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

Abstract

Let $p\ge 3$ be a prime integer and, for $l\ge 1$, let $G\cong {\mathbb{Z}}_p^l$ be a group of conformal automorphisms of some closed Riemann surface $S$ of genus $g\ge 2$. By the Riemann-Hurwitz formula, either $p\le g+1$ 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 $p\ge 7$, then $G$ is the unique $p$-Sylow subgroup of $\mathrm{Aut}(S)$. If $l=1$ and $p=g+1$, then $S/G$ is the sphere with exactly four cone points and, if moreover $p\ge 13$, then $G$ is again the unique $p$-Sylow subgroup. The above unique facts permited many authors to obtain algebraic models and the corresponding groups $\mathrm{Aut}(S)$ in these situations. Now, let us assume $l\ge 2$. If $p\ge 5$, then either (i) $p^l\le g-1$ or (ii) $S/G$ has genus zero, $p^{l-1}(p-3)\le 2(g-1)$ and $2\le l\le r-1$, where $r\ge 3$ 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.