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Jacobi form

From Wikipedia, the free encyclopedia

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 . The theory was first systematically studied by Eichler & Zagier (1985).

Definition

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

  • for all integers λ, μ.
  • has a Fourier expansion

Examples

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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

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  • 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