Effective Andr\'e-Oort for non-compact curves in Hilbert modular varieties

TL;DR

This paper provides an effective proof of the Andre9-Oort conjecture for non-compact curves in Hilbert modular varieties, replacing non-effective steps with explicit endomorphism estimates and G-function techniques.

Contribution

It introduces an effective approach to the Andre9-Oort conjecture for certain Hilbert modular varieties, improving upon previous non-effective methods.

Findings

Effective proof for non-compact curves in Hilbert modular surfaces

Replacement of non-effective steps with explicit endomorphism estimates

Use of G-functions and Q-functions for effectivization

Abstract

In the proofs of most cases of the Andr\'e-Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer-Siegel and the use of Pila-Wilkie. Only the case of curves in ${\bf C}^2$ is currently known effectively (by other methods). We give an effective proof of Andr\'e-Oort for non-compact curves in every Hilbert modular surface and every Hilbert modular variety of odd genus (under a minor generic simplicity condition). In particular we show that in these cases the first step may be replaced by the endomorphism estimates of W\"ustholz and the second author together with the specialization method of Andr\'e via G-functions, and the second step may be effectivized using the Q-functions of Novikov, Yakovenko and the first author.