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21 Multidimensional Theta FunctionsProperties

§21.2 Definitions

Contents
  1. §21.2(i) Riemann Theta Functions
  2. §21.2(ii) Riemann Theta Functions with Characteristics
  3. §21.2(iii) Relation to Classical Theta Functions

§21.2(i) Riemann Theta Functions

21.2.1 θ(𝐳|𝛀)=𝐧ge2πi(12𝐧𝛀𝐧+𝐧𝐳).

This g-tuple Fourier series converges absolutely and uniformly on compact sets of the 𝐳 and 𝛀 spaces; hence θ(𝐳|𝛀) is an analytic function of (each element of) 𝐳 and (each element of) 𝛀. θ(𝐳|𝛀) is also referred to as a theta function with g components, a g-dimensional theta function or as a genus g theta function.

For numerical purposes we use the scaled Riemann theta function θ^(𝐳|𝛀), defined by (Deconinck et al. (2004)),

21.2.2 θ^(𝐳|𝛀)=eπ[𝐳][𝛀]1[𝐳]θ(𝐳|𝛀).

θ^(𝐳|𝛀) is a bounded nonanalytic function of 𝐳. Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. See also §21.10(i).

Example

21.2.3 θ(z1,z2|[i1212i])=n1=n2=eπ(n12+n22)eiπn1n2e2πi(n1z1+n2z2).

With z1=x1+iy1, z2=x2+iy2,

21.2.4 θ^(x1+iy1,x2+iy2|[i1212i])=n1=n2=eπ(n1+y1)2π(n2+y2)2eπi(2n1x1+2n2x2n1n2).

§21.2(ii) Riemann Theta Functions with Characteristics

Let 𝜶,𝜷g. Define

21.2.5 θ[𝜶𝜷](𝐳|𝛀)=𝐧ge2πi(12[𝐧+𝜶]𝛀[𝐧+𝜶]+[𝐧+𝜶][𝐳+𝜷]).

This function is referred to as a Riemann theta function with characteristics [𝜶𝜷]. It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor:

21.2.6 θ[𝜶𝜷](𝐳|𝛀)=e2πi(12𝜶𝛀𝜶+𝜶[𝐳+𝜷])θ(𝐳+𝛀𝜶+𝜷|𝛀),

and

21.2.7 θ[𝟎𝟎](𝐳|𝛀)=θ(𝐳|𝛀).

Characteristics whose elements are either 0 or 12 are called half-period characteristics. For given 𝛀, there are 22g g-dimensional Riemann theta functions with half-period characteristics.

§21.2(iii) Relation to Classical Theta Functions

For g=1, and with the notation of §20.2(i),

21.2.8 θ(z|Ω)=θ3(πz|Ω),
21.2.9 θ1(πz|Ω) =θ[1212](z|Ω),
21.2.10 θ2(πz|Ω) =θ[120](z|Ω),
21.2.11 θ3(πz|Ω) =θ[00](z|Ω),
21.2.12 θ4(πz|Ω) =θ[012](z|Ω).