§21.3 Symmetry and Quasi-Periodicity

Permalink:: http://dlmf.nist.gov/21.3

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

§21.3(i) Riemann Theta Functions

Notes:: See Mumford (1983, pp. 120–122).
Keywords:: Riemann theta functions
Permalink:: http://dlmf.nist.gov/21.3.i

21.3.1 $\mathop{\theta\/}\nolimits\!\left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E1
Encodings:: TeX, pMML, png

21.3.2 $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}+\mathbf{m}_{1}\middle|\boldsymbol{{\Omega}}\right)=\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E2
Encodings:: TeX, pMML, png

when $\mathbf{m}_{1}\in\Integer^{g}.$ Thus $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ is periodic, with period 1, in each element of $\mathbf{z}$ . More generally,

21.3.3 $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle|\boldsymbol{{\Omega}}\right)=e^{{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}\right)}}\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right),$

Symbols:: $\mathop{\theta\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function, $e$ : base of exponential function and $\boldsymbol{{\Omega}}$ : a Riemann matrix
Permalink:: http://dlmf.nist.gov/21.3.E3
Encodings:: TeX, pMML, png

with $\mathbf{m}_{1}$ , $\mathbf{m}_{2}$ $\in\Integer^{g}$ . This is the quasi-periodicity property of the Riemann theta function. It determines the Riemann theta function up to a constant factor. The set of points $\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}$ form a $g$ -dimensional lattice, the period lattice of the Riemann theta function.

§21.3(ii) Riemann Theta Functions with Characteristics

Notes:: See Mumford (1983, pp. 120–122).
Keywords:: Riemann theta functions, Riemann theta functions with characteristics, characteristics
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.3.ii

Again, with $\mathbf{m}_{1}$ , $\mathbf{m}_{2}$ $\in\Integer^{g}$

21.3.4 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{\boldsymbol{{\beta}}+\mathbf{m}_{2}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Referenced by:: Version 1.0.5 (October 1, 2012)
Permalink:: http://dlmf.nist.gov/21.3.E4
Encodings:: TeX, pMML, png
Errata (effective with 1.0.5):: Originally the vector $\mathbf{m}_{2}$ on the right-hand-side was given incorrectly as $\mathbf{m}_{1}$ .
Reported 2012-08-27 by Klaas Vantournhout

Because of this property, the elements of $\boldsymbol{{\alpha}}$ and $\boldsymbol{{\beta}}$ are usually restricted to $[0,1)$ , without loss of generality.

21.3.5 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle|\boldsymbol{{\Omega}}\right)=e^{{2\pi i\left(\boldsymbol{{\alpha}}\cdot\mathbf{m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}\cdot\mathbf{z}\right)}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $e$ : base of exponential function, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E5
Encodings:: TeX, pMML, png

For Riemann theta functions with half-period characteristics,

21.3.6 $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-1)^{{4\boldsymbol{{\alpha}}\cdot\boldsymbol{{\beta}}}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).$

Symbols:: $\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)$ : Riemann theta function with characteristics, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.3.E6
Encodings:: TeX, pMML, png

See also §20.2(iii) for the case $g=1$ and classical theta functions.

§21.3 Symmetry and Quasi-Periodicity

Contents

§21.3(i) Riemann Theta Functions

§21.3(ii) Riemann Theta Functions with Characteristics