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 $\phi (\tau ,z)$ of two complex variables (with τ in the upper half plane) such that

$\phi \left({\frac {a\tau +b}{c\tau +d}},{\frac {z}{c\tau +d}}\right)=(c\tau +d)^{k}e^{{{\frac {2\pi imcz^{2}}{c\tau +d}}}}\phi (\tau ,z){\text{ for }}{a\ b \choose c\ d}\in SL_{2}(Z)$
$\phi (\tau ,z+\lambda \tau +\mu )=e^{{-2\pi im(\lambda ^{2}\tau +2\lambda z)}}\phi (\tau ,z)$ for all integers λ μ.
$\phi$ has a Fourier expansion

\phi (\tau ,z)=\sum _{{n\geq 0}}\sum _{{r^{2}\leq 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, ISBN 978-0-8176-3180-2, MR 781735

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