2003/12/15 by T. Shaska · 1 citation
Mathematics · #math.AG #msc:14xx
published as J. Symbolic Comp. 31 (2001), no. 5, 603-617
arxiv created 2003/12/15 · arxiv updated 2009/12/01
Let C be a curve of genus 2 and ψ1:C \lar E1 a map of degree n, from C to an elliptic curve E1, both curves defined over \bC. This map induces a degree n map ϕ1:\bP1 \lar \bP1 which we call a Frey-Kani covering. We determine all possible ramifications for ϕ1. If ψ1:C \lar E1 is maximal then there exists a maximal map ψ2:C\lar E2, of degree n, to some elliptic curve E2 such that there is an isogeny of degree n2 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.