Quadrangular ${\mathbb Z}_{p}^{l}$-actions on Riemann surfaces

TL;DR

This paper investigates the automorphism groups of certain Riemann surfaces with abelian group actions, focusing on the case of four cone points, providing algebraic models, automorphism descriptions, and Jacobian decompositions.

Contribution

It offers a detailed analysis of Riemann surfaces with ${f Z}_p^l$ automorphism groups, especially for the case of four cone points, including algebraic, group, and Jacobian structures.

Findings

Algebraic curve representations for the surfaces studied.

Descriptions of the automorphism groups of these surfaces.

Analysis of the Jacobian variety decompositions.

Abstract

Let $p \geq 3$ be a prime integer and, for $l \geq 1$, let $G \cong {\mathbb Z}_{p}^{l}$ be a group of conformal automorphisms of some closed Riemann surface $S$ of genus $g \geq 2$. By the Riemann-Hurwitz formula, either $p \leq g+1$ or $p=2g+1$. If $l=1$ and $p=2g+1$, then $S/G$ is the sphere with exactly three cone points and, if moreover $p \geq 7$, then $G$ is the unique $p$-Sylow subgroup of ${\rm Aut}(S)$. If $l=1$ and $p=g+1$, then $S/G$ is the sphere with exactly four cone points and, if moreover $p \geq 13$, then $G$ is again the unique $p$-Sylow subgroup. The above unique facts permited many authors to obtain algebraic models and the corresponding groups ${\rm Aut}(S)$ in these situations. Now, let us assume $l \geq 2$. If $p \geq 5$, then either (i) $p^{l} \leq g-1$ or (ii) $S/G$ has genus zero, $p^{l-1}(p-3) \leq 2(g-1)$ and $2 \leq l \leq r-1$, where $r \geq 3$ is the number of cone points of $S/G$. Let us assume we are in case (ii). If $r=3$, then $l=2$ and $S$ happens to be the classical Fermat curve of degree $p$, whose group of automorphisms is well known. The next case, $r=4$, is studied in this paper. We provide an algebraic curve representation for $S$, a description of its group of conformal automorphisms, a discussion of its field of moduli and an isogenous decomposition of its jacobian variety.