21 Multidimensional Theta Functions Properties 21.2 Definitions 21.4 Graphics

§21.3 Symmetry and Quasi-Periodicity

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Permalink:: http://dlmf.nist.gov/21.3
See also:: Annotations for Ch.21

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

§21.3(i) Riemann Theta Functions

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Keywords:: Riemann theta functions, period lattice, quasi-periodicity, symmetry
Notes:: See Mumford (1983, pp. 120–122).
Permalink:: http://dlmf.nist.gov/21.3.i
See also:: Annotations for §21.3 and Ch.21

21.3.1		$\theta\left(-\mathbf{z}\middle\|\boldsymbol{{\Omega}}\right)=\theta\left(% \mathbf{z}\middle\|\boldsymbol{{\Omega}}\right),$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix Permalink: http://dlmf.nist.gov/21.3.E1 Encodings: TeX, pMML, png See also: Annotations for §21.3(i), §21.3 and Ch.21

21.3.2		$\theta\left(\mathbf{z}+\mathbf{m}_{1}\middle\|\boldsymbol{{\Omega}}\right)=% \theta\left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}\right),$
ⓘ Symbols: $\theta\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function and $\boldsymbol{{\Omega}}$ : a Riemann matrix Permalink: http://dlmf.nist.gov/21.3.E2 Encodings: TeX, pMML, png See also: Annotations for §21.3(i), §21.3 and Ch.21

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

21.3.3		$\theta\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}% \middle\|\boldsymbol{{\Omega}}\right)=e^{-2\pi i\left(\frac{1}{2}\mathbf{m}_{2}% \cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}+\mathbf{m}_{2}\cdot\mathbf{z}% \right)}\theta\left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}\right),$
ⓘ 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, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and $\boldsymbol{{\Omega}}$ : a Riemann matrix Permalink: http://dlmf.nist.gov/21.3.E3 Encodings: TeX, pMML, png See also: Annotations for §21.3(i), §21.3 and Ch.21

with $\mathbf{m}_{1}$ , $\mathbf{m}_{2}$ $\in{\mathbb{Z}}^{g}$ . 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

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Keywords:: Riemann theta functions, Riemann theta functions with characteristics, characteristics, quasi-periodicity, symmetry
Notes:: See Mumford (1983, pp. 120–122).
Referenced by:: §20.11(iv)
Permalink:: http://dlmf.nist.gov/21.3.ii
See also:: Annotations for §21.3 and Ch.21

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

21.3.4		$\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{% \boldsymbol{{\beta}}+\mathbf{m}_{2}}\left(\mathbf{z}\middle\|\boldsymbol{{% \Omega}}\right)=e^{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2}}\theta% \genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(% \mathbf{z}\middle\|\boldsymbol{{\Omega}}\right).$
ⓘ Symbols: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector Referenced by: Erratum (V1.0.5) for Equation (21.3.4) 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 See also: Annotations for §21.3(ii), §21.3 and Ch.21

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		$\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}\mathbf{m}_{2}\middle\|% \boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\boldsymbol{{\alpha}}\cdot\mathbf{% m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}{2}\mathbf{m}_{2}\cdot% \boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}\cdot\mathbf{z}\right)}% \theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}\right).$
ⓘ Symbols: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\boldsymbol{{\Omega}}$ : 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 See also: Annotations for §21.3(ii), §21.3 and Ch.21

For Riemann theta functions with half-period characteristics,

21.3.6		$\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(-\mathbf{z}\middle\|\boldsymbol{{\Omega}}\right)=(-1)^{4\boldsymbol{{% \alpha}}\cdot\boldsymbol{{\beta}}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{% \alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle\|\boldsymbol{{\Omega}}% \right).$
ⓘ Symbols: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle\|\NVar{\boldsymbol{{\Omega}}}\right)$ : 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: Annotations for §21.3(ii), §21.3 and Ch.21

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

21.2 Definitions 21.4 Graphics

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§21.3 Symmetry and Quasi-Periodicity

Contents

§21.3(i) Riemann Theta Functions

§21.3(ii) Riemann Theta Functions with Characteristics