§21.5 Modular Transformations

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

21.5.1
ⓘ Defines: Matrix $\mathbf{\Gamma }$ (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		$$\mathbf{\Gamma }{\mathbf{J}}_{2g}{\mathbf{\Gamma }}^{\mathrm{T}}={\mathbf{J}}_{2g}.$$
ⓘ Symbols: $g$: positive integer and Matrix $\mathbf{\Gamma }$ 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		$$det\mathbf{\Gamma }=1,$$
ⓘ Symbols: $det$: determinant and Matrix $\mathbf{\Gamma }$ 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		$$\begin{array}{l}\theta \left({\left[{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1}\right]}^{\mathrm{T}}\mathbf{z}\right|[\mathbf{A}\mathbf{\Omega }+\mathbf{B}]{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1})\\ =\xi (\mathbf{\Gamma })\sqrt{det[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}{\mathrm{e}}^{\pi \mathrm{i}\mathbf{z}\cdot \left[{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1}\mathbf{C}\right]\cdot \mathbf{z}}\theta \left(\mathbf{z}\right|\mathbf{\Omega }).\end{array}$$
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: 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, $\mathbf{\Omega }$: a Riemann matrix, Matrix $\mathbf{\Gamma }$ and $\xi (\mathbf{\Gamma })$: 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 $\xi (\mathbf{\Gamma })$ is an eighth root of unity, that is, ${(\xi (\mathbf{\Gamma }))}^8=1$. For general $\mathbf{\Gamma }$, it is difficult to decide which root needs to be used. The choice depends on $\mathbf{\Gamma }$, but is independent of $\mathbf{z}$ and $\mathbf{\Omega }$. 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 $\xi (\mathbf{\Gamma })$ is determinate:

21.5.5
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $g$: positive integer, $\mathbf{\Omega }$: a Riemann matrix and Matrix $\mathbf{\Gamma }$ 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
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $g$: positive integer, $\mathbf{\Omega }$: a Riemann matrix and Matrix $\mathbf{\Gamma }$ 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
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: Riemann theta function, $g$: positive integer, $\mathbf{\Omega }$: a Riemann matrix, $\mathrm{diag}\mathbf{A}$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $\mathbf{\Gamma }$ 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 $\mathbf{A}$ we define $\mathrm{diag}\mathbf{A}$, as a column vector with the diagonal entries as elements.

21.5.8		$\mathbf{\Gamma }$	$\Rightarrow$
		$\theta \left({\mathbf{\Omega }}^{-1}\mathbf{z}\right|-{\mathbf{\Omega }}^{-1})$	$=\sqrt{det\left[-\mathrm{i}\mathbf{\Omega }\right]}{\mathrm{e}}^{\pi \mathrm{i}\mathbf{z}\cdot {\mathbf{\Omega }}^{-1}\cdot \mathbf{z}}\theta \left(\mathbf{z}\right|\mathbf{\Omega }),$
ⓘ Symbols: $\theta \left(\mathbf{z}\right|\mathbf{\Omega })$: 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, $\mathbf{\Omega }$: a Riemann matrix and Matrix $\mathbf{\Gamma }$ 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.

21.5.9		$$\begin{array}{l}\theta \left[\genfrac{}{}{0.0pt}{}{\mathbf{D}\mathit{\alpha }-\mathbf{C}\mathit{\beta }+\frac{1}{2}\mathrm{diag}[\mathbf{C}{\mathbf{D}}^{\mathrm{T}}]}{-\mathbf{B}\mathit{\alpha }+\mathbf{A}\mathit{\beta }+\frac{1}{2}\mathrm{diag}[\mathbf{A}{\mathbf{B}}^{\mathrm{T}}]}\right]\left({\left[{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1}\right]}^{\mathrm{T}}\mathbf{z}\right|[\mathbf{A}\mathbf{\Omega }+\mathbf{B}]{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1})\\ =\kappa (\mathit{\alpha },\mathit{\beta },\mathbf{\Gamma })\sqrt{det[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}{\mathrm{e}}^{\pi \mathrm{i}\mathbf{z}\cdot \left[{[\mathbf{C}\mathbf{\Omega }+\mathbf{D}]}^{-1}\mathbf{C}\right]\cdot \mathbf{z}}\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega }),\end{array}$$
ⓘ Symbols: $\theta \left[\genfrac{}{}{0.0pt}{}{\mathit{\alpha }}{\mathit{\beta }}\right]\left(\mathbf{z}\right|\mathbf{\Omega })$: 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, $\mathbf{\Omega }$: a Riemann matrix, $\mathit{\alpha }$: $g$-dimensional vector, $\mathit{\beta }$: $g$-dimensional vector, $\mathrm{diag}\mathbf{A}$: Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$ and Matrix $\mathbf{\Gamma }$ 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 $\kappa (\mathit{\alpha },\mathit{\beta },\mathbf{\Gamma })$ is a complex number that depends on $\mathit{\alpha }$, $\mathit{\beta }$, and $\mathbf{\Gamma }$. However, $\kappa (\mathit{\alpha },\mathit{\beta },\mathbf{\Gamma })$ is independent of $\mathbf{z}$ and $\mathbf{\Omega }$. For explicit results in the case $g=1$, see §20.7(viii).