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Jacobi form - Wikipedia

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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).

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 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.