The Zilber--Pink conjecture for products of curves with highly degenerate reduction

Abstract

We give a proof of the Zilber--Pink conjecture for $n$-fold self-products of a curve $X$ inside the self-product of its Jacobian, when $X$ has appropriate bad reduction, its Jacobian has no extra endomorphisms, and $n$ is sufficiently small. The strategy of proof follows the work of Katz, Rabinoff and Zureick-Brown on explicit Manin--Mumford bounds.