Modularity of abelian varieties over Q with bad reduction in one prime only

We show that certain abelian varieties over Q with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.     

On the rank of abelian varieties over function fields

Let be a smooth projective curve defined over a number field k, A/k() an abelian variety and (τ, B) the k()/k-trace of A. We estimate how the rank of A(k())/τB(k) varies when we take a finite geometrically abelian cover defined over k. This work was partially supported by CNPq research grant 304424/2003-0, Pronex 41.96.0830.00 and CNPq Edital Universal 470099/2003-8. I would like to thank Douglas Ulmer for comments on how to treat the case of arbitrary ramification, but the conductor prime to the ramification locus, in the case of elliptic fibrations. I would also like to thank Marc Hindry for comments on the inequality comparing the conductors of A and A'. Finally, I also thank the referee for his comments and criticisms.     

On CM abelian varieties over imaginary quadratic fields

In this paper, we associate canonically to every imaginary quadratic field K= one or two isogenous classes of CM (complex multiplication) abelian varieties over K, depending on whether D is odd or even (D4). These abelian varieties are characterized as of smallest dimension and smallest conductor, and such that the abelian varieties themselves descend to . When D is odd or divisible by 8, they are the scalar restriction of canonical elliptic curves first studied by Gross and Rohrlich. We prove that these abelian varieties have the striking property that the vanishing order of their L-function at the center is dictated by the root number of the associated Hecke character. We also prove that the smallest dimension of a CM abelian variety over K is exactly the ideal class number of K and classify when a CM abelian variety over K has the smallest dimension.Mathematics Subject Classification (1991): 11G05, 11M20, 14H52Partially supported by a NSF grant DMS-0302043     

Uniform boundedness of level structures on abelian varieties over complex function fields

Let X = Ω/Γ be a smooth quotient of a bounded symmetric domain Ω by an arithmetic subgroup . We prove the following generalization of Nadel's result: for any non-negative integer g, there exists a finite étale cover X_g = Ω/Γ(g) of X determined by a subgroup depending only on g, such that for any compact Riemann surface R of genus g and any non-constant holomorphic map f : R → X_g^* from R into the Satake-Baily-Borel compactification X_g^* of X_g, the image f(R) lies in the boundary ∂X_g: = X^*_g\X_g. Nadel proved it for g = 0 or 1. Moreover, for any positive integer n and any non-negative integer g≥0, we show that there exists a positive number a(n,g) depending only on n and g with the following property: a principally polarized non-isotrivial n-dimensional abelian variety over a complex function field of genus g does not have a level-N structure for N≥a(n,g). This was proved by Nadel for g = 0 or 1, and by Noguchi for arbitrary g under the additional hypothesis that the abelian variety has non-empty singular fibers.     

Extensions of abelian varieties defined over a number field

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with congruences between modular forms and special values of L-functions.     

A modularity criterion for Klein forms, with an application to modular forms of level 13

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ₁(N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ₁(13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).     

On Euler characteristics of Selmer groups for abelian varieties over global function fields

Let F be a global function field of characteristic \({p > 0}\), \({K/F}\) an \({\ell}\)-adic Lie extension (\({ \ell \neq p}\)), and \({A/F}\) an abelian variety. We provide Euler characteristic formulas for the Gal\({(K/F)}\)-module \({Sel_A(K)_\ell}\).     

The groups of points on abelian varieties over finite fields

Let A be an abelian variety with commutative endomorphism algebra over a finite field k. The k-isogeny class of A is uniquely determined by a Weil polynomial f _A without multiple roots. We give a classification of the groups of k-rational points on varieties from this class in terms of Newton polygons of f _A (1 − t).     

On syzygies of projective bundles over abelian varieties

Syzygies or $N_p$-property of an ample line bundles on abelian varieties are well known. In this paper, we study defining equations and syzygies among them of projective bundles over abelian varieties. We prove an analogue of Pareschi's theorem (or Lazarsfeld's conjecture) on abelian varieties, extended to projective bundles over an abelian variety.