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21 Multidimensional Theta FunctionsProperties

§21.2 Definitions

  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).


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),


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).