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

TL;DR

This paper counts zero-sum multisets of vectors over finite fields and applies the results to classify topological types of certain group actions on Riemann surfaces, linking combinatorics with algebraic topology.

Contribution

It provides a novel enumeration of zero-sum multisets under group actions and connects this to the classification of surface automorphisms.

Findings

Count of zero-sum multisets under GL_2(F_p) action

Enumeration of topological types of group actions on Riemann surfaces

Explicit formula relating multisets 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 $1+ Rp(p-1)/2-p^2.$