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Vanishing properties of Fourier coefficients of holomorphic $η$-quotients

Kathrin Bringmann, Guoniu Han, Bernhard Heim, Ben Kane

Abstract

In this paper, we study vanishing of Fourier coefficients of holomorphic $η$-quotients. We investigate examples of two different types: the first one involves integral weight CM newforms, while the second one involves half-integral weight $η$-quotients associated with sums of squares and Hurwitz class numbers.

Vanishing properties of Fourier coefficients of holomorphic $η$-quotients

Abstract

In this paper, we study vanishing of Fourier coefficients of holomorphic -quotients. We investigate examples of two different types: the first one involves integral weight CM newforms, while the second one involves half-integral weight -quotients associated with sums of squares and Hurwitz class numbers.

Paper Structure

This paper contains 27 sections, 56 theorems, 133 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Modular forms
  4. Operators on modular forms
  5. Hecke eigenforms
  6. Eisenstein series
  7. Valence formula
  8. Modularity of eta-quotients
  9. Unary theta functions
  10. Binary quadratic forms
  11. Hurwitz class numbers
  12. The space $S_2(\Gamma_0(36),\chi_{12})$
  13. Proof of \ref{['thm:L52']}
  14. The case $1^{-1}3^34^2$
  15. The case $1^42^{-2}4^4$
  16. ...and 12 more sections

Key Result

Theorem 1.1

We have

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • ...and 53 more