# On Riemann surfaces of genus $g$ with $4g-4$ automorphisms

## TL;DR

This paper classifies compact Riemann surfaces of genus g with automorphism groups of order 4g-4, especially when g-1 is prime, and analyzes their Jacobian varieties.

## Contribution

It provides a complete classification of such surfaces and determines the isogeny decompositions of their Jacobians under specific conditions.

## Key findings

- Classified Riemann surfaces with automorphism group order 4g-4 when g-1 is prime.
- Determined isogeny decompositions of Jacobian varieties for these surfaces.
- Enhanced understanding of automorphism groups in relation to surface genus.

## Abstract

In this article we study compact Riemann surfaces with a non-large group of automorphisms of maximal order; namely, compact Riemann surfaces of genus $g$ with a group of automorphisms of order $4g-4.$ Under the assumption that $g-1$ is prime, we provide a complete classification of them and determine isogeny decompositions of the corresponding Jacobian varieties.