Jacobian variety of generalized Fermat curves

Abstract

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The isogenous decomposition of the Jacobian variety of classical Fermat curve of prime degree p >= 5 has been obtained by Aoki using techniques of number theory, by Barraza and Rojas in terms of decompositions of the algebra of groups, and by Hidalgo and Rodriguez using Kani-Rosen results. In the last, it was seen that all factors in the isogenous decomposition are Jacobian varieties of certain cyclic p-gonal curves. The highest Abelian branched covers of an orbifold of genus 0 with exactly n + 1 branch points, each one of order p, are provided by the so-called generalized Fermat curves of type (p, n)

these being a suitable fiber product of n-1 Fermat curves of degree p. In this paper, we provide a decomposition, up to isogeny, of the Jacobian variety of a generalized Fermat curve S of type (p, n) as a product of Jacobian varieties of certain cyclic p-gonal curves, whose explicit equations 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. KeyWords Plus:GENUS 2