Periods of generalized Fermat curves (Q2196326)

A closed Riemann surface \(S\) is called a generalized Fermat curve of type \((k,n)\), where \(k,n \geq 2\) are integers, if it admits a group \(H \cong {\mathbb Z}_{k}^{n}\) of conformal automorphisms such that the quotient orbifold \(S/H\) has genus zero and exactly \((n+1)\) conical points, each one of order \(k\). An explicit equation for \(S\) is known as a fiber product of \(n-1\) Fermat curves of degree \(k\) [\textit{G. González-Diez} et al., J. Algebra 321, No. 6, 1643--1660 (2009; Zbl 1175.14023)] and it is also known that the group \(H\) is unique in \(\mathrm{Aut}(S)\) [\textit{R. A. Hidalgo} et al., J. Pure Appl. Algebra 221, No. 9, 2312--2337 (2017; Zbl 1400.14080)]. In the paper under review, the author constructs a set of generators for the first homology group of \(S\) and finds a set of generators for the period lattice of the associated Jacobian variety.