On the one-dimensional family of Riemann surfaces of genus $q$ with $4q$ automorphisms

TL;DR

This paper investigates the properties of a specific one-dimensional family of Riemann surfaces of prime genus with a fixed automorphism count, focusing on their geometric and Jacobian structures.

Contribution

It extends previous work by analyzing the detailed properties of Riemann surfaces in the family _q for prime q , including their Jacobian varieties.

Findings

Characterization of automorphism groups for surfaces in _q

Analysis of the Jacobian varieties associated with these surfaces

Identification of special geometric features of the family _q

Abstract

Bujalance, Costa and Izquierdo have recently proved that all those Riemann surfaces of genus $g \ge 2$ different from $3, 6, 12, 15$ and 30, with exactly $4g$ automorphisms form an equisymmetric one-dimensional family, denoted by $\mathcal{F}_g.$ In this paper, for every prime number $q \ge 5,$ we explore further properties of each Riemann surface $S$ in $\mathcal{F}_q$ as well as of its Jacobian variety $JS.$