Theta constant
In mathematics, a theta constant or Thetanullwert (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(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 θm(τ,z) = θa,b(τ,z)is defined by
![{\displaystyle \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]}](../I/m/b01fd70d3489ca592e9ecf2ffcca06415fc9c822.svg)
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 θa,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 θa,b(τ,z) are the four Jacobi theta functions, and the functions θa,b(τ,0) are the classical Jacobi theta constants. The theta constant θ1/2,1/2(τ,0) is identically zero, but the other three can be nonzero.
References
- Igusa, Jun-ichi (1972), Theta functions., Die Grundlehren der mathematischen Wissenschaften, 194, New York-Heidelberg: Springer-Verlag, MR 0325625