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On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms

Baptiste Depouilly

Abstract

Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where these coefficients are rational with bounded denominators and demonstrate divisibility properties under suitable linear combinations.

On the Divisibility Properties of the Fourier Coefficients of Meromorphic Hilbert Modular Forms

Abstract

Following Zagier, this work studies the rationality and divisibility of Fourier coefficients of meromorphic Hilbert modular forms associated with real quadratic fields, using theta lifts and weak Maass forms. We establish conditions where these coefficients are rational with bounded denominators and demonstrate divisibility properties under suitable linear combinations.

Paper Structure

This paper contains 20 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Fourier coefficients of meromorphic Hilbert modular forms
  3. Numerical examples
  4. How this article is organized
  5. Harmonic weak Maass forms for the Weil representation
  6. The Weil representation
  7. Vector-valued harmonic Maass forms
  8. Non-holomorphic Poincaré series
  9. Orthogonal modular forms
  10. The Grassmannian of H2
  11. Hilbert modular forms as orthogonal modular forms
  12. Hilbert meromorphic forms
  13. Hirzebruch-Zagier divisors and Weyl chambers
  14. The Doi-Naganuma theta lift
  15. The theta lift
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $m < 0$ and an even integer $k \geq 4$, if the space of cusp forms $S_{k,\overline{\rho}_L}$ is trivial, then, up to multiplication by an integer that doesn't depend on $\nu$, the $\nu$-th coefficient of $\omega_m$ is divisible by $(D\ell_\nu N(\nu_0))^{k-1}$ for all $\nu \in \partial_{F}^{-1}$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Theorem 3.2: The Fourier expansion of $\omega_m$
  • Definition 4.1
  • Proposition 4.2
  • proof
  • Theorem 5.1
  • Theorem 5.2
  • ...and 6 more