21 Multidimensional Theta Functions Properties 21.4 Graphics 21.6 Products

§21.5 Modular Transformations

ⓘ

Permalink:: http://dlmf.nist.gov/21.5
See also:: Annotations for Ch.21

§21.5(i) Riemann Theta Functions

ⓘ

Keywords:: Riemann theta functions, matrix, modular group, modular transformations, symplectic
Notes:: See Arnol^′d (1997, p. 222) and Mumford (1983, pp. 189–210).
Permalink:: http://dlmf.nist.gov/21.5.i
See also:: Annotations for §21.5 and Ch.21

Let $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ , and $\mathbf{D}$ be $g\times g$ matrices with integer elements such that

21.5.1		$\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}$
ⓘ Defines: Matrix $\boldsymbol{{\Gamma}}$ (locally) Permalink: http://dlmf.nist.gov/21.5.E1 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

is a symplectic matrix, that is,

21.5.2		$\boldsymbol{{\Gamma}}\mathbf{J}_{2g}\boldsymbol{{\Gamma}}^{\mathrm{T}}=\mathbf% {J}_{2g}.$
ⓘ Symbols: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $g$ : positive integer and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E2 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

Then

21.5.3		$\det\boldsymbol{{\Gamma}}=1,$
ⓘ Symbols: $\det$ : determinant and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E3 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

and

21.5.4		$\theta\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{% \mathrm{T}}\mathbf{z}\middle\|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][% \mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right)=\xi(\boldsymbol{{% \Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{\pi i% \mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\mathbf{% C}\right]\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}% \right).$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, Matrix $\boldsymbol{{\Gamma}}$ and $\xi(\boldsymbol{{\Gamma}})$ : eighth root of unity Referenced by: §21.5(i) Permalink: http://dlmf.nist.gov/21.5.E4 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

Here $\xi(\boldsymbol{{\Gamma}})$ is an eighth root of unity, that is, $(\xi(\boldsymbol{{\Gamma}}))^{8}=1$ . For general $\boldsymbol{{\Gamma}}$ , it is difficult to decide which root needs to be used. The choice depends on $\boldsymbol{{\Gamma}}$ , but is independent of $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ . 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 $\xi(\boldsymbol{{\Gamma}})$ is determinate:

21.5.5		$\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{-1}]^{\mathrm{T}}\end{bmatrix}\Rightarrow% \theta\left(\mathbf{A}\mathbf{z}\middle\|\mathbf{A}\boldsymbol{{\Omega}}\mathbf% {A}^{\mathrm{T}}\right)=\theta\left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}% \right).$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E5 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

( $\mathbf{A}$ invertible with integer elements.)

21.5.6		$\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\theta\left(\mathbf% {z}\middle\|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\theta\left(\mathbf{z}% \middle\|\boldsymbol{{\Omega}}\right).$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E6 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

( $\mathbf{B}$ symmetric with integer elements and even diagonal elements.)

21.5.7		$\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\theta\left(\mathbf% {z}\middle\|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\theta\left(\mathbf{z}+% \tfrac{1}{2}\operatorname{diag}\mathbf{B}\middle\|\boldsymbol{{\Omega}}\right).$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\operatorname{diag}\mathbf{A}$ : Transpose of $[A_{11},A_{22},\ldots,A_{gg}]$ and Matrix $\boldsymbol{{\Gamma}}$ Referenced by: (21.5.8), Erratum (V1.0.11) for Clarifications Permalink: http://dlmf.nist.gov/21.5.E7 Encodings: TeX, pMML, png See also: Annotations for §21.5(i), §21.5 and Ch.21

( $\mathbf{B}$ symmetric with integer elements.) See Heil (1995, p. 24). For a $g\times g$ matrix ${\bf A}$ we define $\operatorname{diag}\mathbf{A}$ , as a column vector with the diagonal entries as elements.

21.5.8	$\displaystyle\boldsymbol{{\Gamma}}$	$\displaystyle=\begin{bmatrix}\boldsymbol{{0}}_{g}&-\mathbf{I}_{g}\\ \mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ $\Rightarrow$
21.5.8	$\displaystyle\theta\left(\boldsymbol{{\Omega}}^{-1}\mathbf{z}\middle\|-% \boldsymbol{{\Omega}}^{-1}\right)$	$\displaystyle=\sqrt{\det\left[-i\boldsymbol{{\Omega}}\right]}e^{\pi i\mathbf{z% }\cdot\boldsymbol{{\Omega}}^{-1}\cdot\mathbf{z}}\theta\left(\mathbf{z}\middle\|% \boldsymbol{{\Omega}}\right),$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png Addition (effective with 1.0.11): A sentence was added at the end of (21.5.7) to define a diagonal matrix. Suggested 2015-10-27 by Howard Cohl and Adri Olde Daalhuis See also: Annotations for §21.5(i), §21.5 and Ch.21

where the square root assumes its principal value.

§21.5(ii) Riemann Theta Functions with Characteristics

ⓘ

Keywords:: Riemann theta functions, Riemann theta functions with characteristics, modular transformations
Notes:: See Igusa (1972, pp. 78–85).
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.5.ii
See also:: Annotations for §21.5 and Ch.21

21.5.9		$\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}% \boldsymbol{{\beta}}+\tfrac{1}{2}\operatorname{diag}[\mathbf{C}\mathbf{D}^{% \mathrm{T}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+% \tfrac{1}{2}\operatorname{diag}[\mathbf{A}\mathbf{B}^{\mathrm{T}}]}\left(\left% [[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{\mathrm{T}}\mathbf{% z}\middle\|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{% \Omega}}+\mathbf{D}]^{-1}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{% \beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]}e^{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]^{-1}\mathbf{C}\right]\cdot\mathbf{z}}\theta\genfrac{[}{]}{0.0pt}{}% {\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle\|% \boldsymbol{{\Omega}}\right),$
ⓘ Symbols: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\operatorname{diag}\mathbf{A}$ : Transpose of $[A_{11},A_{22},\ldots,A_{gg}]$ and Matrix $\boldsymbol{{\Gamma}}$ Permalink: http://dlmf.nist.gov/21.5.E9 Encodings: TeX, pMML, png See also: Annotations for §21.5(ii), §21.5 and Ch.21

where $\kappa(\boldsymbol{{\alpha}},\boldsymbol{{\beta}},\boldsymbol{{\Gamma}})$ is a complex number that depends on $\boldsymbol{{\alpha}}$ , $\boldsymbol{{\beta}}$ , and $\boldsymbol{{\Gamma}}$ . However, $\kappa(\boldsymbol{{\alpha}},\boldsymbol{{\beta}},\boldsymbol{{\Gamma}})$ is independent of $\mathbf{z}$ and $\boldsymbol{{\Omega}}$ . For explicit results in the case $g=1$ , see §20.7(viii).

21.4 Graphics 21.6 Products

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov

§21.5 Modular Transformations

Contents

§21.5(i) Riemann Theta Functions

§21.5(ii) Riemann Theta Functions with Characteristics

pFad - (p)hone/(F)rame/(a)nonymizer/(d)eclutterfier! Saves Data!