Curves of genus 2 with (n, n)-decomposable jacobians

الأصل · EN

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. This map induces a degree n map ϕ₁:¹ 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 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.