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 In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theo ...

''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 form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...

s. The

theta function of a lattice In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. Definition One can associate to any (positive-definite) lattice Λ a theta function given by :\Theta_\Lambda(\tau) = \sum_ ...

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_(\tau,z) = \sum_ \exp\left pi (\xi+a)\tau(\xi+a)^t + 2\pi i(\xi+a)(z+b)^t\right /math>

where 

* ''n'' is a positive integer, called the genus or rank. 
* ''m'' = (''a'',''b'') is called the characteristic
* ''a'',''b'' are in R

^''n'' * ''τ'' is a complex ''n'' by ''n'' matrix with positive definite imaginary part * ''z'' is in C^''n'' * ''t'' means the transpose of a row vector. If ''a'',''b'' are in Q^''n'' 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 In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...

, 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

*{{citation, mr=0325625 , last=Igusa, first= Jun-ichi , title=Theta functions. , series=Die Grundlehren der mathematischen Wissenschaften, volume= 194, publisher= Springer-Verlag, place= New York-Heidelberg, year= 1972 Automorphic forms Modular forms