Jacobiform

Jacobiforms are, in function theory, certain extensions of the concept of modular forms in two complex variables (they are automorphic forms). Examples are Jacobi theta functions and the Weierstrasse p-function . Their theory was especially developed by Don Zagier and Martin Eichler and, for example, by Nils-Peter Skoruppa .

While common modular forms (weight k) are defined in the space of the elliptical functions and parameterize them via the j-function (elliptical functions are defined on the complex plane modulo a lattice and the modular forms are defined on the equivalence class of these lattices), Jacobi forms are one Step further and are additional analytical functions on elliptic curves (defined with a grid determined by the upper half-plane ) via a second variable . ${\displaystyle \tau \in \mathbb {H} }$ ${\displaystyle z}$

${\displaystyle f(\tau ,z):\mathbb {H} \times \mathbb {C} \to \mathbb {C} }$

Under the module group, you transform with the following automorphism factors: ${\displaystyle \Gamma }$

${\displaystyle f\left({\frac {a\tau +b}{c\tau +d}},{\frac {z}{c\tau +d}}\right)={(c\tau +d)}^{k}e^{\left({\frac {2\pi iNcz^{2}}{c\tau +d}}\right)}f(\tau ,z)}$

${\displaystyle f(\tau ,z+l\tau +m)=e^{-2\pi iN(l^{2}\tau +2lz)}f(\tau ,z)}$

for , , and . It is the weight and the index of the Jacobi form. ${\displaystyle z\in \mathbb {C} }$ ${\displaystyle \tau \in \mathbb {H} }$ ${\displaystyle l,m\in \mathbb {Z} }$ ${\displaystyle \left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma }$ ${\displaystyle k}$ ${\displaystyle N}$

In addition, as with modular forms, a growth condition is required. It says that the Fourier expansion (it is , ): ${\displaystyle q=\exp {2\pi i\tau }}$ ${\displaystyle \zeta =\exp {2\pi iz}}$

${\displaystyle f(\tau ,z)=\sum _{n=0}^{\infty }\sum _{r\in \mathbb {Z} \atop r^{2}\leq 4Nn}c(n,r)q^{n}\zeta ^{r}}$

with the Fourier coefficients . ${\displaystyle c(n,r)}$

According to Don Zagier, many physically relevant applications of modular forms (in a generalized sense) and theta functions can be classified under the Jacobi forms, for example some characters in irreducible representations of the highest weight of Kac-Moody algebras are Jacobi forms. Important examples are generalized Eisenstein series.

literature

• Don Zagier , Introduction to modular forms, M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson, From Number Theory to Physics, Springer 1995, Chapter 4, pp. 277ff, Online, pdf
• Don Zagier, Martin Eichler, The theory of Jacobi forms, Birkhäuser 1985

Individual evidence

1. ↑ The usual modular forms are special cases of Jacobi forms if the z-dependence is neglected
2. ^ Zagier, Introduction to modular forms in Waldschmidt, u. a. From Number Theory to Physics, Springer 1995, p. 278