Effective bounds on the dimensions of Jacobians covering abelian varieties

Abstract:

We show that any 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 and explicit 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, for simple abelian varieties we prove a better bound. As an application, we show that for any elliptic curve $E$ over a finite field and any $n\in \mathbb {N}$, there exist smooth curves of bounded genus whose Jacobians have a factor isogenous to $E^n$.