# §21.5 Modular Transformations

## §21.5(i) Riemann Theta Functions

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

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

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

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

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

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

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

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

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).