mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Jacobi form is an

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

on the

Jacobi group In mathematics, the Jacobi group, introduced by , is the semidirect product of the symplectic group Sp2''n''(R) and the Heisenberg group R1+2''n''. The concept is named after Carl Gustav Jacob Jacobi. Automorphic forms on the Jacobi group are ...

, which is the

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...

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 grou ...

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 ' ...

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 In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...

, 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 algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...

s. Meromorphic Jacobi forms appear in the theory of

Mock modular form In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight . The first examples of mock theta functions were described by Srinivasa Ramanu ...

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