# The classification of surfaces with p_g=q=1 isogenous to a product of curves

## TL;DR

This paper classifies all algebraic surfaces with geometric genus and irregularity equal to one that are constructed as quotients of products of two curves by a finite group, enriching the understanding of their structure.

## Contribution

It provides a complete classification of surfaces with p_g=q=1 that are isogenous to a product, a previously unresolved problem in algebraic geometry.

## Key findings

- Complete classification of surfaces with p_g=q=1 that are isogenous to a product.
- Identification of all possible group actions on product of curves.
- Structural insights into the geometry of these surfaces.

## Abstract

A projective surface S is said to be isogenous to a product if there exist two smooth curves C, F and a finite group G acting freely on C \times F so that S=(C \times F)/G. In this paper we classify all surfaces with p_g=q=1 which are isogenous to a product.