Isogenous decomposition of the Jacobian of generalized Fermat curves

Original · EN

A closed Riemann surface S is called a generalized Fermat curve of type (p,n), where p,n ≥ 2 are integers, if it admits a group H Zₚⁿ 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.