# Periods of generalized Fermat curves

## TL;DR

This paper studies generalized Fermat curves, providing algebraic models, computing their first homology groups, and determining generators for their Jacobian period lattices.

## Contribution

It introduces algebraic models for generalized Fermat curves and explicitly computes generators for their homology and Jacobian period lattices.

## Key findings

- Explicit generators for the first homology group of generalized Fermat curves.
- Determination of generators for the period lattice of the Jacobian variety.
- Enhanced understanding of the algebraic and topological structure of these curves.

## Abstract

Let $k,n \geq 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface $S$ that admits a subgroup of conformal automorphisms $H \leq \mbox{Aut}(S)$ isomorphic to $\mathbb{Z}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\hat{\mathbb{C}}$ and has $n+1$ branch points, each one of order $k$. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.