On direct summands of products of Jacobians over arbitrary fields

TL;DR

This paper characterizes when a principally polarized abelian variety over any field is a direct summand of Jacobians of curves with a rational point, extending complex case results to arbitrary fields and applying to the Tate conjecture.

Contribution

It generalizes previous complex field results to arbitrary fields, providing new criteria for abelian varieties to be direct summands of Jacobians and addressing the Tate conjecture in new contexts.

Findings

Characterization of abelian varieties as summands of Jacobians over arbitrary fields

Extension of complex results to fields of positive characteristic

Applications to the integral Tate conjecture for divisors and 1-cycles

Abstract

We show that a principally polarized abelian variety over a field $k$ is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a $k$-point if and only if the polarization and the minimal class are both algebraic over $k$. This extends results of Beckmann--de Gaay Fortman and Voisin over the complex numbers to arbitrary fields, and refines an obstruction to the direct summand property over $\mathbb{Q}$ due to Petrov--Skorobogatov. We also give applications to the integral Tate conjecture for divisors and for $1$-cycles on abelian varieties over finitely generated fields; our results also address a $p$-adic version of the integral Tate conjecture over finite fields of characteristic $p$, for the first time beyond the case of divisors.