How to split two-dimensional Jacobians: a geometric construction

Abstract

Let $\pi\colon Y \to X$ be a branched cover of algebraic curves. Assume that there exists a curve $W$ such that $\operatorname{Jac} Y \sim \operatorname{Jac} X \times \operatorname{Jac} W$. 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).