How to split two-dimensional Jacobians: a geometric construction
Andrea Gallese
TL;DR
The paper addresses extracting the elliptic factor in a two-dimensional split Jacobian $\operatorname{Jac} Y \simeq \operatorname{Jac} X \times \operatorname{Jac} W$ for a genus-2 curve $Y$ mapped to a genus-1 curve $X$ via a branched cover $\pi\colon Y\to X$. It introduces a geometric construction using the self-fiber product $Z=Y\times_X Y$, an involution $\eta$, and quotients to produce a complementary genus-1 curve $W$ whose Jacobian completes the isogeny; it also provides a criterion for when a given correspondence has a core and thus fits in a Galois diagram. The approach connects monodromy methods, Hurwitz theory, and pseudo Brauer relations, and it yields an explicit algebraic model for $W$ in at least the generic case, with a framework applicable to non-generic ramification as well. The results advance practical computations of split Jacobians, furnish a constructive path to the Prym-analytic factor, and illuminate when push-outs of correspondences arise from Galois-diagram structures, impacting number theory and algebraic geometry of maps between curves. The methods have potential implications for arithmetic applications involving heights and parity in elliptic settings, and they provide a bridge between topological monodromy data and explicit algebraic realizations of Jacobian decompositions.
Abstract
Let $π\colon Y \to X$ 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.