A SAGE Package for n-Gonal Equisymmetric Stratification of M-g

Abstract

In this work, we present an algorithm running over SAGE, which allows users to deal with group actions on Riemann surfaces up to topological equivalence. Our algorithm allows us to study the equisymmetric stratification of the branch locus B-g of the moduli space M-g of compact Riemann surfaces of genus g >= 2, corresponding to group actions with orbit genus 0. That is, it works for actions on surfaces of any genus in the case the genus of the quotient surface is zero, except for obvious hardware constraints. Our approach is toward studying inclusions and intersections of (closed) strata of B-g. We apply our algorithm to describe part of the geometry of the branch locus B-9, in terms of equisymmetric stratification. We also use it to compute all group actions up to topological equivalence for genus 5-10, this completes the lists. Finally, we add an optimized version of an algorithm, which allows us to identify Jacobian varieties of CM-type. As a byproduct, we obtain a Jacobian variety of dimension 11 which is isogenous to E-i(9) x E-i root 3(2), where E-i and E-i root 3 are elliptic curves with complex multiplication.