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).

Definition

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

• ${\displaystyle \phi \left({\frac {a\tau +b}{c\tau +d)),{\frac {z}{c\tau +d))\right)=(c\tau +d)^{k}e^{\frac {2\pi imcz^{2)){c\tau +d))\phi (\tau ,z){\text{ for )){a\ b \choose c\ d}\in \mathrm {SL} _{2}(\mathbb {Z} )}$
• ${\displaystyle \phi (\tau ,z+\lambda \tau +\mu )=e^{-2\pi im(\lambda ^{2}\tau +2\lambda z)}\phi (\tau ,z)}$ for all integers λ, μ.
• ${\displaystyle \phi }$ has a Fourier expansion

${\displaystyle \phi (\tau ,z)=\sum _{n\geq 0}\sum _{r^{2}\leq 4mn}C(n,r)e^{2\pi i(n\tau +rz)}.}$

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.