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

Let C be a curve of genus 2 and ψ₁: C  −  → E ₁  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  −  → P ¹  which we call a Frey–Kani covering. We determine all possible ramifications for φ₁. If ψ₁:C  −  → E ₁  is maximal then there exists a maximal map ψ₂: C  −  → E ₂  , of degree n, to some elliptic curveE₂ such that there is an isogeny of degree n²from the JacobianJ_C to E₁ × E₂. We say thatJ_C 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.