21 Multidimensional Theta Functions Properties 21.4 Graphics 21.6 Products

§21.5 Modular Transformations

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

§21.5(i) Riemann Theta Functions
§21.5(ii) Riemann Theta Functions with Characteristics

§21.5(i) Riemann Theta Functions

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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

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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).

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§21.5 Modular Transformations

Contents

§21.5(i) Riemann Theta Functions

§21.5(ii) Riemann Theta Functions with Characteristics