On large prime actions on Riemann surfaces

TL;DR

This paper classifies compact Riemann surfaces of genus g with prime order automorphisms, describing their algebraic structure, automorphism groups, moduli fields, and Jacobian properties, including explicit period matrices.

Contribution

It provides a comprehensive classification and detailed analysis of Riemann surfaces with prime order automorphisms, including algebraic, automorphic, and Jacobian aspects.

Findings

Classification of surfaces with automorphism of order g+1

Explicit description of automorphism groups and moduli fields

Computed period matrix of the genus four Accola-Maclachlan curve

Abstract

In this article we study compact Riemann surfaces of genus $g$ with an automorphism of prime order $g+1.$ The main result provides a classification of such surfaces. In addition, we give a description of them as algebraic curves, determine and realise their full automorphism groups and compute their fields of moduli. We also study some aspects of their Jacobian varieties such as isogeny decompositions and complex multiplication. Finally, we determine the period matrix of the Accola-Maclachlan curve of genus four.