A two dimensional arithmetic Andr\'e-Oort problem

TL;DR

This paper explores an integral version of the André-Oort conjecture within Shimura varieties, proving a significant case for modular curves over Z using advanced equidistribution techniques.

Contribution

It introduces an integral analogue of the André-Oort conjecture and proves it for modular curves over Z, employing novel equidistribution methods.

Findings

Proved the integral André-Oort conjecture for modular curves over Z.

Established unconditional results using equidistribution and subconvexity estimates.

Demonstrated the conjecture's non-triviality even in low-dimensional cases.

Abstract

We state and investigate an integral analogue of the Andr\'e-Oort conjecture (in integral models of Shimura varieties). We establish an instance of this conjecture: the case of a modular curve, as a scheme over Z. It is a scheme of dimension two and, already in this case, our conjecture is highly non-trivial. Our approach relies on equidistribution estimates related to subconvexity in analytic number theory and our result is unconditional.