Weyl invariant Jacobi forms: a new approach

Abstract

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type $E_8$. This approach is also applicable to $E_8$. Even if the algebra of $E_8$ Jacobi forms is known to be non-free, we still derive a new structure result.

Final version

References in corpus (1)

• The D_8-tower of weak Jacobi forms and applications

Cited by in corpus (4)

• Weyl invariant $E_8$ Jacobi forms and $E$-strings
• Algebraic construction of Weyl invariant $E_8$ Jacobi forms
• Sums of four polygonal numbers: precise formulas
• Conway invariant Jacobi forms on the Leech lattice