A Diophantine Frobenius problem related to Riemann surfaces

TL;DR

This paper derives bounds and exact formulas for a specific four-dimensional Frobenius number linked to prime pairs, with applications to the genus of Riemann surfaces and their automorphisms.

Contribution

It provides sharp bounds and exact formulas for a special Frobenius number related to prime pairs, connecting number theory with the topology of Riemann surfaces.

Findings

Sharp upper and lower bounds on the Frobenius number.

Exact formulas for two subclasses of prime pairs.

Applications to the genus and automorphisms of Riemann surfaces.

Abstract

We obtain sharp upper and lower bounds on a certain four-dimensional Frobenius number determined by a prime pair $(p,q)$, $2<p<q$, including exact formulae for two infinite subclasses of such pairs. Our work is motivated by the study of compact Riemann surfaces which can be realized as a semi-regular $pq$-fold coverings of surfaces of lower genus. In this context, the Frobenius number is (up to an additive translation) the largest genus in which no surface is such a covering. In many cases it is also the largest genus in which no surface admits an automorphism of order $pq$. The general $t$-dimensional Frobenius problem ($t \geq 3$) is $NP$-hard, and it may be that our restricted problem retains this property.