Equations of Riemann surfaces with automorphisms

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 $4\le g_X\le 7$. 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 Gröbner basis techniques for computing flattening stratifications.

16 pages