Complex Multiplication - Singular Moduli

Singular Moduli

The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers. The corresponding modular invariants j(τ) are the singular moduli, coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.

The modular function j(τ) is algebraic on imaginary quadratic numbers τ: these are the only algebraic numbers in the upper half-plane for which j is algebraic.

If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in a ring of integers O of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus. The values j(a) are then real algebraic integers, and generate the Hilbert class field H of K: the field extension degree = h is the class number of K and the H/K is a Galois extension with Galois group isomorphic to the ideal class group of K. The class group acts on the values j(a) by : j(a) → j(ab).

In particular, if K has class number one, then j(a) = j(O) is a rational integer: for example, j(Z) = j(i) = 1728.