# §21.5 Modular Transformations

## §21.5(i) Riemann Theta Functions

Let , , , and be matrices with integer elements such that

21.5.1

is a symplectic matrix, that is,

21.5.2

Then

21.5.3

and

Here is an eighth root of unity, that is, . For general , it is difficult to decide which root needs to be used. The choice depends on , but is independent of and . 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 is determinate:

( invertible with integer elements.)

( symmetric with integer elements and even diagonal elements.)

( symmetric with integer elements.) See Heil (1995, p. 24).

where the square root assumes its principal value.

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

where is a complex number that depends on , , and . However, is independent of and . For explicit results in the case , see §20.7(viii).