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Modular forms of half-integral weight on exceptional groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 3 / March 2024
- Published online by Cambridge University Press:
- 22 February 2024, pp. 657-707
- Print publication:
- March 2024
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- Article
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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.
