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21 Multidimensional Theta FunctionsProperties

§21.6 Products

Contents
  1. §21.6(i) Riemann Identity
  2. §21.6(ii) Addition Formulas

§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)=1𝒟g𝐀𝒦𝐁𝒦e2πij=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=12[1111111111111111].

Then

21.6.6 θ(𝐱+𝐲+𝐮+𝐯2|𝛀)θ(𝐱+𝐲𝐮𝐯2|𝛀)θ(𝐱𝐲+𝐮𝐯2|𝛀)θ(𝐱𝐲𝐮+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+𝐜2𝐜3𝐜4]12[𝐝1+𝐝2𝐝3𝐝4]](𝐱+𝐲𝐮𝐯2|𝛀)×θ[12[𝐜1𝐜2+𝐜3𝐜4]12[𝐝1𝐝2+𝐝3𝐝4]](𝐱𝐲+𝐮𝐯2|𝛀)θ[12[𝐜1𝐜2𝐜3+𝐜4]12[𝐝1𝐝2𝐝3+𝐝4]](𝐱𝐲𝐮+𝐯2|𝛀)=12g𝜶12g/g𝜷12g/ge2πi𝜷[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[𝜶𝜷+𝝂]𝜸𝜹](𝐳1𝐳2|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).