Slopes and Abelian Subvariety Theorem

49 Pages·2005·0.5 MB·English

Journal of Number Theory 112 (2005) 67–115
www.elsevier.com/locate/jnt
Slopes and Abelian Subvariety Theorem
∗
Evelina Viada
ETH Zürich, D-Math, Rämistrasse 101, CH-8092 Zürich, Switzerland
Received 8 April 2004; revised 17 May 2004
Communicated by J.-B. Bost
Abstract
In this article, we show how to modify the proof of theAbelian Subvariety Theorem by Bost
(Périodes et isogénies des variétés abeliennes sur les corps de nombres, Séminaire Bourbaki,
1994–95,Theorem5.1)inordertoimprovetheboundsinaquantitativerespectandtoextendthe
theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number
field (cid:1) and a non-trivial period (cid:2) in a proper subspace W of tAK with K a finite extension
of (cid:1), the Abelian Subvariety Theorem shows the existence of a proper abelian subvariety B of
A , whose degree is bounded in terms of the height of W, the norm of (cid:2), the degree of (cid:1)
Q
and the degree and dimension of A. If A is principally polarized then the theorem is explicit.
© 2004 Elsevier Inc. All rights reserved.
Keywords: Abelian varieties; Slopes Theory; Transcendence Theory
0. Introduction
In1990MasserandWüstholzstartedaseriesofpapersonperiodrelationsforabelian
varieties [24–29]. As an application they obtained a proof of the Tate Conjecture,
different from the proof originally given by Faltings in 1983. One of the central results
in the work of Masser and Wüstholz states that, given an abelian variety A defined
over a number field (cid:1), there exists only a finite number of (cid:1)-isomorphism classes of
abelian varieties defined over (cid:1) which are isogenous to A. Their approach consists in
∗
Corresponding author. Tu Darmstadt, 214 Schlossgartenstrasse 7, D-64289 Darmstadt, Germany. Fax:
[phone].
E-mail address: [email protected]
0022-314X/$-see front matter © 2004 Elsevier Inc. All rights reserved.
doi:10.1016/j.jnt.2004.07.015
68 E. Viada/Journal of Number Theory 112 (2005) 67–115
giving a bound for the degree of a minimal abelian subvariety B of A whose tangent
space at the origin contains a given period of the lattice associated to A. The proof
is a completely effective version of the analytic subgroup theorem by Wüstholz [46]
in the special case where the group is an abelian variety and where torsion points are
considered. No other known method gives such precise quantitative results.
In 1995 Bost [3] gave a Bourbaki talk on the work of Masser and Wüstholz. In 1997
and 1999 he also gave lectures at the Institut Henri Poincaré in Paris [5], where he
explained many details of the proofs not presented in [3]. One of the interesting aspects
of his approach is the intrinsic and geometric version of the argument originally given
by Masser and Wüstholz. Several new tools were introduced. For example, the use
of Arakelov geometry which had meanwhile become available. Arithmetic intersection
theory allows, among others, to define the height of an algebraic variety in general [7].
This height has nice functorial properties. Other geometric ingredients are hermitian
vector bundles on the spectrum of the ring of integers of a number field and the related
concept ofArakelov degree and slopes.An interesting aspect is the use of semistability
in transcendence. His approach avoids theta functions, the study of the moduli space of
polarized abelian varieties and the construction of auxiliary functions as they appear in
the work of Masser and Wüstholz. As a consequence proofs and effective computation
are more direct.
In this paper, we present the proof of theAbelian Subvariety Theorem. We focus our
attention on the analytic estimates.1 We improve the bounds given by Bost [3] in a
quantitative respect and we extend the theorem to subspaces rather than hyperplanes.
We consider an abelian variety A of dimension g defined over a number field (cid:1) of
absolute degree d =[(cid:1):Q]. Let L be a symmetric ample line bundle on A. We denote
by tAK the stalk at zero of the tangent bundle of AK where K is a finite extension
of (cid:1). Let W be a proper subspace of tAK of codimension s. Let (cid:3)0: K → C be an
embedding and (cid:2) a non-trivial period of A(cid:3) (C) such that (cid:2) ∈ W(cid:3) . We consider on
0 0
tthAe(cid:3)0etxhpeonnoernmtia|l|·m||a(cid:3)p0.inWdeucdeednboytethbeyfihr(sAt C)htherenFcalaltsisngosf hLe(cid:3)i0g.hWt oefcaAllaenxdp:bytA(cid:3)h0(W→)Ath(cid:3)e0
height of the space W, (see 1.1 and 1.5 for notation). We prove the following
Theorem 1 (Abelian Subvariety Theorem). There exists a proper, non-trivial abelian
subvariety B of A such that
Q
(g−1)((cid:4)−(cid:4) )(cid:1)s(cid:4), (1)
1
degL(cid:3) B(cid:1)C(g)(degLA)(cid:4)−s(cid:4)1 max(cid:1)degLA,hdr,dr log(dr)(cid:2)(cid:4)−gs((cid:4)−(cid:4)1), (2)
0
where
h = max(1,h(A),logdegLA,h(W)),
r = max(1,||(cid:2)||2(cid:3) ),
0
1The author intends to thank Bost for hand-written notes in which the main arguments of this section
were given.
E. Viada/Journal of Number Theory 112 (2005) 67–115 69
(cid:4) = dimB,
(cid:4)1 = dim(B∩expW(cid:3)0)
and C(g) is a constant depending only on g. If L is a principal polarization for A we
can set
C(g)=(5g)g5.
Note that condition (1) ensures that dim tB ∩ W(cid:3)0(cid:2)1 (if not (cid:4)1 = 0) and that
tB + W(cid:3)0 (cid:13)= tA(cid:3)0 (if not s = (cid:4) − (cid:4)1). If (cid:4)1 = 0 or s = (cid:4) − (cid:4)1 relation (1) is
contradicted.
Using an idea given by Masser and Wüstholz [26], we can deduce the following:
Corollary 1. UnderthesamehypothesesoftheAbelianSubvarietyTheoremthereexists
a proper abelian subvariety B of A such that
Q
(cid:2)∈t (cid:13)=t
B A(cid:3)
0
and
(cid:1) (cid:2)
degL(cid:3) B(cid:1)C(g)max degLA,hdr (cid:17)(g)
0
where C(g) and (cid:17)(g) are constants depending only on g. In particular we can set
(cid:17)(g)=5g.
As a nice application of our Abelian Subvariety Theorem in [45], we improve in a
quantitative respect the estimate given by Masser and Wüstholz for a minimal elliptic
isogenies, see [24]. Our proof is effective and puts in light the geometric aspects of the
problem. Estimates of this type have been studied by David [9] and later by Pellarin
[39]. However, their bounds are weaker in the dependence on the degree d of the field
of definition of A and on the constants. This simplest case has been a test for the more
general case of abelian varieties. The geometrical techniques used here can possibly
also be used to improve the result of Masser andWüstholz for abelian varieties. Further
technical difficulties are expected. For instance the more complicate structure of the
ring of endomorphisms for an abelian variety of dimension larger than 2, or the bigger
range for dimensions of a proper abelian subvariety may cause problems.
A very ambitious conjecture is that the degree of the isogeny, at least in the case
of elliptic curves, does not depend on the elliptic curve at all, but just on its field of
definition. This would imply for instance that an elliptic curve defined over a number
field (cid:1) has only finitely many torsion points defined over (cid:1). This result was proven by
Mazur [32,33], in the case that the field of definition is the field of rational numbers.
It was later generalized by Merel [34] for number fields. However, how to extend the
result to an arbitrary abelian variety seems to be unknown. In order to deduce the
conjecture with the classical approach one shall prove that, under the same hypotheses
70 E. Viada/Journal of Number Theory 112 (2005) 67–115
of the Abelian Subvariety Theorem, there exists a proper, non-trivial abelian subvariety
B of A whose degree is bounded only in terms of dimension, degree and field of
Q
definition of A.
We now say something about the structure of this work. In the first chapter we fix
notation and we recall the main properties of abelian varieties, Moret–Bailly models,
non-reduced subschemes and heights. In Chapter 2–4 we prove the Abelian Subvariety
Theorem. We give a proof of the Slope Inequality which gives a relation between
the analytic and algebraic quantities attached to a morphism of modules. We define
non-reduced subschemes of A of dimension zero and an arithmetic filtration in order
to be able to apply the Slope Inequality to our problem, (Chapter 2). We estimate
the norms of the evaluation operators. Here the classical Schwarz Lemma, which is
fundamental for all modern transcendence proofs, is replaced, on Bost’s suggestion,
by the Phragmen–Lindelöf Theorem to obtain a better lemma, (Chapter 3). For the
algebraic estimates we refer to [3,15]. Using the Slope Inequality and these estimates,
we prove that there exists a section of a line bundle on A which vanishes on a non-
reduced torsion subscheme. By the Multiplicity Estimate, Theorem 3, we conclude that
there exists a proper abelian subvariety of A of bounded degree, (Chapter 4). Finally,
Q
in the appendix we recall basic properties of the Arakelov degree.
1. Preliminaries and notation
Let OK be the ring of integers of a number field K. We denote by S = SpecOK
and by S = SpecK. If E is an OK-module and R an OK-algebra, we denote by EK
and RK the tensor product E⊗O K and R⊗O K, respectively. If X is a scheme
K K
over S, we denote by XK the fiber product of X and S over S. If K is fixed we might
denote XK simply by X. If (cid:5) is a finite subscheme of X we denote by (cid:5)K the fiber
product of (cid:5) and S over S. If (cid:3): K →C is an embedding we will write X(cid:3) for the
fiber product X ×(cid:3)C. Let E be a sheaf of OX-modules on X, we denote by E(cid:3) the
sheaf on X(cid:3) given by pull-back E(cid:3) =p1∗E, here p1 is the canonical projection of the
fiber product X(cid:3) on X. We denote by X(K) = HomZ(S,X) the set of points of X
with value in K.
1.1. Abelian varieties
Let A be an abelian variety of dimension g, defined over a number field K and L
an ample line bundle on A. The Euler–Poincaré characteristic (cid:6)(A,L) is defined as
the alternating sum of the Hi(A,L) dimensions. The degree of A with respect to L
is defined as the intersection number of the first Chern class c (L) with itself g-times.
1
The Riemann–Roch theorem [22, Theorem 3.10] gives the relation
1
(cid:6)(A,L)= g!degLA. (3)
E. Viada/Journal of Number Theory 112 (2005) 67–115 71
If L is an ample line bundle then the ith cohomology group vanishes for every i (cid:13)=0,
so (cid:6)(A,L)=dim H0(A,L), (see [22, Corollary 3.11]). One says that L is a principal
polarization for A if (cid:6)(A,L) = 1. We denote the stalk of the tangent bundle of the
variety A at zero by tA. We indicate the sheaf of differentials by (cid:7)A and its stalk at
zero by tˇA. On an abelian variety the global forms are translation invariant, this is a
consequence of the fact that the translation maps are isomorphisms, [37, Paragraph 11,
Proposition, p. 98].
Let (cid:3): K → C be an embedding. Let exp(cid:3): Lie(A(cid:3)) = tA(cid:3) → A(cid:3)(C) be the
exponential map of A(cid:3), we denote by (cid:8)A(cid:3) its kernel. The first Chern class c1(L(cid:3))
of L(cid:3) is an element of H2(A(cid:3),Z) = H2((cid:8)A(cid:3),Z). There exists a unique translation
invariant representative of c1(L(cid:3)). It defines an alternating 2-form E(cid:3)((cid:2)1,(cid:2)2) on (cid:8)A(cid:3)
with values in Z such that E(cid:3)(ix,iy) = E(cid:3)(x,y) and E(cid:3) = ImH(cid:3) where H(cid:3) is a
hermitian form on tA(cid:3) ×tA(cid:3) [22, Lemma 3.1, 3.4]; [37, I.2 pp. 18–21]. Since L(cid:3) is
ample H(cid:3) turns out to be positive defined, thus it can be seen as a positive element
(cid:9)(cid:3) in (cid:7)1,1(tˇA(cid:3)), we define the g-exterior product by d(cid:10)(cid:3) = (cid:9)(cid:3)∧···∧(cid:9)(cid:3) [22, 3.3].
The determinant of E(cid:3) does not depend on the embedding (cid:3), we denote it by det E.
From [22, Theorem 2.3] we have the following relation:
(cid:3)
√
det E =(cid:6)(A,L)= d(cid:10)(cid:3), (4)
F
A(cid:3)
where FA(cid:3) is a fundamental domain with respect to the lattice (cid:8)A(cid:3).
1.1.1. Trivialization
In this section, to avoid heavy notation, we identify A(cid:3) with A and a line bundle
L(cid:3) on A(cid:3) with L.
We consider on tA the hermitian metric induced by the first Chern class of L. We
denote the related norm ||·||. We denote by |·| the standard Euclidean norm on C.
We endow the trivial bundle OtA on tA with the norm
||f(z)||z∗L =|f(z)|em(z), (5)
where f is a section of OtA and m(z)=−(cid:11)2||z||2.
The line bundle exp∗L is trivial on tA. By theAppel–Humbert theorem [37, p. 20] a
trivialization (cid:12): OtA →exp∗L is given identifying (z,f(z)) and (z+(cid:2),e(cid:2)(z)f(z)) for
(cid:2)∈(cid:8)A and automorphic factor e(cid:2)(z) =(cid:13)(z)e−(cid:11)H((cid:2),z)+(cid:11)2H((cid:2),(cid:2)) where (H,(cid:13)) is a Appel–
Humbert couple. Since (5) is compatible with this identification it defines a metric on
L whose first Chern form is equal to (cid:9). Thus the metric induced by L on exp∗L and
the one defined by (5) are equal up to a multiplicative constant. Then we can choose
a trivialization such that the isomorphism (cid:12): OtA → exp∗L is an isometry, (see also
[13, p. 70]).
72 E. Viada/Journal of Number Theory 112 (2005) 67–115
Let (cid:7) be an open set of tA on which exp is a homeomorphism. Let LD be the D
tensor product of L and s a section of LD(exp((cid:7))). For every z∈(cid:7) we have
||s(expz)||LD =|f(z)|em(z), (6)
with f =(cid:12)(exp∗s) and m(z)=−(cid:11)D||z||2.
2
Let FA be(cid:4) a fundamental domain for the lattice (cid:8)A = ker(exp). Then (4) gives
(cid:6)(A,LD)= d(cid:10) with d(cid:10)=(cid:9)D ∧···∧(cid:9)D the g-exterior power of the translation
F
A
invariant representative (cid:9)D of c (LD).
1
By definition the normalized Haar-measure on A satisfies
(cid:3) (cid:3)
1= d(cid:14)= exp∗ d(cid:14). (7)
A FA
Since the Haar-measure is unique we can write
d(cid:10)
=(cid:6)(A,LD). (8)
exp∗d(cid:14)
1.1.2. Injectivity radius
The radius of injectivity of A(cid:3) with respect to the metric on tA(cid:3) induced by L(cid:3)
is the largest real number (cid:15)i(A(cid:3),L(cid:3)) such that the restriction of the exponential map
to the open ball with center in zero and radius (cid:15)i(A(cid:3),L(cid:3)) is a homeomorphism. By
definition
(cid:15)i(A(cid:3),L(cid:3))= 210(cid:13)=m(cid:10)(cid:3)i∈n(cid:8)(cid:3) ||(cid:10)(cid:3)||(cid:3). (9)
Minkowski’s theorem (see [8, VIII.4.3]) yields
(cid:15)i(A(cid:3),L(cid:3))(cid:1)(cid:11)−21(degLA)21g. (10)
An important estimate for the radius of injectivity is due to Masser and Wüstholz [25,
Lemma 8.6]. For every line bundle L on A one has
(cid:6) (cid:7)
(cid:5)
1 1
[K :Q] (cid:15)i(A(cid:3),L(cid:3))−2(cid:1)c(g)max 1,h(A)+ 2log(cid:6)(A,L) (11)
(cid:3): K→C
where c(g) is a positive real constant depending only on g and h(A) is the Faltings
height of A (see Section 1.5). Effective versions, when L is a principal polarization for
A, have been given by Graftieaux [15, Proposition 2.11] and by David and Philippon
[10, Appendix, Lemma 6.7, 6.8]. In this case one can set c(g)=2g4.
E. Viada/Journal of Number Theory 112 (2005) 67–115 73
1.2. Arithmetic varieties
Definition 1. An arithmetic variety X over OK is a scheme over SpecOK such that
(cid:11): X →SpecOK is a quasi-projective flat morphism of schemes. Moreover we require
that there exists a section P: SpecOK →X and that the generic fiber is smooth and
proper.
The set of complex p(cid:8)oints X(C)=HomZ(SpecC,X) is the disjoint union of com-
plex varieties X(C) = (cid:3): K→CX(cid:3)(C). Indeed, if P: SpecC → XK is a complex
point, then the composition (cid:11)K ◦P: SpecC → SpecK induces an embedding (cid:3) =
((cid:11)P)" from K to C.
The notion of hermitian vector bundle is given in Appendix A.1.
Definition 2. Let X be an arithmetic variety over SpecOK. A hermitian vector bundle
E on X is a pair (E,h) where E is a locally free sheaf of finite rank on X, and
(E(C),h) is a hermitian vector bundle on X(C) invariant under conjugation. If E has
rank 1 one says that E is a hermitian line bundle.
Invariant under conjugation means that if (cid:3) and (cid:3)c are conjugated embeddings of
K, then for every open set U of X the map id⊗c: E(U)⊗(cid:3)C→E(U)⊗(cid:3)c C is an
isometry, in particular we get ||s||(cid:3) =||s||(cid:3)c.
(cid:8)
Example. In the special case of S :=SpecOK, we have that SC = {(cid:3): K→C} SpecC.
In fact SC =Spec(OK⊗ZC)=Spec(K⊗QC). Let f(x) be an element of Q[x] such
that K(cid:1)Q[x]/(f(x)). Then K⊗QC is isomorphic to C[x]/((cid:9)f(x)). In the field C the
polynomial f(x) splits in d =[K :Q] linear factors f(x)= di=1(x−(cid:13)i). By Galois
theory there exists an isomorphism be(cid:9)tween the embeddings {(cid:3): K →C} and the roots
of f(x). Thus we can write f(x(cid:9))= (cid:3)(x−(cid:13)(cid:3))(cid:9). By the Chinese Reminder Theorem
we g(cid:9)et the isomorphism C(cid:8)[x]/( (cid:3)(x −(cid:13)(cid:3)))(cid:1) (cid:3)C[x]/(x −(cid:13)(cid:3)). Therefore, SC =
Spec (cid:3)C[x]/(x −(cid:13)(cid:3)) = (cid:3): K→CSpecC and we have the following commutative
diagram:
SC =Spec(C[x]/(f(x))) −−−−→ SpecC
 
 
(cid:11) (cid:11)
S =SpecK −−−−→ SpecQ
 
 
(cid:11) (cid:11)
S =SpecOK −−−−→ SpecZ.
(cid:8)
In an analogous way, if X is a S-scheme then XC = (cid:3): K→CX(cid:3), where X(cid:3) is the
fiber product of X and SpecC over SpecK through the embedding (cid:3).
74 E. Viada/Journal of Number Theory 112 (2005) 67–115
A sheaf on an affine variety is locally free if and only if its global sections are a
projective module, (see [18, Chapter II, Paragraph 5].
For a finitely generated module over a Dedekind-domain the notions of torsion free,
flat and projective module coincide, (see [12,Theorem 13, p. 95].Therefore there exists
an isomorphism of categories between the category of locally free sheaves of finite rank
over SpecOK and the category of finitely generated torsion free OK-modules. For this
reason we will identify the objects of the two categories.
1.2.1. The push-forward of a Hermitian vector bundle
Let E be a hermitian vector bundle on an arithmetic variety (cid:11): X →SpecOK.
The work of Moret–Bailly [36, Lemma 1.4.2] shows that the push forward (cid:11)∗E of
a locally free sheaf E on X to SpecOK is still locally free.
We want to induce a metric on the vector bundle E =H0(S,(cid:11)∗E)=H0(X,E) on
SpecOK. For each section s ∈E we define
(cid:3)
||s||2E,(cid:3) = X(cid:3)(C)||sx||2E(cid:3)d(cid:14)(cid:3)(x), (12)
with d(cid:14)(cid:3) a measure on X(cid:3) and ||·||2E = h(cid:3),x(·,·). In the special case of an abelian
variety we will choose d(cid:14)(cid:3) to be the no(cid:3)rmalized Haar measure, i.e. the only normalized
measure, invariant under the group law. In the case of a projective space one can use
the Fubini-Study metric [16, Example 2, p. 30].
1.3. Moret–Bailly models
Definition 3. A semiabelian scheme (cid:11): A → SpecOK is a smooth group scheme
(separated and of finite type), such that the components of its fibers are extensions of
abelian varieties by tori (semiabelian group), and its generic fiber is an abelian variety.
A semiabelian scheme A is in particular an arithmetic variety, thus for any line
bundle L on A the direct image (cid:11)∗L is a locally free sheaf on SpecOK, (Section 1.2).
If L is a hermitian ample line bundle, we can endow the vector bundle (cid:11)∗L with the
push-forward metric (1.2.1). The bundle (cid:11)∗L = ((cid:11)∗L,||s||2L) is an hermitian vector
bundle on SpecOK of rank equal to dim H0(A,L).
We are going to recall the definition of MB-models given in [4, 4.3.1].
Let A be an abelian variety over Q, L an ample symmetric line bundle over A and
(cid:5) a finite subset of A(Q). A MB-model of (A,L,(cid:5) ) over a number field K is
Q Q
defined as the data
• a semiabelian scheme (cid:11): A→SpecOK,
• an isomorphism i :A(cid:1)A of abelian varieties over Q,
Q
• a cubist hermitian line bundle L on A (in the sense of [35]),
• an isomorphism L(cid:1)L ,
Q
E. Viada/Journal of Number Theory 112 (2005) 67–115 75
• for each point P ∈ (cid:5)Q a section P: SpecOK → A of (cid:11) such that the geometric
point P coincides with i(P),
Q
which satisfy the following condition: there exists a subscheme K of A flat and finite
over SpecOK such that i−1(KQ) coincides with the Mumford group K(L2) (see [22,
4.1]).
The properties of MB-model we are going to use are summarized in the following
theorem (see [4, Theorem 4.10]; [3, 4.2] for the semistability in (iii)).
Theorem 2. (i) Existence: There exists a MB-model for the data (A,L,(cid:5) ) defined
Q
over a finite extension of the field of definition of A and (cid:5) .
Q
(ii) Néron-Tate height: For any MB-model and any P ∈ (cid:5) the Arakelov degree
Q
d(cid:12)egnP∗L coincides with the normalized Néron-Tate height of P associated to L.
(iii) Semistability: The vector bundle (cid:11)∗L is semistable and its slope is
1 1 (cid:6)(A,L)
(cid:14)ˆ((cid:11)∗L)=−2h(A)+ 4 log (2(cid:11))g ,
where h(A) is the Faltings height of A (see Section 1.5).
(iv) Compatibility of a MB-model with scalar extension. If we have a MB-model of
(A,L,(cid:5) ) over some number field K and K is a finite extension of K, then the model
Q
got by extension of scalar from OK to OK is a MB-model of (A,L,(cid:5)Q) over the
number field K.
1.4. Non-reduced subschemes
If X is a scheme we denote by Xtop its underlying topological space and by OX its
structural sheaf. If j: Y → X is a closed immersion, we denote by IY = ker j" the
ideal sheaf of Y and by It its tth power.
Y
1.4.1. Subschemes of multiplicity t
The cokernel OY,t of the natural inclusion IYt → OX defines a scheme Yt =
(Ytop,j−1OY,t), (see [18, Example II.3.2.5]). We call Yt the subscheme of X of mul-
tiplicity t at Y.
The exact sequence
0 −−−−→ IY/IYt −−−−→ OX/IYt −−−−→ OX/IY −−−−→ 0
induces a closed immersion it: Y →Yt.
1.4.2. Schematic image and closure of a scheme
Let f: Z →X be a morphism of schemes. Then there is a unique closed subscheme
Y of X with the following property: the morphism f factors throughY, and if Y(cid:25) is any
76 E. Viada/Journal of Number Theory 112 (2005) 67–115
other closed subscheme of X through which f factors, then Y → X factors trough Y(cid:25)
too. The scheme Y is called the schematic image of Z in X. If f is an immersion
then the scheme Y is called the schematic closure of Z in X (see [18, Exercise
II.3.11.d]).
Example. A point P of A with value in OK is defined as a morphism of schemes
P: S →A. Since A is a scheme over S, the morphism P is a closed immersion. We
denote by Pt, meaning Pt: St →A, the subscheme of A of multiplicity t at P.
The scheme A is smooth over S thus the push forward via (cid:11)◦Pt of the structural
sheaf OP,t of Pt is a locally free sheaf over S.
If we consider an abelian variety over S =SpecK and a point P: S →A defined
over K, we can define the subscheme Pt: St → A of multiplicity t at P. If A,P
is a MB-model of A,P then the subscheme Pt is the schematic closure in A of the
subscheme Pt.
1.4.3. Multiplicity along a subspace
Let A be an abelian variety of dimension g defined over a number field (cid:1), and let
L be an ample symmetric hermitian line bundle over A. We denote by S the scheme
SpecK with K a finit(cid:8)e extension of (cid:1). We consider a reduced closed subscheme of
dimension zero (cid:5)K: S → A with values in A(K), i.e. a disjoint union of points
P: S →A defined over K. We suppose that the neutral element 0A: SpecK →A of
A belongs to (cid:5)K.
Since A is smooth we have the isomorphism OA/I02A(cid:1)K⊕I0A/I02A =K⊕tˇA. We
denote by SW,1 the spectrum of K ⊕Wˇ, with W a subspace of tAK. The inclusion
W )→ tA induces a surjection of algebras OA /I2 → K ⊕Wˇ and hence a closed
0 0A
embedding of schemes
0A,W,1: SW,1 )→A.
We consider the schematic image SW,t of the scheme SW,1 ×···×SW,1 under the
addition morphism
+t: A(cid:13) ×·(cid:14)·(cid:15)·×A(cid:16)−→A.
t−times
We call 0A,W,t: SW,t →A the subscheme of A of multiplicity t at 0A along W.
If P is any point of A different from the origin we consider the translation by −P
(cid:16)−P: A → A. We define the subscheme PW,t of multiplicity t at P along W as the
pull-back via the isomorphism (cid:16)−P of the scheme 0A,W,t.
Let t: (cid:5)K → N+ be a multipl(cid:8)icity map that associates to any P ∈ (cid:5)K the multi-
plicity t(P). We denote (cid:5)K,W,t = P∈(cid:5)K PW,t(P) the scheme of multiplicity t alongW
at (cid:5)K. In the following work we will suppose 0A ∈(cid:5)K and to every natural number
m∈N we associate the multiplicity function m: (cid:5)→{2m,m} such that m(0A)=2m
and m(P)=m for 0A (cid:13)=P ∈(cid:5)K.