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(n;R) and the Heisenberg group ${\displaystyle H_{R}^{(n,h)}}$. The theory was first systematically studied by Eichler & Zagier (1985).

A Jacobi form of level 1, weight k and index m is a function ${\displaystyle \phi (\tau ,z)}$ of two complex variables (with τ in the upper half plane) such that

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.

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