Curves of Genus 2 with (N, N) Decomposable Jacobians

Abstract

Let C be a curve of genus 2 and ψ1: C −→E1 a map of degree n, from C to an elliptic curveE1 , both curves defined over C. This map induces a degree n map φ1:P1 −→P1 which we call a Frey–Kani covering. We determine all possible ramifications for φ1. If ψ1:C −→E1 is maximal then there exists a maximal map ψ2: C −→E2 , of degree n, to some elliptic curveE2 such that there is an isogeny of degree n2from the JacobianJC to E1×E2. We say thatJC is (n, n)-decomposable. If the degree n is odd the pair (ψ2, E2) is canonically determined. For n= 3, 5, and 7, we give arithmetic examples of curves whose Jacobians are (n, n)-decomposable.