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On the Vanishing and Cuspidality of $D_4$ Modular Forms

Finn McGlade

TL;DR

The paper develops vanishing and cuspidality criteria for quaternionic modular forms on the split group G=Spin(4,4) via scalar Fourier coefficients from a Fourier-Jacobi framework. It proves a primitive-coefficient vanishing criterion: a level-one quaternionic form on G vanishes if all primitive Fourier coefficients vanish, and it derives a Hecke-bound based cuspidality criterion for weights $\ell\\ge 5$. The analysis connects degenerate Fourier-Jacobi coefficients to holomorphic Fourier data of constant terms for cubic-norm-structure groups G_J, yielding a broad cuspidality-growth equivalence in weights $\\ell\ge 5$. Applications include refinements of the quaternionic Maass Spezialschar and a characterization of quaternionic Saito-Kurokawa lifts via slice-primitive Fourier data. Overall, the work establishes a novel coefficient-growth criterion for cuspidality in the D_4/quaternionic setting and ties together Fourier-Jacobi theory, theta liftings, and Hecke theory with implications for automorphic forms on multiple classical and exceptional groups.

Abstract

We develop vanishing and cuspidality criteria for quaternionic modular forms on $G=\mathrm{Spin}(4,4)$ using a theory of scalar Fourier coefficients. By analyzing a Fourier-Jacobi expansion for these forms, we prove that a level one quaternionic modular form on $G$ vanishes if and only if its primitive Fourier coefficients are zero. Using this criterion, we characterize Pollack's quaternionic Saito-Kurokawa subspace by imposing a system of linear relation among certain primitive Fourier coefficients. This characterization strengthens earlier work of the author with Johnson-Leung, Negrini, Pollack, and Roy. We also study quaternionic modular forms in the more general setting of a group $G_J$ associated to a cubic norm structure $J$. Here we establish a new relationship between the degenerate Fourier coefficients of quaternionic modular forms, and the Fourier coefficients of the holomorphic modular forms associated to their constant terms. As a consequence, we prove that in weights $\ell\geq 5$, a level one quaternionic modular form on $G$ is cuspidal if and only if its non-degenerate Fourier coefficients satisfy a polynomial growth condition.

On the Vanishing and Cuspidality of $D_4$ Modular Forms

TL;DR

The paper develops vanishing and cuspidality criteria for quaternionic modular forms on the split group G=Spin(4,4) via scalar Fourier coefficients from a Fourier-Jacobi framework. It proves a primitive-coefficient vanishing criterion: a level-one quaternionic form on G vanishes if all primitive Fourier coefficients vanish, and it derives a Hecke-bound based cuspidality criterion for weights . The analysis connects degenerate Fourier-Jacobi coefficients to holomorphic Fourier data of constant terms for cubic-norm-structure groups G_J, yielding a broad cuspidality-growth equivalence in weights . Applications include refinements of the quaternionic Maass Spezialschar and a characterization of quaternionic Saito-Kurokawa lifts via slice-primitive Fourier data. Overall, the work establishes a novel coefficient-growth criterion for cuspidality in the D_4/quaternionic setting and ties together Fourier-Jacobi theory, theta liftings, and Hecke theory with implications for automorphic forms on multiple classical and exceptional groups.

Abstract

We develop vanishing and cuspidality criteria for quaternionic modular forms on using a theory of scalar Fourier coefficients. By analyzing a Fourier-Jacobi expansion for these forms, we prove that a level one quaternionic modular form on vanishes if and only if its primitive Fourier coefficients are zero. Using this criterion, we characterize Pollack's quaternionic Saito-Kurokawa subspace by imposing a system of linear relation among certain primitive Fourier coefficients. This characterization strengthens earlier work of the author with Johnson-Leung, Negrini, Pollack, and Roy. We also study quaternionic modular forms in the more general setting of a group associated to a cubic norm structure . Here we establish a new relationship between the degenerate Fourier coefficients of quaternionic modular forms, and the Fourier coefficients of the holomorphic modular forms associated to their constant terms. As a consequence, we prove that in weights , a level one quaternionic modular form on is cuspidal if and only if its non-degenerate Fourier coefficients satisfy a polynomial growth condition.
Paper Structure (42 sections, 39 theorems, 197 equations)

This paper contains 42 sections, 39 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Quaternionic Modular Forms on $G$
  3. Primitivity in the Fourier Expansion of $D_4$ Modular Forms
  4. An Application to The Quaternionic Maass Spezialschar
  5. Characterizing Quaternionic Cusp Forms via a Hecke Bound
  6. The structure of paper
  7. Acknowledgments
  8. Algebraic Preliminaries
  9. Standard Notation
  10. The Quadratic Space $V$
  11. The Group $G$
  12. The Heisenberg Parabolic Subgroup $P$
  13. The Orthogonal Parabolic Subgroup $R$
  14. The Siegel Parabolic Subgroup $Q'$
  15. Compact Subgroups
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $\varphi$ be quaternionic modular forms on $G$ of weight $\ell>0$ and level one such that $\Lambda_{\varphi}[B]=0$ for all primitive $B\in \mathrm{M}_2({\mathbf Z})^{\oplus 2}$. Then $\varphi=0$.

Theorems & Definitions (86)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • Lemma 4.4
  • ...and 76 more