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

Rubén A. Hidalgo
arXiv: Algebraic Geometry
May 2021
Let $S$ be a closed Riemann surface of genus $g\geq 2$ and let $p$ be a prime integer dividing the order of ${\rm Aut}(S)$. The Riemann-Hurwitz formula asserts that either $p \leq g+1$ or $p=2g+1$. Let $G \cong {\mathbb Z}_{p}$ be a group of conformal automorphisms of $S$. If $p=2g+1$, then $S/G$ is the sphere with exactly three cone points and, if $p \geq 7$, then it happens that $G$ is the unique $p$-Sylow subgroup. If $p=g+1$, then $S/G$ is the sphere with exactly four cone points and, if $p \geq 13$, then it happens that $G$ is 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 $G \cong {\mathbb Z}_{p}^{l}$, where $l \geq 2$. If $p \geq 5$, then either (i) $p^{l} \leq g-1$ or (ii) $S/G$ has genus zero and $p^{l-1}(p-3) \leq 2(g-1)$. Moreover, in case (ii), $2 \leq l \leq r-1$, where $r \geq 3$ is the number of cone points. 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 (i) an algebraic curve representation for $S$, (ii) a description of its group of conformal automorphisms, (iii) a discussion of it field of moduli and (iv) an isogenous decomposition of its jacobian variety.