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21 Multidimensional Theta FunctionsProperties

§21.2 Definitions


§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)=e-π[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|[i-12-12i])=n1=-n2=-e-π(n12+n22)e-iπn1n2e2πi(n1z1+n2z2).

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

21.2.4 θ^(x1+iy1,x2+iy2|[i-12-12i])=n1=-n2=-e-π(n1+y1)2-π(n2+y2)2eπi(2n1x1+2n2x2-n1n2).

§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).