Jacobi form

Short description: Class of complex vector function

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).

Definition

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

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

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