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The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$

Jennifer Johnson-Leung, Finn McGlade, Isabella Negrini, Aaron Pollack, Manami Roy

Abstract

The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two, i.e., on $\mathrm{Sp}_4$, cut out by certain linear relations between the Fourier coefficients. It is a theorem of Andrianov, Maass, and Zagier, that the classical Maass Spezialschar is exactly equal to the space of Saito-Kurokawa lifts. We study an analogous space of quaternionic modular forms on split $\mathrm{SO}_8$, and prove the analogue of the Andrianov-Maass-Zagier theorem. Our main tool for proving this theorem is the development of a theory of a Fourier-Jacobi expansion of quaternionic modular forms on orthogonal groups.

The quaternionic Maass Spezialschar on split $\mathrm{SO}(8)$

Abstract

The classical Maass Spezialschar is a Hecke-stable subspace of the level one holomorphic Siegel modular forms of genus two, i.e., on , cut out by certain linear relations between the Fourier coefficients. It is a theorem of Andrianov, Maass, and Zagier, that the classical Maass Spezialschar is exactly equal to the space of Saito-Kurokawa lifts. We study an analogous space of quaternionic modular forms on split , and prove the analogue of the Andrianov-Maass-Zagier theorem. Our main tool for proving this theorem is the development of a theory of a Fourier-Jacobi expansion of quaternionic modular forms on orthogonal groups.
Paper Structure (36 sections, 68 theorems, 171 equations)

This paper contains 36 sections, 68 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. The Fourier-Jacobi coefficient
  3. $D_4$ modular forms
  4. The quaternionic Maass spezialschar
  5. Periods of quaternionic modular forms
  6. Acknowledgments
  7. The classical Saito-Kurokawa lift and Maass Spezialschar
  8. Preliminaries
  9. The underlying quadratic space
  10. Octonions
  11. Algebraic groups, their Lie algebras, and parabolic subgroups
  12. Compact subgroups
  13. Holomorphic and Quaternionic Modular Forms
  14. Holomorphic modular forms on orthogonal groups
  15. Two Definitions of Quaternionic Modular Forms
  16. ...and 21 more sections

Key Result

Theorem 1.1

Let the notation be as above, so that $\varphi$ is a weight $\ell$ quaternionic modular form on $G$. Then $\mathrm{FJ}_{\varphi}$ is the automorphic function corresponding to a weight $\ell$ holomorphic modular form on $H$. Moreover, the classical Fourier coefficients of $\mathrm{FJ}_{\varphi}$ are

Theorems & Definitions (134)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Remark 3.2
  • ...and 124 more