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

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

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

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

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

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

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

 21.5.8 $\displaystyle\boldsymbol{{\Gamma}}$ $\displaystyle=\begin{bmatrix}\boldsymbol{{0}}_{g}&-\mathbf{I}_{g}\\ \mathbf{I}_{g}&\boldsymbol{{0}}_{g}\end{bmatrix}$$\Rightarrow$ $\displaystyle\mathop{\theta\/}\nolimits\!\left(\boldsymbol{{\Omega}}^{-1}% \mathbf{z}\middle|-\boldsymbol{{\Omega}}^{-1}\right)$ $\displaystyle=\sqrt{\det\left[-i\boldsymbol{{\Omega}}\right]}e^{\pi i\mathbf{z% }\cdot\boldsymbol{{\Omega}}^{-1}\cdot\mathbf{z}}\mathop{\theta\/}\nolimits\!% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

where the square root assumes its principal value.

# §21.5(ii) Riemann Theta Functions with Characteristics

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

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).