Abelian Subvariety Research Articles

HYPERGEOMETRIC SERIES AND HODGE CYCLES OF FOUR DIMENSIONAL CUBIC HYPERSURFACES

In this article we find connections between the values of Gauss hypergeometric functions and the dimension of the vector space of Hodge cycles of four-dimensional cubic hypersurfaces. Since the Hodge conjecture is well-known for those varieties we calculate values of hypergeometric series on certain CM points. Our methods are based on the calculation of the Picard–Fuchs equations in higher dimensions, reducing them to the Gauss equation and then applying the Abelian Subvariety Theorem to the corresponding hypergeometric abelian varieties.

Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

We relate the existence of Frobenius morphisms into the Jacobians of Drinfeld modular curves to the existence of congruences between cusp forms.

Polarizations of Prym varieties of pairs of coverings

To any pair of coverings f i :X → X i , i=1, 2, of smooth projective curves one can associate an abelian subvariety of the Jacobian J X , the Prym variety P(f 1, f 2) of the pair (f 1, f 2). In some cases we can compute the type of the restriction of the canonical principal polarization of J X . We obtain 2 families of Prym-Tyurin varieties of exponent 6.

Heights and Degrees: A Note on a Paper of C. Liebendörfer and G. Rémond

We verify in the rational quaternionic case a conjecture of C. Liebendorfer relating the degrees of abelian subvarieties of certain polarized abelian varieties to the (quaternionic) heights of their defining equations.

A curve algebraically but not rationally uniformized by radicals

Zariski proved the general complex projective curve of genus g > 6 is not rationally uniformized by radicals, that is, admits no map to P 1 whose Galois group is solvable. We give an example of a genus seven complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z ′ → Z with Z ′ rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci W d ( C ) in the Jacobian of a curve C may contain translates of abelian subvarieties not arising from maps from C to other curves.

Slopes and Abelian Subvariety Theorem

In this article, we show how to modify the proof of the Abelian Subvariety Theorem by Bost (Périodes et isogénies des variétés abeliennes sur les corps de nombres, Séminaire Bourbaki, 1994–95, Theorem 5.1) in order to improve the bounds in a quantitative respect and to extend the theorem to subspaces instead of hyperplanes. Given an abelian variety A defined over a number field κ and a non-trivial period γ in a proper subspace W of t A K with K a finite extension of κ , the Abelian Subvariety Theorem shows the existence of a proper abelian subvariety B of A Q ¯ , whose degree is bounded in terms of the height of W, the norm of γ , the degree of κ and the degree and dimension of A. If A is principally polarized then the theorem is explicit.

On the order of the reduction of a point on an abelian variety

Consider a point of infinite order on an abelian variety over a number field. Then its reduction at any place v of good reduction is a torsion point. For most of this paper we fix a rational prime ℓ and study how the ℓ-part of this reduction varies with v. Under suitable conditions we prove various statements on this ℓ-part for all v in a set of positive Dirichlet density: for example that its order is a fixed power of ℓ, that its order is non-trivial for the reductions of finitely many points, or that its order is larger than a certain explicit value that varies with v. By similar methods we prove that for all v in a set of positive Dirichlet density the reduction of a given abelian variety possesses no non-trivial supersingular abelian subvariety.

A conjecture of Beauville and Catanese revisited

A theorem of Green, Lazarsfeld and Simpson (formerly a conjecture of Beauville and Catanese) states that certain naturally defined subvarieties of the Picard variety of a smooth projective complex variety are unions of translates of abelian subvarieties by torsion points. Their proof uses analytic methods. We refine and give a completely new proof of their result. Our proof combines galois-theoretic methods and algebraic geometry in positive characteristic. When the variety has a model over a function field and its Picard variety has no isotrivial factors, we show how to replace the galois-theoretic results we need by results from model theory (mathematical logic). Furthermore, we prove partial analogs of the conjecture of Beauville and Catanese in positive characteristic.

A generalized Bloch's theorem and the hyperbolicity of the complement of an ample divisor in an Abelian variety

Bloch's theorem on the hyperbolicity of nonlinear subvarieties of abelian varieties states that the Zariski closure of the image of a holomorphic map from C to an abelian variety is precisely the translate of an abelian subvariety [B26,KS0,O77,W80,GG79] (for a survey on the proofs of Bloch's theorem see e.g., [Si95,Chapter 7]). The purpose of this note is to prove the following generalization of Bloch's theorem. If the image of a holomorphic map f from C to an abelian variety is Zariski dense, then the Zariski closure of the image of the differential of any order of the map f is invariant under any translation of the abelian variety (Theorem(2.2) below). This generalization has as a corollary the conjecture of Lang [La91] that the complement of an ample divisor in an abelian variety is hyperbolic. The following slightly more general statement, which together with Bloch's theorem implies Lang 's conjecture (see Corollary(3.2)), is a corollary of our generalized Bloch's theorem. The image from C to an abelian variety with Zariski dense image must intersect any complex hypersurface of the abelian variety (Theorem(3. !)).

Problème de Mordell-Lang modulo certaines sous-variétés abéliennes

Following a result of Bombieri, Masser, and Zannier on tori, the second author proved that the intersection of a transversal curve C in a power Eg of a C. M. elliptic curve with the union of all algebraic subgroups of Eg of codimension 2 is finite. Here transversal means that C is not contained in any translate of an algebraic subgroup of codimension 1. We merge this result with Faltings' theorem that C∩Γ is finite when Γ is a finite rank subgroup of Eg. We obtain the finiteness of the intersection of C with the union of all Γ+B for Ban abelian subvariety of codimension 2. As a corollary, we generalize the previous result to a curve C not contained in any proper algebraic subgroup, but possibly contained in a translate. We also have weaker analog results in the non C. M. case.

Formal subgroups of abelian varieties

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Neron model of A over the ring of integers O k of k. Completing ? along its zero section defines a formal group $\widehat{\mathcal{A}}$ over O k . We prove that any formal subgroup of the generic fiber of $\widehat{\mathcal{A}}$ whose closure in $\widehat{\mathcal{A}}$ is smooth over an open subset of Spec O k arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3].

Décompte dans une conjecture de Lang

Faltings has proven the following conjecture of Lang: if A is an abelian variety over a number field and X any subvariety then all rational points of X lie on a finite number N of translates, contained in X, of abelian subvarieties of A. We provide an upper bound for N whose main feature is uniformity in X since it does not depend on the height of X. Moreover, the bound is completely explicit. Together with the result of a previous paper, the heart of the proof is a suitable generalization of Mumford’s theorem for curves.

On invisible elements of the Tate-Shafarevich group

Mazur [7] has introduced the concept of visible elements in the Tate-Shafarevich group of optimal modular elliptic curves. We generalized the notion to arbitrary abelian subvarieties of abelian varieties and found, based on calculations that assume the Birch-Swinnerton-Dyer conjecture, that there are elements of the Tate-Shafarevich group of certain sub-abelian varieties of J 0 ( p) and J 1 ( p) that are not visible.

Parametrizations of elliptic curves by Shimura curves and by classical modular curves.

Fix an isogeny class of semistable elliptic curves over Q. The elements A of have a common conductor N, which is a square-free positive integer. Let D be a divisor of N which is the product of an even number of primes--i.e., the discriminant of an indefinite quaternion algebra over Q. To D we associate a certain Shimura curve X(0)D(N/D), whose Jacobian is isogenous to an abelian subvariety of J0(N). There is a unique A [symbol; see text] A in for which one has a nonconstant map piD : X(0)D(N/D) --> A whose pullback A --> Pic0(X(0)D(N/D)) is injective. The degree of piD is an integer deltaD which depends only on D (and the fixed isogeny class A). We investigate the behavior of deltaD as D varies.

Duality on tori and multiplicative dependence relations

AbstractIn this paper, we give multihomogeneous estimates for the group of relations linking multiplicatively dependent algebraic numbers. In the process, we raise a question in the style of Lehmer's problem, concerning multidimensional covolumes in the lattice of units. The proofs are based on the Brill-Gordan duality theorem on orthogonal lattices, and the paper closes with an algebraic version of the theorem, concerning orthogonal abelian subvarieties of an arbitrarily polarized abelian variety.

Principal bundles on elliptic fibrations

A central role in recent investigations of the duality of F-theory and heterotic strings is played by the moduli of principal bundles, with various structure groups G, over an elliptically fibered Calabi-Yau manifold on which the heterotic theory is compactified. In this note we propose a simple algebro-geometric technique for studying the moduli spaces of principal G-bundles on an arbitrary variety X which is elliptically fibered over a base S: The moduli space itself is naturally fibered over a weighted projective base parametrizing spectral covers $\tilde{S}$ of S, and the fibers are identified as translates of distinguished Pryms of these covers. In nice situations, the generic Prym fiber is isogenous to the product of a finite group and an abelian subvariety of $Pic(\tilde{S})$.

A Lie theoretic Galois theory for the spectral curves of an integrable system. II

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group W W and the Hecke algebra of double cosets of a parabolic subgroup of W . W. For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

Seshadri constants on abelian varieties

We show that on a complex abelian variety of dimension two or greater the Seshadri constant of an ample line bundle is at least one. Moreover, the Seshadri constant is equal to one if and only if the polarized abelian variety splits as a product of a principally polarized elliptic curve and a polarized abelian subvariety of codimension one. We also examine the case when the Seshadri constant is not one and obtain lower bounds when the dimension of the abelian variety is small.

Abelian Subvarieties

LetKbe a field and letXbe an abelian variety overK. The group AutXofK-automorphisms ofXacts in a natural way on the set of abelian subvarieties ofXthat are defined overK. In this paper it is proved that the number of orbits is finite. The proof makes use of a finiteness result about semisimple algebras.

An explicit version of Faltings' Product Theorem and an improvement of Roth's lemma

Introduction. In his paper Diophantine approximation on abelian varieties [1], Faltings proved, among other things, the following conjecture of Weil and Lang: if A is an abelian variety over a number field k and X a subvariety of A not containing a translate of a positive dimensional abelian subvariety of A, then X contains only finitely many k-rational points. One of Faltings’ basic tools was a new non-vanishing result of his, also proved in [1], the so-called (arithmetic version of the) Product Theorem. It has turned out that this Product Theorem has a much wider range of applicability in Diophantine approximation. For instance, recently Faltings and Wustholz gave an entirely new proof [2] of Schmidt’s Subspace Theorem [15] based on the Product Theorem. Faltings’ Product Theorem is not only very powerful for deriving new qualitative finiteness results in Diophantine approximation but, in an explicit form, it can be used also to derive significant improvements of existing quantitative results. In the present paper, we work out an explicit version of the arithmetic version of the Product Theorem; except for making explicit some of Faltings’ arguments from [1] this did not involve anything new. By using the same techniques we improve Roth’s lemma from [12]. Roth’s lemma was used by Roth in his theorem on the approximation of algebraic numbers by rationals [12] and later by Schmidt in his proof of the Subspace Theorem [15]. In two subsequent papers we shall apply our improvement of Roth’s lemma to derive significant improvements on existing explicit upper bounds for the number of subspaces in the Subspace Theorem, due to Schmidt [16] and Schlickewei [14] and for the number of solutions of norm form equations [17] and S-unit equations [13].