Equations of Riemann surfaces with automorphisms

TL;DR

This paper introduces an algorithm to compute equations of Riemann surfaces with automorphisms, enabling explicit descriptions of surfaces with large automorphism groups for genera 4 to 7.

Contribution

It develops a new algorithm combining trace formulas, group representation, and Gr"obner basis techniques to explicitly compute equations of Riemann surfaces with automorphisms.

Findings

Successfully computed equations for surfaces with large automorphism groups

Extended the range of genus for which explicit equations are available

Demonstrated the effectiveness of the algorithm with heuristic improvements

Abstract

We present an algorithm for computing equations of canonically embedded Riemann surfaces with automorphisms. A variant of this algorithm with many heuristic improvements is used to produce equations of Riemann surfaces $X$ with large automorphism groups (that is, $|\mathrm{Aut}(X)| > 4(g_X-1)$) for genus $4 \leq g_X \leq 7$. The main tools are the Eichler trace formula for the character of the action of $\mathrm{Aut}(X)$ on holomorphic differentials, algorithms for producing matrix generators of a representation of a finite group with a specified irreducible character, and Gr\"obner basis techniques for computing flattening stratifications.