Jacobi form

 ( Modular Forms ) 

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 $H^\{(n,h)\}\_R$. The theory was first systematically studied by .

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

• $\backslash phi\backslash left(\backslash frac\{a\backslash tau+b\}\{c\backslash tau+d\},\backslash frac\{z\}\{c\backslash tau+d\}\backslash right)\; =\; (c\backslash tau+d)^ke^\{\backslash frac\{2\backslash pi\; i\; mcz^2\}\{c\backslash tau+d\}\}\backslash phi(\backslash tau,z)\backslash text\{\; for\; \}\{a\backslash \; b\backslash choose\; c\backslash \; d\}\backslash in\; \backslash mathrm\{SL\}\_2(\backslash mathbb\{Z\})$
• $\backslash phi(\backslash tau,z+\backslash lambda\backslash tau+\backslash mu)\; =\; e^\{-2\backslash pi\; i\; m(\backslash lambda^2\backslash tau+2\backslash lambda\; z)\}\backslash phi(\backslash tau,z)$ for all integers λ, μ.
• $\backslash phi$ has a Fourier expansion

: $\backslash phi(\backslash tau,z)\; =\; \backslash sum\_\{n\backslash ge\; 0\}\; \backslash sum\_\{r^2\backslash le\; 4mn\}\; C(n,r)e^\{2\backslash pi\; i\; (n\backslash tau+rz)\}.$

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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.

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