§21.3 Symmetry and Quasi-Periodicity

when ${𝐦}_1\in {\mathbb{Z}}^g.$ Thus $\theta \left(𝐳\right|𝛀)$ is periodic, with period $1$, in each element of $𝐳$. More generally,

with ${𝐦}_1$, ${𝐦}_2$ $\in {\mathbb{Z}}^g$. 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 ${𝐦}_1+𝛀{𝐦}_2$ form a $g$-dimensional lattice, the period lattice of the Riemann theta function.

Again, with ${𝐦}_1$, ${𝐦}_2$ $\in {\mathbb{Z}}^g$

Because of this property, the elements of $𝜶$ and $𝜷$ are usually restricted to $[0,1)$, without loss of generality.

For Riemann theta functions with half-period characteristics,

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