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
θ a , b ( τ , z ) = ∑ ξ ∈ Z n exp [ π i ( ξ + a ) τ ( ξ + a ) t + 2 π i ( ξ + a ) ( z + b ) t ] {\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]}
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, vol. 194, New York-Heidelberg: Springer-Verlag, MR 0325625
