# Involutions on hyperelliptic curves and Prym maps

## TL;DR

This paper studies the geometry of hyperelliptic curves with extra involutions, focusing on Prym theory, and proves the injectivity of the Prym map for certain hyperelliptic coverings.

## Contribution

It establishes the injectivity of the Prym map for hyperelliptic ^2-coverings over hyperelliptic curves of positive genus, advancing understanding of Prym varieties.

## Key findings

- Proves injectivity of the Prym map in the specified setting
- Analyzes involutions on hyperelliptic curves from a Prym perspective
- Enhances understanding of Prym varieties associated with hyperelliptic curves

## Abstract

We investigate the geometry of smooth hyperelliptic curves that possess additional involutions, especially from the point of view of the Prym theory. Our main result is the injectivity of the Prym map for hyperelliptic $\mathbb{Z}_2^2$-coverings over hyperelliptic curves of positive genus.