Related papers: Curves in abelian varieties over finite fields

Heavenly abelian varieties are abelian varieties defined over number fields that exhibit constrained $\ell$-adic Galois representations for some rational prime $\ell$. At the ICMS Workshop held in November 2024, we presented evidence for…

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…

We discuss methods for using the Weil polynomial of an isogeny class of abelian varieties over a finite field to determine properties of the curves (if any) whose Jacobians lie in the isogeny class. Some methods are strong enough to show…

We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial…

We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of…

We count the number of isomorphism classes of degree $d$-twists of some polarized abelian varieties over finite fields of odd prime dimension. This can be seen as a higher dimensional analogue of the counting problem for elliptic curves…

We find all the possible torsion groups of $\Q$-curves over quadratic fields and determine which groups appear finitely and which appear infinitely often.

In this survey, we discuss the problem of the maximum number of points of curves of genus 1,2 and 3 over finite fields

In this note some properties of the sum of element orders of a finite abelian group are studied.

We determine in this paper the distribution of the number of points on the cyclic covers of $\mathbb{P}^1(\mathbb{F}_q)$ with affine models $C: Y^r = F(X)$, where $F(X) \in \mathbb{F}_q[X]$ and $r^{th}$-power free when $q$ is fixed and the…

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers, the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over…

We discuss, in a non-Archimedean setting, the distribution of the coefficients of $L$-polynomials of curves of genus $g$ over $\mathbb{F}_q$. Among other results, this allows us to prove that the $\mathbb{Q}$-vector space spanned by such…

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…

We give a presentation of abelian class field theory.

We prove that the torsion points of an abelian variety are equidistributed over the corresponding berkovich space with respect to the canonical measure.

We survey new results on finite groups of birational transformations of algebraic varieties.

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the…

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields.

A new family of maximal curves over a finite field is presented and some of their properties are investigated.

Given a finite field $\mathbb{F}_{q}$, we study the distribution of the number of $\mathbb{F}_{q}$-points on (possibly singular) affine curves given by the polynomial equations of the form $C_{f} : y^{m} = f(x)$, where $f$ is randomly…