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

摘要

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: 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 curve E2 such that there is an isogeny of degree n2 from the Jacobian JC to E1 x 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.