Isogenous decomposition of the Jacobian of generalized Fermat curves

Abstract

A closed Riemann surface $S$ is called a generalized Fermat curve of type $(p,n)$, where $p,n \geq 2$ are integers, if it admits a group $H \cong {\mathbb Z}_{p}^{n}$ of conformal automorphisms so that $S/H$ is an orbifold of genus zero with exactly $n+1$ cone points, each one of order $p$. It is known that $S$ is a fiber product of $(n-1)$ classical Fermat curves of degree $p$ and, for $(p-1)(n-1)>2$, that it is a non-hyperelliptic Riemann surface. In this paper, assuming $p$ to be a prime integer, we provide a decomposition, up to isogeny, of the Jacobian variety $JS$ as a product of Jacobian varieties of certain cyclic $p$-gonal curves. Explicit equations for these $p$-gonal curves are provided in terms of the equations for $S$. As a consequence of this decomposition, we are able to provide explicit positive-dimensional families of closed Riemann surfaces whose Jacobian variety is isogenous to the product of elliptic curves.