§23.3 Differential Equations

§23.3(i) Invariants, Roots, and Discriminant

The lattice roots satisfy the cubic equation

23.3.3

and are denoted by . The discriminant1.11(ii)) is given by

23.3.4

In consequence,

23.3.5
23.3.6
23.3.7

Let , or equivalently be nonzero, or be distinct. Given and there is a unique lattice such that (23.3.1) and (23.3.2) are satisfied. We may therefore define

23.3.8

Similarly for and . As functions of and , and are meromorphic and is entire.

Conversely, , , and the set are determined uniquely by the lattice independently of the choice of generators. However, given any pair of generators , of , and with defined by (23.2.1), we can identify the individually, via

In what follows, it will be assumed that (23.3.9) always applies.

23.3.10
23.3.11
23.3.12
23.3.13