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The classification of free algebras of orthogonal modular forms
- Part of
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 9 / September 2021
- Published online by Cambridge University Press:
- 06 August 2021, pp. 2026-2045
- Print publication:
- September 2021
-
- Article
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We prove a necessary and sufficient condition for the graded algebra of automorphic forms on a symmetric domain of type IV being free. From the necessary condition, we derive a classification result. Let
$M$ be an even lattice of signature 
$(2,n)$ splitting two hyperbolic planes. Suppose 
$\Gamma$ is a subgroup of the integral orthogonal group of 
$M$ containing the discriminant kernel. It is proved that there are exactly 26 groups 
$\Gamma$ such that the space of modular forms for 
$\Gamma$ is a free algebra. Using the sufficient condition, we recover some well-known results.
