Isogenies of Jacobians

TL;DR

This paper investigates conditions under which Jacobian varieties are not isogenous to each other, using Hodge theory and degeneration techniques, with implications for the structure of moduli spaces of curves.

Contribution

The authors establish new bounds on when Jacobians of generic curves are not isogenous, extending classical results with novel degeneration methods.

Findings

Jacobian of a generic element in a codimension k subvariety is not isogenous to a different Jacobian for g > 3k+4.

Extended the non-isogeny result to the case k=1, g≥5 using degeneration techniques.

Provided a generalized understanding of the isogeny classes of Jacobians in moduli space.

Abstract

We prove by means of the study of the infinitesimal variation of Hodge structure and a generalization of the classical Babbage-Enriques-Petri theorem that the Jacobian variety of a generic element of a $k$ codimensional subvariety of $\mathcal M_g$ is not isogenous to a distinct Jacobian if $g>3k+4$. We extend this result to $k=1, g\ge 5$ by using degeneration methods.