Special points on the product of two modular curves

TL;DR

Under the assumption of the Generalized Riemann Hypothesis, the paper characterizes irreducible curves with infinitely many CM points in the product of two modular curves, showing they are either Hecke correspondences or fibers.

Contribution

It provides a conditional proof supporting Oort's conjecture about the structure of subvarieties containing CM points in Shimura varieties.

Findings

Irreducible curves with infinitely many CM points are either Hecke correspondences or fibers.

Supports Oort's conjecture on the structure of CM point sets in Shimura varieties.

Conditional proof assuming the Generalized Riemann Hypothesis.

Abstract

We prove, assuming the Generalized Riemann Hypothesis for imaginary quadratic fields, that irreducible curves in the product of two modular curves that contain infinitely many complex multiplication points are either a Hecke correspondence or a fibre for one of the two projections. This gives evidence for a conjecture of Oort that says that irreducible components of the Zariski closure of a set of CM points in a Shimura variety are sub Shimura varieties.