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Theta constant

In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θ<sub>m</sub>(τ)&nbsp;=&nbsp;θ<sub>m</sub>(τ,0) of a theta function θ<sub>m</sub>(τ,z) with rational characteristic m to z&nbsp;=&nbsp;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<sub>m</sub>(τ,z) = θ<sub>a,b</sub>(τ,z)is defined by

where

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

If a,b are in Q<sup>n</sup> then θ<sub>a,b</sub>(τ,0) is called a theta constant.

Examples

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

References