About the Project
21 Multidimensional Theta FunctionsProperties

§21.3 Symmetry and Quasi-Periodicity

Contents
  1. §21.3(i) Riemann Theta Functions
  2. §21.3(ii) Riemann Theta Functions with Characteristics

§21.3(i) Riemann Theta Functions

21.3.1 θ(𝐳|𝛀)=θ(𝐳|𝛀),
21.3.2 θ(𝐳+𝐦1|𝛀)=θ(𝐳|𝛀),

when 𝐦1g. Thus θ(𝐳|𝛀) is periodic, with period 1, in each element of 𝐳. More generally,

21.3.3 θ(𝐳+𝐦1+𝛀𝐦2|𝛀)=e2πi(12𝐦2𝛀𝐦2+𝐦2𝐳)θ(𝐳|𝛀),

with 𝐦1, 𝐦2 g. This is the quasi-periodicity property of the Riemann theta function. It determines the Riemann theta function up to a constant factor. The set of points 𝐦1+𝛀𝐦2 form a g-dimensional lattice, the period lattice of the Riemann theta function.

§21.3(ii) Riemann Theta Functions with Characteristics

Again, with 𝐦1, 𝐦2 g

21.3.4 θ[𝜶+𝐦1𝜷+𝐦2](𝐳|𝛀)=e2πi𝜶𝐦2θ[𝜶𝜷](𝐳|𝛀).

Because of this property, the elements of 𝜶 and 𝜷 are usually restricted to [0,1), without loss of generality.

21.3.5 θ[𝜶𝜷](𝐳+𝐦1+𝛀𝐦2|𝛀)=e2πi(𝜶𝐦1𝜷𝐦212𝐦2𝛀𝐦2𝐦2𝐳)θ[𝜶𝜷](𝐳|𝛀).

For Riemann theta functions with half-period characteristics,

21.3.6 θ[𝜶𝜷](𝐳|𝛀)=(1)4𝜶𝜷θ[𝜶𝜷](𝐳|𝛀).

See also §20.2(iii) for the case g=1 and classical theta functions.