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21 Multidimensional Theta FunctionsProperties

§21.3 Symmetry and Quasi-Periodicity


§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)=e-2πi(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(αm1-βm2-12m2Ωm2-m2z)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.