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Equidistribution of CM points on Shimura Curves and ternary theta series

Francesco Maria Saettone

TL;DR

The paper proves that reductions of CM points on Shimura curves over a totally real field $F$ become equidistributed in the appropriate loci (supersingular, superspecial, or smooth) as discriminants and conductors go to infinity, accommodating both split and ramified reduction at a place $v$. The approach ties CM data to weight $\tfrac{3}{2}$ Hilbert modular forms via Gross lattices and ternary theta series, then leverages subconvexity and genus–spinor genus analysis within a metaplectic/Weil representation framework to establish weak-* convergence with explicit error terms. A key novelty is the ramified case, handled through Cerednik–Drinfeld uniformisation and the associated moduli interpretation of Drinfeld spaces, enabling a complete equidistribution statement and an integral André–Oort-like result for Shimura curves. The work connects CM-point distribution to automorphic forms, theta correspondences, and arithmetic geometry, yielding a robust tool for diophantine applications and potential extensions to products of Shimura curves and arithmetic pencils.

Abstract

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field, considering both the split and the ramified case. The main novelty of the ramified case consists in the use of the moduli interpretation of the Cerednik--Drinfeld uniformisation. Our result is achieved by associating to the reduction of CM points certain Hilbert modular forms of weight $3/2$ and by analyzing their Fourier coefficients. Moreover, we also deduce the Shimura curves case of the integral version of the André--Oort conjecture.

Equidistribution of CM points on Shimura Curves and ternary theta series

TL;DR

The paper proves that reductions of CM points on Shimura curves over a totally real field become equidistributed in the appropriate loci (supersingular, superspecial, or smooth) as discriminants and conductors go to infinity, accommodating both split and ramified reduction at a place . The approach ties CM data to weight Hilbert modular forms via Gross lattices and ternary theta series, then leverages subconvexity and genus–spinor genus analysis within a metaplectic/Weil representation framework to establish weak-* convergence with explicit error terms. A key novelty is the ramified case, handled through Cerednik–Drinfeld uniformisation and the associated moduli interpretation of Drinfeld spaces, enabling a complete equidistribution statement and an integral André–Oort-like result for Shimura curves. The work connects CM-point distribution to automorphic forms, theta correspondences, and arithmetic geometry, yielding a robust tool for diophantine applications and potential extensions to products of Shimura curves and arithmetic pencils.

Abstract

We prove an equidistribution statement for the reduction of Galois orbits of CM points on the special fiber of a Shimura curve over a totally real field, considering both the split and the ramified case. The main novelty of the ramified case consists in the use of the moduli interpretation of the Cerednik--Drinfeld uniformisation. Our result is achieved by associating to the reduction of CM points certain Hilbert modular forms of weight and by analyzing their Fourier coefficients. Moreover, we also deduce the Shimura curves case of the integral version of the André--Oort conjecture.

Paper Structure

This paper contains 41 sections, 26 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Conventions
  3. Shimura Curves
  4. Quaternion Algebras and Ramification
  5. Ramification Setting
  6. Indefinite Case
  7. Definite Case
  8. Special Points and Quaternion Algebras
  9. Indefinite Case
  10. Definite Case
  11. Reductions of Integral Models
  12. Universal $v$-Divisible Group
  13. Unramified Case
  14. Ramified case
  15. Eichler Orders associated to Supersingular and Superspecial Points.
  16. ...and 26 more sections

Key Result

Theorem A

The reductions at $v$ of the Galois orbits of CM points are equidistributed for the discriminants and the conductors varying, i.e., for their absolute norms going to infinity:

Figures (1)

  • Figure 1: The illustration (drawn with the help of Francesco Beccuti) represents the Shimura curve over $\mathbb{Q}$ of discriminant $77$ and maximal level structure, whose CM points reduced modulo $7$ land in the superspecial locus (on the left) and to the supersingular locus for a prime $p\neq 7,11$. Note that on the right hand side the two curves should intersect $6$ times, since the special fiber is of genus $5$.

Theorems & Definitions (54)

  • Theorem A
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 44 more