Communications in Number Theory and Physics

Volume 15 (2021)

Number 3

Weyl invariant $E_8$ Jacobi forms

Pages: 517 – 573



Haowu Wang (Max-Planck-Institut für Mathematik, Bonn, Germany)


We investigate the Jacobi forms for the root system $E_8$ invariant under the Weyl group. This type of Jacobi forms has significance in Frobenius manifolds, Gromov–Witten theory and string theory. In 1992, Wirthmüller proved that the space of Jacobi forms for any irreducible root system not of type $E_8$ is a polynomial algebra. But very little has been known about the case of $E_8$. In this paper we show that the bigraded ring of Weyl invariant $E_8$ Jacobi forms is not a polynomial algebra and prove that every such Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent Jacobi forms introduced by Sakai with coefficients which are meromorphic $\mathrm{SL}_2 (\mathbb{Z})$ modular forms. The latter result implies that the space of Weyl invariant $E_8$ Jacobi forms of fixed index is a free module over the ring of $\mathrm{SL}_2 (\mathbb{Z})$ modular forms and that the number of generators can be calculated by a generating series. We determine and construct all generators of small index. These results give a proper extension of the Chevalley type theorem to the case of $E_8$.


Jacobi forms, root systems, $E_8$ lattice, Weyl groups, invariant theory

2010 Mathematics Subject Classification

Primary 11F50. Secondary 17B22.

Received 14 January 2020

Accepted 8 March 2021

Published 15 July 2021