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Jacobi form

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Sander zwegers fourier coefficients of meromorphic jacobi 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 R ( n , h ) . The theory was first systematically studied by Eichler & Zagier (1985).

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Definition

A Jacobi form of level 1, weight k and index m is a function ϕ ( τ , 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 )
  • ϕ ( τ , z + λ τ + μ ) = e − 2 π i m ( λ 2 τ + 2 λ z ) ϕ ( τ , z ) for all integers λ μ.
  • ϕ has a Fourier expansion

    • 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

    Jacobi form Wikipedia
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