Simplicity of some Jacobians with many automorphisms

TL;DR

This paper investigates a family of Jacobian varieties derived from cyclic coverings of hyperelliptic curves, proving generic simplicity, describing their endomorphism algebra, and analyzing their automorphisms, with applications to the Prym map.

Contribution

It introduces an explicit family of Jacobians, proves their generic simplicity, characterizes their endomorphism algebra, and establishes the injectivity of the Prym map for certain coverings.

Findings

Generic Jacobians in the family are simple.

Endomorphism algebra of these Jacobians is fully described.

The Prym map is generically injective under specified conditions.

Abstract

We study an explicit $(2g-1)$-dimensional family of Jacobian varieties of dimension $\frac{d-1}2(g-1)$, arising from quotient curves of unramified cyclic coverings of prime degree $d$ of hyperelliptic curves of genus $g\ge 2$. By using a deformation argument, we prove that the generic element of the family is simple. Furthermore, we completely describe their endomorphism algebra, and we show that they admit a rank $\frac{d-1}2-1$ group of non-polarized automorphisms. As an application of these results, we prove the generic injectivity of the Prym map for \'etale cyclic coverings of hyperelliptic curves of odd prime degree under some slight numerical restrictions. This result generalizes in several directions previous results on genus 2.