Eigenforms of hyperelliptic curves with many automorphisms

TL;DR

This paper constructs infinitely many examples of hyperelliptic curves with many automorphisms supporting collections of translation surfaces, linking algebraic curves and translation surface eigenforms.

Contribution

It introduces a method to generate infinite collections of translation surfaces supported on the same hyperelliptic curve with rich automorphism groups.

Findings

Infinite examples of translation surfaces on hyperelliptic curves

Automorphism of maximal order acts on holomorphic 1-forms

Correspondence between eigenforms and translation surfaces

Abstract

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.