On certain permutation groups and sums of two squares

TL;DR

This paper explores the existence of ramified covers over the projective line with specific ramification conditions, connecting algebraic geometry, permutation groups, and classical number theory related to sums of two squares.

Contribution

It introduces alternative approaches using elliptic curves and universal covers to establish existence results, linking geometric problems with permutation groups and number theory.

Findings

Established new methods for proving existence of ramified covers

Connected geometric ramification problems with permutation groups

Related the problem to classical results on sums of two squares

Abstract

We consider the question of existence of ramified covers over P_1 matching certain prescribed ramification conditions. This problem has already been faced in a number of papers, but we discuss alternative approaches for an existence proof, involving elliptic curves and universal ramified covers with signature. We also relate the geometric problem with finite permutation groups and with the Fermat-Euler Theorem on the representation of a prime as a sum of two squares.