Jacobians with with automorphisms of prime order

TL;DR

This paper investigates the structure of principally polarized abelian varieties with prime order automorphisms, identifying conditions under which they are not Jacobians of curves, thus advancing understanding of their automorphism properties.

Contribution

It introduces specific conditions on automorphism actions that determine when such abelian varieties are not Jacobians, providing new insights into their classification.

Findings

Certain automorphism multiplicity conditions exclude abelian varieties from being Jacobians.

Automorphisms of prime order influence the geometric structure of abelian varieties.

The study clarifies the relationship between automorphisms and the Jacobian property.

Abstract

In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves.