An explicit Andr\'e-Oort type result for P^1(C) x G_m(C)

TL;DR

This paper proves an explicit Andre9-Oort type result for the product of the complex projective line and the multiplicative group, characterizing special points as singular moduli and roots of unity using class field theory.

Contribution

It provides a new explicit proof of an Andre9-Oort type statement for ^1(a3) d7 a3_m(a3) leveraging class field theory.

Findings

Characterization of special points as singular moduli and roots of unity

Explicit description of special points in the product space

Application of class field theory to Diophantine geometry

Abstract

Using class field theory we prove an explicit result of Andr\'e-Oort type for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$. In this variation the special points of $\mathbb{P}^1(\mathbb{C})$ are the singular moduli, while the special points of $\mathbb{G}_m(\mathbb{C})$ are defined to be the roots of unity.