Theta constant


In mathematics, a theta constant or Thetanullwert is the restriction θm = θm of a theta function θm 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 = θa,bis defined by
where
If a,b are in Qn then θa,b is called a theta constant.

Examples

If n = 1 and a and b are both 0 or 1/2, then the functions θa,b are the four Jacobi theta functions, and the functions θa,b are the classical Jacobi theta constants. The theta constant θ1/2,1/2 is identically zero, but the other three can be nonzero.