Clebsch graph

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 Template:Harvtxt.

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

• $\varphi (\frac{a\tau +b}{c\tau +d},\frac{z}{c\tau +d})=(c\tau +d{)}^k{e}^{\frac{2\pi imc{z}^2}{c\tau +d}}\varphi (\tau ,z)\text{ for }(\genfrac{}{}{0ex}{}{ab}{cd})\in S{L}_2(Z)$
• $\varphi (\tau ,z+\lambda \tau +\mu )=e^{-2\pi im({\lambda }^2\tau +2\lambda z)}\varphi (\tau ,z)$ for all integers λ μ.
• $\varphi$ has a Fourier expansion

$\varphi (\tau ,z)=\sum _{n\ge 0}\sum _{r^2\le 4mn}c(n,r)e^{2\pi i(n\tau +rz)}.$

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.

References

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