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

Let C be a curve of genus 2 and ψ1 : C −→ E1 a map of degree n, from C to an elliptic curve E1, both curves defined over C. This map induces a degree n map φ1 : P 1 −→ P 1 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 curve E2 such that there is an isogeny of degree n 2 from the Jacobian JC to E1 × E2. We say that JC 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.