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The Gross--Kohnen--Zagier theorem via $p$-adic uniformization

Lea Beneish, Henri Darmon, Lennart Gehrmann, Martí Roset

Abstract

This article gives a new proof of the Gross--Kohnen--Zagier theorem for Shimura curves which exploits the $p$-adic uniformization of Cerednik--Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross--Kohnen--Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a $p$-adic family of positive definite ternary theta series.

The Gross--Kohnen--Zagier theorem via $p$-adic uniformization

Abstract

This article gives a new proof of the Gross--Kohnen--Zagier theorem for Shimura curves which exploits the -adic uniformization of Cerednik--Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross--Kohnen--Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a -adic family of positive definite ternary theta series.
Paper Structure (25 sections, 32 theorems, 180 equations, 2 tables)

This paper contains 25 sections, 32 theorems, 180 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Cerednik--Drinfeld theorem
  3. $p$-adic uniformization of $X$
  4. $p$-adic analytic description of Heegner divisors
  5. Hecke action on divisors
  6. Modularity of degrees of Heegner divisors
  7. The Abel--Jacobi map
  8. Definition and properties of the Abel--Jacobi map
  9. Divisors of strong degree $0$
  10. Reduction of the main theorem to convenient Schwartz--Bruhat functions
  11. Values of $p$-adic theta functions
  12. The quotient $\gamma^\mathbb{Z} \backslash \mathcal{T}$
  13. Computation of $j_{\mathscr D}(\gamma)$ for divisors of strong degree $0$
  14. Abel--Jacobi images of Heegner divisors
  15. Values of theta functions associated to Heegner divisors
  16. ...and 10 more sections

Key Result

Theorem 1.2

The generating series $G(q) \in \mathrm{Pic}(X)(\mathbb{Q})_\mathbb{Q}[[q]]$ of eqn:gkz is a modular form of weight $3/2$ and level $\Gamma_0(4N)$.

Theorems & Definitions (75)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Cerednik--Drinfeld
  • proof
  • Lemma 2.4
  • proof
  • ...and 65 more