How to split two-dimensional Jacobians: a geometric construction

Abstract

Let $\ddot{I}€:Y\to X$ be a branched cover of algebraic curves. Assume that there exists a curve $W$ such that $\mathrm{Jac}Y\sim \mathrm{Jac}X\times \mathrm{Jac}W$. We conjecture that every such isogeny decomposition is induced by an algebraic correspondence of curves that fits in a Galois diagram, and we prove this conjecture when $g(Y)=2$ and $g(X)=1$. Our proof yields a geometric construction of the complementary curve $W$, an explicit correspondence inducing the isogeny, and a general criterion for deciding when an algebraic correspondence of curves fits in a Galois diagram (admits a push-out).

22 pages, comments are welcome! v3: updated construction