Jacobi form

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

  • ϕ ( a τ + b c τ + d , z c τ + d ) = ( c τ + d ) k e 2 π i m c z 2 c τ + d ϕ ( τ , z )  for  ( a   b c   d ) ∈ S L 2 ( Z ) {\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} )}
  • ϕ ( τ , z + λ τ + μ ) = e − 2 π i m ( λ 2 τ + 2 λ z ) ϕ ( τ , 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.
  • ϕ {\displaystyle \phi } has a Fourier expansion
ϕ ( τ , z ) = ∑ n ≥ 0 ∑ r 2 ≤ 4 m n C ( n , r ) e 2 π i ( n τ + r z ) . {\displaystyle \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, vol. 55, Boston, MA: Birkhäuser Boston, doi:10.1007/978-1-4684-9162-3, ISBN 978-0-8176-3180-2, MR 0781735