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Automatic convergence for holomorphic modular forms

Aaron Pollack

Abstract

We prove an automatic convergence theorem for holomorphic modular forms on tube domains. The argument works in some generality, and covers in particular the case of orthogonal groups, symplectic groups, unitary and quaternion unitary groups, and the exceptional group $E_7$.

Automatic convergence for holomorphic modular forms

Abstract

We prove an automatic convergence theorem for holomorphic modular forms on tube domains. The argument works in some generality, and covers in particular the case of orthogonal groups, symplectic groups, unitary and quaternion unitary groups, and the exceptional group .
Paper Structure (14 sections, 14 theorems, 89 equations, 1 table)

This paper contains 14 sections, 14 theorems, 89 equations, 1 table.

Table of Contents

  1. Introduction
  2. Holomorphic automorphic forms
  3. Formal modular forms
  4. Related work
  5. Proof idea
  6. Outline of paper
  7. Tube domains and holomorphic modular forms
  8. The Jordan algebra
  9. The action on the tube domain
  10. Fourier coefficients of holomorphic modular forms
  11. Reduction theory
  12. Basic reduction theory
  13. Quantitative Sturm bound
  14. Automatic convergence

Key Result

Theorem 1.4

Let $U \subseteq G({\mathbf A}_f)$ be a compact open subgroup and suppose $X \subseteq G({\mathbf A}_f)$ is a subset satisfying Then the map $M_\rho(U) \rightarrow M_\rho^{f}(X,U)$ is a ${\mathbf C}$-linear isomorphism.

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 26 more