§21.5 Modular Transformations

Permalink:: http://dlmf.nist.gov/21.5

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

§21.5(i) Riemann Theta Functions

Notes:: See Arnol ${}^{{\prime}}$ d (1997, p. 222) and Mumford (1983, pp. 189–210).
Keywords:: Riemann theta functions, matrix
Permalink:: http://dlmf.nist.gov/21.5.i

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

21.5.1 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}$

Symbols:: Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E1
Encodings:: TeX, pMML, png

is a symplectic matrix, that is,

21.5.2 $\boldsymbol{{\Gamma}}\mathbf{J}_{{2g}}\boldsymbol{{\Gamma}}^{{\mathrm{T}}}=\mathbf{J}_{{2g}}.$

Symbols:: $g$ : positive integer and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E2
Encodings:: TeX, pMML, png

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

and

21.5.4 $\mathop{\theta\/}\nolimits\!\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\right]^{{\mathrm{T}}}\mathbf{z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\right)=\xi(\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\mathbf{C}\right]\cdot\mathbf{z}}}\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\det$ : determinant, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : 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

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 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{A}&\boldsymbol{{0}}_{g}\\ \boldsymbol{{0}}_{g}&[\mathbf{A}^{{-1}}]^{{\mathrm{T}}}\end{bmatrix}\Rightarrow\mathop{\theta\/}\nolimits\!\left(\mathbf{A}\mathbf{z}\middle|\mathbf{A}\boldsymbol{{\Omega}}\mathbf{A}^{{\mathrm{T}}}\right)=\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E5
Encodings:: TeX, pMML, png

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

21.5.6 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : 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

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

21.5.7 $\boldsymbol{{\Gamma}}=\begin{bmatrix}\mathbf{I}_{g}&\mathbf{B}\\ \boldsymbol{{0}}_{g}&\mathbf{I}_{g}\end{bmatrix}\Rightarrow\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}+\mathbf{B}\right)=\mathop{\theta\/}\nolimits\!\left(\mathbf{z}+\tfrac{1}{2}\diag\mathbf{B}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E7
Encodings:: TeX, pMML, png

( $\mathbf{B}$ symmetric with integer elements.) See Heil (1995, p. 24).

21.5.8

$\boldsymbol{{\Gamma}}=\begin{bmatrix}\boldsymbol{{0}}_{g}&-\mathbf{I}_{g}\\ \mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$ $\Rightarrow$

$\mathop{\theta\/}\nolimits\!\left(\boldsymbol{{\Omega}}^{{-1}}\mathbf{z}\middle|-\boldsymbol{{\Omega}}^{{-1}}\right)=\sqrt{\det\left[-i\boldsymbol{{\Omega}}\right]}e^{{\pi i\mathbf{z}\cdot\boldsymbol{{\Omega}}^{{-1}}\cdot\mathbf{z}}}\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $\det$ : determinant, $e$ : base of exponential function, $g$ : 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

where the square root assumes its principal value.

§21.5(ii) Riemann Theta Functions with Characteristics

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

21.5.9 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}\boldsymbol{{\beta}}+\tfrac{1}{2}\diag[\mathbf{C}\mathbf{D}^{{\mathrm{T}}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+\tfrac{1}{2}\diag[\mathbf{A}\mathbf{B}^{{\mathrm{T}}}]}\/}\nolimits\!\left(\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\right]^{{\mathrm{T}}}\mathbf{z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{\beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]}e^{{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{{-1}}\mathbf{C}\right]\cdot\mathbf{z}}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\det$ : determinant, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E9
Encodings:: TeX, pMML, png

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).

§21.5 Modular Transformations

Contents

§21.5(i) Riemann Theta Functions

§21.5(ii) Riemann Theta Functions with Characteristics