Effective Bounds on the Dimensions of Jacobians Covering Abelian Varieties

TL;DR

This paper establishes explicit bounds on the dimensions of Jacobians covering polarized abelian varieties over finite fields, using an effective Bertini theorem and applications to elliptic curves.

Contribution

It provides the first effective bounds on Jacobian dimensions covering abelian varieties over finite fields, including improved bounds for simple varieties.

Findings

Bounded dimension Jacobians cover all polarized abelian varieties over finite fields.

Existence of smooth curves with Jacobians having factors isogenous to powers of elliptic curves.

Effective version of Poonen's Bertini theorem over finite fields.

Abstract

We show that any polarized abelian variety over a finite field is covered by a Jacobian whose dimension is bounded by an explicit constant. We do this by first proving an effective version of Poonen's Bertini theorem over finite fields, which allows us to show the existence of smooth curves arising as hypersurface sections of bounded degree and genus. Additionally, we show that for simple abelian varieties a better bound is possible. As an application of these results we show that if $E$ is an elliptic curve over a finite field then for any $n\in \mathbb{N}$ there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $E^n$.