On orientably-regular maps of Euler characteristic $-2p^2$

TL;DR

This paper classifies certain orientably-regular maps and Riemann surfaces with specific automorphism groups related to prime number p, focusing on Euler characteristic -2p^2 and automorphism groups of order 10p^2.

Contribution

It provides a classification of orientably-regular maps and Riemann surfaces with automorphism groups of order 10p^2 and genus 1+p^2, respectively, for prime p.

Findings

Classified orientably-regular maps with Euler characteristic -2p^2.

Identified Riemann surfaces of genus 1+p^2 with automorphism group of order 5p^2.

Abstract

In this article, we study orientably-regular maps of Euler characteristic $-2p^2$ and classify those that admit a group of orientation-preserving automorphisms of order $10p^2$, where $p$ is a prime number. Along the way, we classify all compact Riemann surfaces (or complex algebraic curves) of genus $1+p^2$ endowed with a group of conformal automorphisms of order $5p^2$.