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Antisymmetry of real quadratic singular moduli

Sören Sprehe

Abstract

We confirm a conjecture of Darmon-Vonk on the antisymmetry of real quadratic singular moduli. The proof relies on a careful analysis of rigid meromorphic cocycles à la Darmon-Gehrmann-Lipnowski for the split orthogonal group on four variables. Moreover, we prove the modularity of a generating series of Kudla-Millson divisors in the spirit of Gross-Kohnen-Zagier.

Antisymmetry of real quadratic singular moduli

Abstract

We confirm a conjecture of Darmon-Vonk on the antisymmetry of real quadratic singular moduli. The proof relies on a careful analysis of rigid meromorphic cocycles à la Darmon-Gehrmann-Lipnowski for the split orthogonal group on four variables. Moreover, we prove the modularity of a generating series of Kudla-Millson divisors in the spirit of Gross-Kohnen-Zagier.
Paper Structure (24 sections, 56 theorems, 294 equations)

This paper contains 24 sections, 56 theorems, 294 equations.

Table of Contents

  1. Introduction
  2. Divisor-valued cocycles
  3. Symmetric spaces
  4. Special cycles and divisor-valued cocycles
  5. The three-dimensional case
  6. The four-dimensional case
  7. Lifting obstructions
  8. Hecke operators
  9. Lifting obstructions
  10. The Darmon--Vonk case
  11. The four-dimensional case
  12. The modularity theorem
  13. The split case
  14. The non-split case
  15. Intersection with an RM point
  16. ...and 9 more sections

Key Result

Proposition 2.2

There is an isomorphism of $\mathbb{Q}$-varieties which commutes with the $B^\times \times B^\times$-action on both sides.

Theorems & Definitions (112)

  • Remark 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['prosplit']}:
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 102 more