mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Jacobi form is an automorphic form on the Jacobi group, which is the

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

of the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...

Sp(n;R) and the

Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b' ...

H^_R

. The theory was first systematically studied by .

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,\frac\right) = (c\tau+d)^ke^\phi(\tau,z)\text\in \mathrm_2(\mathbb)

\phi(\tau,z+\lambda\tau+\mu) = e^\phi(\tau,z)

for all integers λ, μ. *

\phi

has a Fourier expansion ::

\phi(\tau,z) = \sum_ \sum_ C(n,r)e^.

Examples

Examples in two variables include Jacobi theta functions, the Weierstrass ℘ function, and Fourier–Jacobi coefficients of

Siegel modular form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...

s 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

*{{Citation , last1=Eichler , first1=Martin , last2=Zagier , first2=Don , title=The theory of Jacobi forms , publisher=Birkhäuser Boston , location=Boston, MA , series=Progress in Mathematics , isbn=978-0-8176-3180-2 , mr=781735 , year=1985 , volume=55 , doi=10.1007/978-1-4684-9162-3 Modular forms Theta functions