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 ...
''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 for ...
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) = \su ...
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
: