# Covering of elliptic curves and the kernel of the Prym map

## TL;DR

This paper investigates the properties of coverings of elliptic curves and their Prym maps, providing criteria for minimal kernel dimension of the period map, which helps in constructing counterexamples to Xiao's conjecture.

## Contribution

It offers a new characterization of coverings with minimal kernel dimension of the period map, aiding in the exclusion of certain fibrations and analyzing Pirola's counterexample.

## Key findings

- Characterization of coverings with minimal kernel of the period map
- Counterexamples to Xiao's conjecture constructed using this framework
- Analysis of Pirola's counterexample within the new framework

## Abstract

Motivated by a conjecture of Xiao, we study families of coverings of elliptic curves and their corresponding Prym map $\Phi$. More precisely, we describe the codifferential of the period map $P$ associated to $\Phi$ in terms of the residue of meromorphic $1$-forms and then we use it to give a characterization for the coverings for which the dimension of $\ker(dP)$ is the least possibile. This is useful in order to exclude the existence of non isotrivial fibrations with maximal relative irregularity and thus also in order to give counterexamples to the Xiao's conjecture mentioned above. The first counterexample to the original conjecture, due to Pirola, is then analysed in our framework.