Theta constant

In mathematics, a theta constant or Thetanullwert (German for theta zero value; plural Thetanullwerte) is the restriction θ_m(τ) = θ_m(τ,0) of a theta function θ_m(τ,z) with rational characteristic m to z = 0. The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant.

Definition

The theta function θ_m(τ,z) = θ_a,b(τ,z)is defined by

$\theta _{a,b}(\tau ,z)=\sum _{\xi \in Z^{n}}\exp \left[\pi {\rm {i}}(\xi +a)\tau (\xi +a)^{t}+2\pi i(\xi +a)(z+b)^{t}\right]$

where

• n is a positive integer, called the genus or rank.
• m = (a,b) is called the characteristic
• a,b are in Rⁿ
• τ is a complex n by n matrix with positive definite imaginary part
• z is in Cⁿ
• t means the transpose of a row vector.

If a,b are in Qⁿ then θ_a,b(τ,0) is called a theta constant.

Examples

If n = 1 and a and b are both 0 or 1/2, then the functions θ_a,b(τ,z) are the four Jacobi theta functions, and the functions θ_a,b(τ,0) are the classical Jacobi theta constants. The theta constant θ_1/2,1/2(τ,0) is identically zero, but the other three can be nonzero.

References

• Igusa, Jun-ichi (1972), Theta functions., Die Grundlehren der mathematischen Wissenschaften, vol. 194, New York-Heidelberg: Springer-Verlag, MR 0325625