to some generalization of Zhang's inequality [36, Thm. 5.2], [1, Thm. 2.4] for the height functions with respect to pre- log-log hermitian line bundles. This will be applied to prove bounds for

cycles in moduli spaces of abelian varieties, in terms of heights of con- veniently chosen integral points.

THEOREM3.5. LetX,Y,W:Y !X be as above. LetL be a pre-log- log hermitian line bundle onX with semi-positive first Chern form. As- sume that L is ample and the unbounded locus of the metric is exactly F(C).⁹Then there is an inequality

eh_L(X)dsup

VU inf

x2V(K)

eh_L(x);

(3:1)

where d is the Krull dimension ofX and the supremum runs over all open Zariski subsets V of UˆXKnF. Moreover the right hand side of the inequality is a finite quantity.

REMARK3.6. In the statement of the theorem, the normalized height of an algebraic pointx2V(K) has to be understood as follows. The pointx induces a 1-dimensional horizontal cyclezinX. Then we put

eh_L(x):ˆeh_L(z)

ˆ h_L(z)

[K:Q]deg_LKz_K ˆ h_L(z) [K(x):Q]:

To prove Theorem 3.5, we will truncate the pre-log-log hermitian metric and work with continuous hermitian metrics with semi-positive first Chern current. We will need two lemmas.

LEMMA 3.7. Let k k₁ (resp. k k) be a continuous (resp. pre-log-log) hermitian metric on the ample line bundleL, whose first Chern current is semi-positive. Assume thatk k₁ k kon U(C). Then:

i. WL₁ is an integrable hermitian line bundle;

ii. for every effective cycle z2Zp;U(X)we have the inequality h_L(z)h_L1(z):

PROOF. L et A be a hermitian ample line bundle on Y, with a smooth hermitian metric whose first Chern form is strictly positive. For every integerN1, the continuous hermitian line bundleW(L₁)^NA

(⁹) These hypotheses could be slightly weakened, but this formulation is enough for our purposes.

is ample on the generic fiber and has strictly positive first Chern cur- rent. By [21, Thm. 4.6.1],W(L1)^NA is admissible [21, Def. 4.5.5] and there exists an increasing sequence of smooth metrics k k_k uniformly converging to the metric on W(L₁^N)A, and with semi-positive first Chern forms. In particular, in the case Nˆ1, it follows that WL₁ˆ (WL₁A)A ¹ is integrable, thus provingi. Also, as the sequence is increasing, these smooth metrics are smaller than the pre-log-log metric on W(L^N)A, which has positive first Chern current. From this and the definition of height easily follows the inequality

h_W(L^N_)A(z)h_W(LN)Ak(z)

forz2Z_p;U(X) effective. The right hand side of the inequality converges toh_W(L1)^NA(z) by [21, Thm. 5.5.2]. Working on the definition of height, we thus obtain

N^dh_WL(z)N^dh_WL1(z)‡O(N^{d 1});

where theOterm only depends on several arithmetic intersection numbers on z. The assertion ii now follows by dividing these heights by N^d and

lettingNconverge to1. p

LEMMA 3.8. The statement of Theorem3.5 holds for continuous her- mitian metricsk k₁onL with semi-positive first Chern current.

PROOF. The original argument of [36] can be adapted to this case, becauseL is ample and the curvature current ofL1is semi-positive. For this, one needs to refer to the version of the arithmetic Hilbert-Samuel theorem proven by Randriambololona [26]. Observe that this work already deals with the case when the generic fiberX_K is just reduced, non-nec- essarily smooth. By construction, the arithmetic intersection number in the dominant term of the main theorem inloc. cit., coincides with our height

with respect toL₁. p

We are now ready to prove Theorem 3.5.

PROOF OFTHEOREM3.5. It suffices to show the inequality withL re- placed by some positive powerL^N. Thus wecan suppose thatLis very ample and that there exists a global sections2H⁰(X;L) withF(C)(divs)(C).

EndowL with a smooth hermitian metrick k₀with semi-positive first Chern form and such thatksk₀1. Writek k ˆe ^g=2k k₀. Observe that the conditiondd^cW(g)‡c1(WL0)0 and the surjectivity ofWguarantee

that g is locally bounded above. Since X(C) is compact, g has a global upper bound. Because the inequality we aim to prove does not change after rescaling the metric, we can suppose that g0. Since k kadmits loga- rithmic singularities along F(C), there exists a positive constant A such that gAlog logksk₀¹. For every integerj3, define

g_jˆmaxfg; Alog logjg:

Because we assume that the unbounded locus of g is exactly F(C), the functions g_j are continuous onX(C). Because the maximum of two psh functions is already psh, we havedd^cW(g_j)‡c₁(WL₀)0. It is clear that the sequenceg_jdecreases towardsg. We define the continuous metrics onL

k k_j ˆe ^gj=2k k₀:

They form an increasing sequence pointwise converging tok konU(C), with semi-positive first Chern currents. By Lemma 3.7 we know that

h_L(z)h_Lj(z) (3:2)

for any effective cycle z2Z_p;U(X). Observe that by Lemma 3.8, this already guarantees the finiteness of the right hand side of inequality (3.1).

To establish (3.1), we reason by contradiction. Suppose the inequality (3.1) does not hold. Then, for somee>0 we can find a generic sequence of pointsfxkg_k1 in (X ndiv (s))(K) with

deh_L(x_k)eh_L(X) e:

(3:3)

By Theorem 3.2 and (3.3) we deduce thatfeh_L0(x_k)g_kis bounded above by some constantB. On the other hand (3.2) applied toX and (3.3) give

deh_Lj(x_k)deh_Lj(x_k) deh_L(x_k)‡eh_Lj(X) e:

(3:4)

Fixk1 and letLbe a field of definition ofxk. There are inequalities eh_Lj(x_k) eh_L(x_k)ˆ 1

[L:Q]

X

s:L,!C

g_j g 2 (x_k;s)

ˆ 1 [L:Q]

X

s:L,!C

maxf0; g Alog logjg

2 (x_k;s)

A 2[L:Q]

X

s:L,!C

maxf0;log logksk₀¹ log logjg(x_k;s)

A

2log 1‡ 1 logj

1 [L:Q]

X

s:L,!C

logksk₀¹(x_k;s)

! :

To derive the last inequality, we made use of the Jensen's formula, again inspired by Faltings [12, Lemma 3]. By definition of height and because eh_L0(x_k)B, we finally obtain

eh_Lj(x_k) eh_L(x_k)A

2 log 1‡eh_L0(x_k) logj

!

A

2 log 1‡ B logj

: Fixj3 large enough so that

A

2 log 1‡ B logj

e 2d: Together with (3.4), we derive

deh_L

j(x_k)eh_L

j(X) e

(3:5) 2

for every point of the generic sequencefx_kg_k. But for the continuous metric k k_j, Zhang's inequality holds (Lemma 3.8). Hence (3.5) provides a con-

tradiction, and the proof is complete. p

3.3. As an illustration of the preceding constructions, in the final part of this section we present the geometric framework to which the theory of heights will be applied in the sequel.

Let`be a positive integer with`ˆ`1`2, where`1; `23 are relatively prime. We first construct a suitable model over SpecZ[z<sub>`</sub>] of the minimal compactification of the fine moduli space of principally polarizedg-dimen- sional abelian varieties with level`structure. We will appeal to the arithmetic minimal compactifications of Faltings-Chai [14, Chap. V, Thm. 2.3, Thm. 2.5].

LEMMA 3.9. There exists a normal integral schemeA_{g;`</sub>, proper over SpecZ[z<sub>`}], and an ample line bundlevonA_g;`, characterized by the fol- lowing properties:

i. the restrictions ofA_{g;`</sub>andvtoSpecZ[z<sub>`};1=`]coincide with the mini- mal compactification and canonical ample line bundle of Faltings-Chai;

ii. the restriction of A_{g;`</sub> to SpecZ[z<sub>`};1=`<sub>i</sub>] admits a natural finite morphism toA<sub>g;`i</sub>(iˆ1;2), extending the canonical morphismA_{g;`</sub>!A<sub>g;`i}; iii. the restriction of v toSpecZ[z<sub>`</sub>;1=`i]arises as a pull-back of the canonical line bundle of Faltings-Chai onA_g;`i, by the natural morphism, for iˆ1;2.

PROOF. We may assume g2, since the result easily follows from Deligne-Rapoport [9, Chap. V, Sec. 4] when gˆ1. Let us first construct A_g;`, satisfying the requirements of the lemma. We recall that there is a natural finite and eÂtale morphism of fine moduli schemes

Ag;` !Ag;`i[1=`]:

(3:6)

The notation Ag;`i[1=`] means we inverted the primes dividing ` in the structure sheaf ofAg;`i. In particularAg;` is obtained by normalization of A<sub>g;`i</sub>[1=`] in the function field ofA<sub>g;`</sub>. We defineAe⁽ⁱ⁾_{g;`</sub>as the normalization of A<sub>g;`i}in the function field ofA_{g;`</sub>. By construction, this is a normal integral scheme, proper of finite type over SpecZ[z<sub>`};1=`i]. Moreover the natural morphismpi:Ae<sup>(i)</sup><sub>g;`</sub>!A_{g;`i</sub> is finite because the base is an integral normal scheme and we are in generic characteristic 0. We also defineve<sub>i</sub>ˆp<sub>i</sub>v<sub>i</sub>, wherev<sub>i</sub>is the canonical ample sheaf onA<sub>g;`i}of Faltings-Chai [14, Thm. 2.5 (1)]. It will be enough to show that the restrictions of theAe⁽ⁱ⁾_{g;`</sub>(resp.ve<sub>i</sub>) to SpecZ[z<sub>`};1=`] coincide with the arithmetic minimal compactification A<sub>g;`</sub>

(resp. the canonical sheafvonA_g;`). This will also give the necessary gluing condition.

We claim that the arithmetic minimal compactificationA_{g;`</sub>(of Faltings- Chai) is obtained fromA<sub>g;`i}[1=`] by normalization too. This follows from the Koecher principle. Indeed, recall that by construction

A_g;`ˆProj M

k

G(A_g;`;v^k)

!

;

whereAg;`is any arithmetic toroidal compactification andvits canonical semi-ample sheaf [14, Thm. 2.5 (3)]. The Koecher principle [14, Prop. 1.5ii]

asserts that the restriction morphism

G(A_{g;`</sub>;v<sup>k</sup>) !G(A<sub>g;`};v^k)

is an isomorphism. In the application of the Koecher principle we used the assumption g2. Finally, from the finite eÂtale canonical projection morphism (3.6) we derive an integral inclusion of algebras

p_i :M

k

G(Ag;`i[1=`];v^k_i ),!M

k

G(Ag;`;v^k):

This establishes the claim. Therefore, sinceAe⁽ⁱ⁾_{g;`</sub>is constructed fromA<sub>g;`i}by normalization, we obtain a chain of finite morphisms

Ae⁽ⁱ⁾<sub>g;`</sub>[1=`]!^pi A_{g;`</sub>!A<sub>g;`i}[1=`]: