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Modular forms of half-integral weight on exceptional groups
Published online by Cambridge University Press: 22 February 2024
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.
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- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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.