Zero-sum multisets mod p with an application to surface automorphisms

Abstract

We solve a problem in enumerative combinatorics which is equivalent to counting topological types of certain group actions on compact Riemann surfaces. Let $V_2(F_p)$ be the two-dimensional vector space over $F_p$, the field with $p$ elements, $p$ an odd prime. We count orbits of the general linear group $G{L}_2(F_p)$ on certain multisets consisting of $R\ge 3$ non-zero columns from $V_2(F_p)$. The $R$-multisets are `zero-sum,' that is, the sum (mod $p$) over the columns in the multiset is $[\begin{array}{c}0\\ 0\end{array}]$. The orbit count yields the number of topological types of fully ramified actions of the elementary abelian $p$-group of rank $2$ on compact Riemann surfaces of genus $1+Rp(p-1)/2-p^2.$

This version clarifies the proof of Theorem 1. It supersedes and replaces all earlier versions, including the version titled "Counting topological types of elementary abelian p group actions on surfaces."