Linear Equations in Singular Moduli

TL;DR

This paper proves an effective version of the Andre9-Oort conjecture for linear subspaces in a complex Shimura variety, enabling the theoretical determination of CM-points in higher-dimensional algebraic subvarieties.

Contribution

It provides the first effective method to identify CM-points in algebraic subvarieties of dimension greater than one in a Shimura variety.

Findings

Established an effective Andre9-Oort conjecture for linear subspaces.

Enabled theoretical determination of CM-points in higher-dimensional subvarieties.

Extended understanding of special points in complex algebraic varieties.

Abstract

We establish an effective version of the Andr\'e-Oort conjecture for linear subspaces of $Y(1)^n_{\mathbb{C}} \approx \mathbb{A}_{\mathbb{C}}^n$. Apart from the trivial examples provided by weakly special subvarieties, this yields the first algebraic subvarieties in a Shimura variety of dimension $> 1$ whose CM-points can be (theoretically) determined.