Intersections of Class Fields

TL;DR

This paper applies class field theory to restrict intersections of maximal abelian extensions, improving results on Heegner points' independence and providing effective bounds for special points on modular curves, with implications for the André-Oort conjecture.

Contribution

It introduces new restrictions on intersections of class fields and enhances understanding of special points on modular curves, advancing number theory and arithmetic geometry.

Findings

Restricted intersections of class fields are characterized.

Improved linear independence results for Heegner points.

Derived effective bounds for special points on modular curves.

Abstract

Using class field theory, we prove a restriction on the intersection of the maximal abelian extensions associated with different number fields. This restriction is then used to improve a result of Rosen and Silverman about the linear independence of Heegner points. In addition, it yields effective restrictions for the special points lying on an algebraic subvariety in a product of modular curves. The latter application is related to the Andr\'e-Oort conjecture.