[Paper Review] Special points on the product of two modular curves

Bas Edixhoven|ArXiv.org|Oct 11, 1996

Analytic Number Theory Research4 references4 citations

TL;DR

This paper proves, under the Generalized Riemann Hypothesis for imaginary quadratic fields, that irreducible curves in the product of two modular curves containing infinitely many complex multiplication (CM) points must be either Hecke correspondences or fibers of the projection maps. This provides strong evidence for Oort's conjecture on the sub-Shimura nature of Zariski closures of CM points in Shimura varieties.

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.

Motivation & Objective

• To investigate the structure of irreducible curves in the product of two modular curves that contain infinitely many complex multiplication (CM) points.
• To test and provide evidence for Oort's conjecture, which posits that Zariski closures of CM points in Shimura varieties are themselves sub-Shimura varieties.
• To determine the precise geometric nature of such curves under number-theoretic assumptions.
• To establish a classification of special subvarieties in the product of two modular curves based on CM point distribution.

Proposed method

• The analysis relies on the Generalized Riemann Hypothesis (GRH) for imaginary quadratic fields to control the distribution and density of CM points on curves.
• The proof uses arithmetic geometry techniques, particularly the theory of Hecke correspondences and the geometry of modular curves.
• It applies results from the theory of complex multiplication and the action of Hecke operators on modular curves.
• The argument proceeds by contradiction, assuming a curve contains infinitely many CM points but is not a Hecke correspondence or a fiber, and derives a contradiction using GRH.
• The classification is based on the Zariski closure of CM points and their behavior under the projections to the factors.
• The paper uses modularity and Galois representation techniques to analyze the arithmetic of points on the curve.

Experimental results

Research questions

• RQ1 What geometric types of curves in the product of two modular curves can contain infinitely many CM points?
• RQ2 Under what conditions do such curves fail to be Hecke correspondences or fibers of the projections?
• RQ3 How does the Generalized Riemann Hypothesis constrain the distribution of CM points on curves in modular curve products?
• RQ4 To what extent do the Zariski closures of CM points in such products reflect the sub-Shimura structure predicted by Oort's conjecture?
• RQ5 Can the classification of special subvarieties in products of modular curves be reduced to Hecke correspondences and fibers?

Key findings

• Under the assumption of the Generalized Riemann Hypothesis for imaginary quadratic fields, any irreducible curve in the product of two modular curves containing infinitely many CM points must be either a Hecke correspondence or a fiber of one of the two projections.
• The result provides strong conditional evidence for Oort's conjecture on the sub-Shimura nature of Zariski closures of CM points in Shimura varieties.
• The classification is complete under the GRH assumption, showing that no other types of curves can contain infinitely many CM points.
• The proof establishes a sharp dichotomy: only Hecke correspondences and fibers can support infinite CM point sets in this setting.
• The result is conditional on GRH, but it represents a significant step toward understanding the arithmetic geometry of special points in Shimura varieties.
• The work demonstrates the power of GRH in controlling the distribution of CM points and their Zariski closures in higher-dimensional moduli spaces.