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21 Multidimensional Theta FunctionsProperties

§21.6 Products

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§21.6(i) Riemann Identity

Let T=[Tjk] be an arbitrary h×h orthogonal matrix (that is, TTT=I) with rational elements. Also, let Z be an arbitrary g×h matrix. Define

21.6.1 𝒦=g×hT/(g×hTg×h),

that is, 𝒦 is the set of all g×h matrices that are obtained by premultiplying T by any g×h matrix with integer elements; two such matrices in 𝒦 are considered equivalent if their difference is a matrix with integer elements. Also, let

21.6.2 𝒟=|TTh/(TThh)|,

that is, 𝒟 is the number of elements in the set containing all h-dimensional vectors obtained by multiplying TT on the right by a vector with integer elements. Two such vectors are considered equivalent if their difference is a vector with integer elements. Then

21.6.3 j=1hθ(k=1hTjkzk|Ω)=1𝒟gA𝒦B𝒦e2πitr[12ATΩA+AT[Z+B]]j=1hθ(zj+Ωaj+bj|Ω),

where zj, aj, bj denote respectively the jth columns of Z, A, B. This is the Riemann identity. On using theta functions with characteristics, it becomes

21.6.4 j=1hθ[k=1hTjkckk=1hTjkdk](k=1hTjkzk|Ω)=1𝒟gA𝒦B𝒦e-2πij=1hbjcjj=1hθ[aj+cjbj+dj](zj|Ω),

where cj and dj are arbitrary h-dimensional vectors. Many identities involving products of theta functions can be established using these formulas.

Example

Let h=4 and

21.6.5 T=[111111-1-11-11-11-1-11].

Then

21.6.6 θ(x+y+u+v2|Ω)θ(x+y-u-v2|Ω)θ(x-y+u-v2|Ω)θ(x-y-u+v2|Ω)=12gα12g/gβ12g/ge2πi(2αΩα+α[x+y+u+v])×θ(x+Ωα+β|Ω)θ(y+Ωα+β|Ω)×θ(u+Ωα+β|Ω)θ(v+Ωα+β|Ω),

and

21.6.7 θ[12[c1+c2+c3+c4]12[d1+d2+d3+d4]](x+y+u+v2|Ω)θ[12[c1+c2-c3-c4]12[d1+d2-d3-d4]](x+y-u-v2|Ω)×θ[12[c1-c2+c3-c4]12[d1-d2+d3-d4]](x-y+u-v2|Ω)θ[12[c1-c2-c3+c4]12[d1-d2-d3+d4]](x-y-u+v2|Ω)=12gα12g/gβ12g/ge-2πiβ[c1+c2+c3+c4]θ[c1+αd1+β](x|Ω)×θ[c2+αd2+β](y|Ω)θ[c3+αd3+β](u|Ω)θ[c4+αd4+β](v|Ω).

§21.6(ii) Addition Formulas

Let α, β, γ, δg. Then

21.6.8 θ[αγ](z1|Ω)θ[βδ](z2|Ω)=νg/(2g)θ[12[α+β+ν]γ+δ](z1+z2|2Ω)θ[12[α-β+ν]γ-δ](z1-z2|2Ω).

Thus ν is a g-dimensional vector whose entries are either 0 or 1. For this result and a generalization see Koizumi (1976) and Belokolos et al. (1994, pp. 38–41). For addition formulas for classical theta functions see §20.7(ii).