How to split two-dimensional Jacobians: a geometric construction

TL;DR

This paper provides a geometric method to decompose the Jacobian of a genus 2 curve into a product of elliptic curves and introduces a criterion for algebraic correspondence push-outs.

Contribution

It offers an explicit geometric construction of the complementary curve in Jacobian splitting and a criterion for algebraic correspondence push-outs.

Findings

Explicit construction of the complementary curve W.

Criterion for algebraic correspondence push-outs.

Decomposition of Jacobians for specific branched covers.

Abstract

Let $\pi\colon Y \to X$ be a branched cover of complex algebraic curves of respective genera $g(Y)=2$ and $g(X)=1$. The Jacobian of $Y$ is isogenous to the product of two elliptic curves: $\operatorname{Jac} Y \sim \operatorname{Jac} X \times \operatorname{Jac} W$. We present an explicit geometric construction of the complementary curve $W$. Furthermore, we establish a criterion to decide whether an algebraic correspondence of curves admits a push-out.