21 Multidimensional Theta Functions Properties 21.5 Modular Transformations 21.7 Riemann Surfaces

§21.6 Products

Keywords:: Riemann theta functions
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.6

§21.6(i) Riemann Identity
§21.6(ii) Addition Formulas

§21.6(i) Riemann Identity

Notes:: See Mumford (1983, pp. 211–216).
Keywords:: Riemann identity, Riemann theta functions, Riemann theta functions with characteristics, matrix, vector
Permalink:: http://dlmf.nist.gov/21.6.i

Let $\mathbf{T}=[T_{{jk}}]$ be an arbitrary $h\times h$ orthogonal matrix (that is, $\mathbf{T}\mathbf{T}^{{\mathrm{T}}}=\mathbf{I}$ ) with rational elements. Also, let $\mathbf{Z}$ be an arbitrary $g\times h$ matrix. Define

21.6.1 $\mathcal{K}=\Integer^{{g\times h}}\mathbf{T}/(\Integer^{{g\times h}}\mathbf{T}% \cap\Integer^{{g\times h}}),$

Symbols:: $\Integer$ : set of all integers, $\cap$ : intersection, : positive integer, : positive integer and $\mathcal{K}$ : set of matrices
Permalink:: http://dlmf.nist.gov/21.6.E1
Encodings:: TeX, pMML, png

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

21.6.2 $\mathcal{D}=|\mathbf{T}^{{\mathrm{T}}}\Integer^{h}/(\mathbf{T}^{{\mathrm{T}}}% \Integer^{h}\cap\Integer^{h})|,$

Symbols:: $\Integer$ : set of all integers, $\cap$ : intersection, : positive integer and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E2
Encodings:: TeX, pMML, png

that is, $\mathcal{D}$ is the number of elements in the set containing all -dimensional vectors obtained by multiplying $\mathbf{T}^{{\mathrm{T}}}$ 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 $\prod_{{j=1}}^{h}\mathop{\theta\/}\nolimits\!\left(\sum_{{k=1}}^{h}T_{{jk}}% \mathbf{z}_{k}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}% \sum_{{\mathbf{A}\in\mathcal{K}}}\sum_{{\mathbf{B}\in\mathcal{K}}}e^{{2\pi i% \trace\left[\frac{1}{2}\mathbf{A}^{{\mathrm{T}}}\boldsymbol{{\Omega}}\mathbf{A% }+\mathbf{A}^{{\mathrm{T}}}[\mathbf{Z}+\mathbf{B}]\right]}}\*\prod_{{j=1}}^{h}% \mathop{\theta\/}\nolimits\!\left(\mathbf{z}_{j}+\boldsymbol{{\Omega}}\mathbf{% a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\in$ : element of, : base of exponential function, $\trace\mathbf{A}$ : trace of matrix $\mathbf{A}$ , : positive integer, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{{jk}}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E3
Encodings:: TeX, pMML, png

where $\mathbf{z}_{j}$ , $\mathbf{a}_{j}$ , $\mathbf{b}_{j}$ denote respectively the th columns of $\mathbf{Z}$ , $\mathbf{A}$ , $\mathbf{B}$ . This is the Riemann identity. On using theta functions with characteristics, it becomes

21.6.4 $\prod_{{j=1}}^{h}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\sum_{{k=1}}^{h}T_{{% jk}}\mathbf{c}_{k}}{\sum_{{k=1}}^{h}T_{{jk}}\mathbf{d}_{k}}\/}\nolimits\!\left% (\sum_{{k=1}}^{h}T_{{jk}}\mathbf{z}_{k}\middle|\boldsymbol{{\Omega}}\right)=% \frac{1}{\mathcal{D}^{g}}\sum_{{\mathbf{A}\in\mathcal{K}}}\sum_{{\mathbf{B}\in% \mathcal{K}}}e^{{-2\pi i\sum_{{j=1}}^{h}\mathbf{b}_{j}\cdot\mathbf{c}_{j}}}% \prod_{{j=1}}^{h}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\mathbf{a}_{j}+% \mathbf{c}_{j}}{\mathbf{b}_{j}+\mathbf{d}_{j}}\/}\nolimits\!\left(\mathbf{z}_{% j}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, : base of exponential function, : positive integer, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $T_{{jk}}$ : element, $\mathcal{K}$ : set of matrices and $\mathcal{D}$ : number of elements
Permalink:: http://dlmf.nist.gov/21.6.E4
Encodings:: TeX, pMML, png

where $\mathbf{c}_{j}$ and $\mathbf{d}_{j}$ are arbitrary -dimensional vectors. Many identities involving products of theta functions can be established using these formulas.

¶ Example

Let and

21.6.5 $\mathbf{T}=\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{bmatrix}.$

Permalink:: http://dlmf.nist.gov/21.6.E5
Encodings:: TeX, pMML, png

Then

21.6.6 $\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}\nolimits\!% \left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol% {{\Omega}}\right)\mathop{\theta\/}\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}% +\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/% }\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}\middle% |\boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum_{{\boldsymbol{{\alpha}}\in% \frac{1}{2}\Integer^{g}/\Integer^{g}}}\,\sum_{{\boldsymbol{{\beta}}\in\frac{1}% {2}\Integer^{g}/\Integer^{g}}}e^{{2\pi i\left(2\boldsymbol{{\alpha}}\cdot% \boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+\boldsymbol{{\alpha}}\cdot[% \mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]\right)}}\*\mathop{\theta\/}% \nolimits\!\left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}% \nolimits\!\left(\mathbf{y}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}% \nolimits\!\left(\mathbf{u}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\/}% \nolimits\!\left(\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\in$ : element of, : base of exponential function, $\Integer$ : set of all integers, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : -dimensional vector and $\boldsymbol{{\beta}}$ : -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E6
Encodings:: TeX, pMML, png

and

21.6.7 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}% _{2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2% }+\mathbf{d}_{3}+\mathbf{d}_{4}]}\/}\nolimits\!\left(\frac{\mathbf{x}+\mathbf{% y}+\mathbf{u}+\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta% \!\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c% }_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{% 3}-\mathbf{d}_{4}]}\/}\nolimits\!\left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\*\mathop{\theta\!\genfrac{[% }{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}+\mathbf{c}_{3}-% \mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}+\mathbf{d}_{3}-% \mathbf{d}_{4}]}\/}\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\!\genfrac{[}{% ]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+\mathbf{% c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+\mathbf{d}_% {4}]}\/}\nolimits\!\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}% \middle|\boldsymbol{{\Omega}}\right)\\ =\frac{1}{2^{g}}\sum_{{\boldsymbol{{\alpha}}\in\frac{1}{2}\Integer^{g}/% \Integer^{g}}}\,\sum_{{\boldsymbol{{\beta}}\in\frac{1}{2}\Integer^{g}/\Integer% ^{g}}}e^{{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_{2}+% \mathbf{c}_{3}+\mathbf{c}_{4}]}}\*\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{% \mathbf{c}_{1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\/}% \nolimits\!\left(\mathbf{x}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta% \!\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_{2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}% +\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{y}\middle|\boldsymbol{{% \Omega}}\right)\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_{3}+% \boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\/}\nolimits\!\left% (\mathbf{u}\middle|\boldsymbol{{\Omega}}\right)\mathop{\theta\!\genfrac{[}{]}{% 0.0pt}{}{\mathbf{c}_{4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{v}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, : base of exponential function, $\Integer$ : set of all integers, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : -dimensional vector and $\boldsymbol{{\beta}}$ : -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E7
Encodings:: TeX, pMML, png

§21.6(ii) Addition Formulas

Notes:: See Dubrovin (1981, pp. 22–23).
Keywords:: Riemann theta functions with characteristics
Permalink:: http://dlmf.nist.gov/21.6.ii

Let $\boldsymbol{{\alpha}}$ , $\boldsymbol{{\beta}}$ , $\boldsymbol{{\gamma}}$ , $\boldsymbol{{\delta}}\in\Real^{g}$ . Then

21.6.8 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \gamma}}}\/}\nolimits\!\left(\mathbf{z}_{1}\middle|\boldsymbol{{\Omega}}\right% )\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\beta}}}{\boldsymbol{{% \delta}}}\/}\nolimits\!\left(\mathbf{z}_{2}\middle|\boldsymbol{{\Omega}}\right% )=\sum_{{\boldsymbol{{\nu}}\in\Integer^{g}/\left(2\Integer^{g}\right)}}\mathop% {\theta\!\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[\boldsymbol{{\alpha}}+\boldsymbol% {{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{\gamma}}+\boldsymbol{{\delta}}}\/}% \nolimits\!\left(\mathbf{z}_{1}+\mathbf{z}_{2}\middle|2\boldsymbol{{\Omega}}% \right)\*\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\frac{1}{2}[\boldsymbol{{% \alpha}}-\boldsymbol{{\beta}}+\boldsymbol{{\nu}}]}{\boldsymbol{{\gamma}}-% \boldsymbol{{\delta}}}\/}\nolimits\!\left(\mathbf{z}_{1}-\mathbf{z}_{2}\middle% |2\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{% \beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\in$ : element of, $\Integer$ : set of all integers, : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : -dimensional vector, $\boldsymbol{{\beta}}$ : -dimensional vector, $\boldsymbol{{\delta}}$ : -dimensional vector and $\boldsymbol{{\gamma}}$ : -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E8
Encodings:: TeX, pMML, png

Thus $\boldsymbol{{\nu}}$ is a -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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 21.5 Modular Transformations 21.7 Riemann Surfaces

§21.6 Products

Contents

§21.6(i) Riemann Identity

¶ Example

§21.6(ii) Addition Formulas