${\mathbb Z}_{k}^{m}$-actions of signature $(0;k,\stackrel{n+1}{\ldots},k)$
Rubén A. Hidalgo, Sebastián Reyes-Carocca
TL;DR
The paper develops a complete framework for classifying ${\mathbb Z}_k^m$-actions of signature $(0;k^{n+1})$ on compact Riemann surfaces by reducing to generalized Fermat curves. Actions are encoded via kernels $K$ in a generalized Fermat group $H$, with topological equivalence governed by orbits of $K$ under the geometric automorphism group $Aut_g(H)$, and extended to include extra automorphisms through liftings to a finite group $\mathcal Q_{S,G}$. In the prime-k, $m=2$ case, explicit descriptions are given for ${\mathcal F}(p,n,2)$, including fiber-product algebraic models for $S_K$ and isogeny decompositions of Jacobians; the results are illustrated by detailed analyses of the cases $n=3$ and $n=5$, unveiling rich families of curves and their automorphism structures. The methods connect Fuchsian uniformization, generalized Fermat curves, and invariant subgroup data to yield concrete classifications and algebraic descriptions, with implications for moduli stratification and Jacobian decompositions across families of curves.
Abstract
In this article we consider group actions on compact Riemann surfaces and their topological classification. We address this problem for pairs $(S, N)$ where $S$ is a compact Riemann surface endowed with a group of automorphisms $N \cong \mathbb{Z}_k^m$ such $S/N$ has signature $(0;k,\stackrel{n+1}{\ldots},k)$, where $n, k \geqslant 2$ and $1 \leqslant m \leqslant n$ are integers. We further assume the existence of extra automorphisms, namely, a group $G$ with $N \lhd G \leqslant \mathrm{Aut}(S)$ and analyze the induced permutational action of $G/N$ on the cone points of $S/N$. To describe such actions up to topological equivalence, we employ the generalized Fermat curves $(X,H)$ and their automorphism groups, showing that every triple $(S,N, G)$ as before is determined by a class of subgroups of $H$ that satisfy certain invariance property. This approach establishes a correspondence between topological equivalence classes and an appropriate quotient set. As an application, we specialize our results to the case $k$ prime and $m=2$, including algebraic models and isogeny decompositions of their Jacobian varieties. We then discuss some examples for the cases $n=3$ and $n=5$, which are interesting in their own right.