# §21.3 Symmetry and Quasi-Periodicity

## §21.3(i) Riemann Theta Functions

when Thus is periodic, with period 1, in each element of . More generally,

with , . 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 form a -dimensional lattice, the period lattice of the Riemann theta function.

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

Again, with ,

21.3.4

Because of this property, the elements of and are usually restricted to , without loss of generality.

For Riemann theta functions with half-period characteristics,