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 $H_{R}^{(n, h)}$ . 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 $ϕ (τ, z)$ of two complex variables (with τ in the upper half plane) such that

$ϕ (\frac{a τ + b}{c τ + d}, \frac{z}{c τ + d}) = (c τ + d)^{k} e^{\frac{2 π i m c z^{2}}{c τ + d}} ϕ (τ, z) for (\binom{a b}{c d}) \in {S L}_{2} (ℤ)$
$ϕ (τ, z + λ τ + μ) = e^{- 2 π i m (λ^{2} τ + 2 λ z)} ϕ (τ, z)$ for all integers λ, μ.
$ϕ$ has a Fourier expansion

ϕ (τ, z) = \sum_{n \geq 0} \sum_{r^{2} \leq 4 m n} C (n, r) e^{2 π i (n τ + r z)} .

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.

References

Eichler, Martin; Zagier, Don (1985), The theory of Jacobi forms, Progress in Mathematics, vol. 55, Boston, MA: Birkhäuser Boston, doi:10.1007/978-1-4684-9162-3, ISBN 978-0-8176-3180-2, MR 0781735