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Modular forms of half-integral weight on exceptional groups

Part of: Discontinuous groups and automorphic forms Linear algebraic groups and related topics

Published online by Cambridge University Press:  22 February 2024

Spencer Leslie  and
Aaron Pollack Open the ORCID record for Aaron Pollack [Opens in a new window]
Spencer Leslie
Affiliation:
Department of Mathematics, Boston College, Chestnut Hill, MA 02467, USA spencer.leslie@bc.edu
Aaron Pollack
Affiliation:
Department of Mathematics, The University of California San Diego, La Jolla, CA 92093, USA apollack@ucsd.edu
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Abstract

We define a notion of modular forms of half-integral weight on the quaternionic exceptional groups. We prove that they have a well-behaved notion of Fourier coefficients, which are complex numbers defined up to multiplication by ${\pm }1$. We analyze the minimal modular form $\Theta _{F_4}$ on the double cover of $F_4$, following Loke–Savin and Ginzburg. Using $\Theta _{F_4}$, we define a modular form of weight $\tfrac {1}{2}$ on (the double cover of) $G_2$. We prove that the Fourier coefficients of this modular form on $G_2$ see the $2$-torsion in the narrow class groups of totally real cubic fields.

Keywords

automorphic minimal representationquaternionic modular formshalf-integral weight modular formscovering groups

MSC classification

Primary: 11F03: Modular and automorphic functions 11F30: Fourier coefficients of automorphic forms
Secondary: 20G41: Exceptional groups
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 3 , March 2024 , pp. 657 - 707
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

SL has been supported by an AMS-Simons Travel Award and by NSF grant DMS-1902865. AP has been supported by the Simons Foundation via Collaboration Grant number 585147, by the NSF via grant numbers 2101888 and 2144021, and by an AMS Centennial Research Fellowship.

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