Related papers: Control Theorems for Abelian Varieties over Functi…
We prove Hilbert's irreducibility theorem for abelian varieties over function fields of characteristic zero.
We study isogeny classes of abelian varieties over a function field in one variable over the field of complex numbers.
We study the distribution of algebraic points on curves in abelian varieties over finite fields.
The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.
We prove a Torelli-like theorem for higher-dimensional function fields, from the point of view of "almost-abelian" anabelian geometry.
In this paper, we consider Abelian varieties over function fields that arise as twists of Abelian varieties by cyclic covers of irreducible quasi-projective varieties. Then, in terms of Prym varieties associated to the cyclic covers, we…
We give a presentation of abelian class field theory.
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 special values of L-functions and congruences between modular forms.
In this paper we improve our previous results on classification of groups of points on abelian varieties over finite fields. The classification is given in terms of the Weil polynomial of abelian varieties in a given $k$-isogeny class.
We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial P(T) modulo l. As an application we find the proportion of isogeny classes of abelian varieties with a…
In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…
This paper surveys Abelian and Tauberian theorems for long-range dependent random fields. We describe a framework for asymptotic behaviour of covariance functions or variances of averaged functionals of random fields at infinity and…
We provide a characterization of almost ordinary abelian varieties over finite fields, and use this characterization to provide lower bounds for the sizes of some almost ordinary isogeny classes.
In this letter we show that for certain infinite families of modular forms of growing level it is possible to have a control result for the exceptional primes of the attached Galois representations. As an application, a uniform version of a…
We give upper and lower bounds on the number of points on abelian varieties over finite fields, and lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian…
We prove the existence of abelian varieties not isogenous to Jacobians over characterstic $p$ function fields. Our methods involve studying the action of degree $p$ Hecke operators on hypersymmetric points, as well as their effect on the…
In this paper we will prove that Tate conjecture of abelian varieties over finite field is equivalent to the finiteness of isomorphism classes of abelian varieties with a fixed dimension. We give a different approach with Zarhin's result.
In this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic $p>0$ is isogenous to another one defined over a finite field. We also show that the category of…
We use knowledge of local fields to adapt Jonathan Lubin and Michael Rosen's proof of Mazur's Proposition 4.39. This changes the result about abelian varieties from only working over local fields with a finite residue field to working with…
In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are…