Let , , , and be matrices with integer elements such that
is a symplectic matrix, that is,
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). For a matrix we define , as a column vector with the diagonal entries as elements.
where the square root assumes its principal value.
where is a complex number that depends on , , and . However, is independent of and . For explicit results in the case , see §20.7(viii).