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 $GL_2(F_p)$ on certain multisets consisting of $R \geq 3$ 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 $[\begin{smallmatrix} 0 \ 0 \end{smallmatrix}]$. 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 $1+ Rp(p-1)/2-p^2.$