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21 Multidimensional Theta FunctionsProperties

§21.5 Modular Transformations


§21.5(i) Riemann Theta Functions

Let A, B, C, and D be g×g matrices with integer elements such that

21.5.1 Γ=[ABCD]

is a symplectic matrix, that is,

21.5.2 ΓJ2gΓT=J2g.


21.5.3 detΓ=1,


21.5.4 θ([[CΩ+D]-1]Tz|[AΩ+B][CΩ+D]-1)=ξ(Γ)det[CΩ+D]eπiz[[CΩ+D]-1C]zθ(z|Ω).

Here ξ(Γ) is an eighth root of unity, that is, (ξ(Γ))8=1. For general Γ, it is difficult to decide which root needs to be used. The choice depends on Γ, but is independent of z and Ω. Equation (21.5.4) is the modular transformation property for Riemann theta functions.

The modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ξ(Γ) is determinate:

21.5.5 Γ=[A0g0g[A-1]T]θ(Az|AΩAT)=θ(z|Ω).

(A invertible with integer elements.)

21.5.6 Γ=[IgB0gIg]θ(z|Ω+B)=θ(z|Ω).

(B symmetric with integer elements and even diagonal elements.)

21.5.7 Γ=[IgB0gIg]θ(z|Ω+B)=θ(z+12diagB|Ω).

(B symmetric with integer elements.) See Heil (1995, p. 24). For a g×g matrix A we define diagA, as a column vector with the diagonal entries as elements.

21.5.8 Γ =[0g-IgIg0g]  
θ(Ω-1z|-Ω-1) =det[-iΩ]eπizΩ-1zθ(z|Ω),

where the square root assumes its principal value.

§21.5(ii) Riemann Theta Functions with Characteristics

21.5.9 θ[Dα-Cβ+12diag[CDT]-Bα+Aβ+12diag[ABT]]([[CΩ+D]-1]Tz|[AΩ+B][CΩ+D]-1)=κ(α,β,Γ)det[CΩ+D]eπiz[[CΩ+D]-1C]zθ[αβ](z|Ω),

where κ(α,β,Γ) is a complex number that depends on α, β, and Γ. However, κ(α,β,Γ) is independent of z and Ω. For explicit results in the case g=1, see §20.7(viii).