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Journal Article

Weyl invariant Jacobi forms: a new approach


Wang,  Haowu
Max Planck Institute for Mathematics, Max Planck Society;

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Wang, H. (2021). Weyl invariant Jacobi forms: a new approach. Advances in Mathematics, 384: 107752. doi:10.1016/j.aim.2021.107752.

Cite as: https://hdl.handle.net/21.11116/0000-0008-8836-2
For any irreducible root system not of type $E_8$, Wirthm\"{u}ller proved in
1992 that the bigraded algebra of weak Jacobi forms invariant under the Weyl
group is a polynomial algebra. In this paper we give a new automorphic proof of
this result based on the general theory of Jacobi forms. We proved in a
previous paper that the space of weak Jacobi forms for $E_8$ is not a
polynomial algebra and every $E_8$ Jacobi form can be expressed uniquely as a
polynomial in nine algebraically independent Jacobi forms introduced by Sakai
with coefficients which are meromorphic $SL_2(Z)$ modular forms. In this paper
we further show that these coefficients are the quotients of some $SL_2(Z)$
modular forms by a certain power of a fixed $SL_2(Z)$ modular form which is
completely determined by the nine generators.