Jacobi form

In mathematics, a Jacobi form is an automorphic form on the Jacobi group, which is the semidirect product of the symplectic group Sp and the Heisenberg group. The theory was first systematically studied by.

Definition

A Jacobi form of level 1, weight k and index m is a function of two complex variables such that

for all integers λ, μ.
has a Fourier expansion

Examples

Examples in two variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of Siegel modular forms of genus 2. Examples with more than two variables include characters of some irreducible highest-weight representations of affine Kac–Moody algebras. Meromorphic Jacobi forms appear in the theory of Mock modular forms.