## The Set of CM Elliptic Curves with Given CM

Definition 3.6 (Fractional Ideal)   A fractional ideal of a number field is an -submodule of that is isomorphic to as an abelian group. In particular, is nonzero.

If is a fractional ideal, the inverse of , which is the set of such that , is also a fractional ideal. Moreover, .

Fix a quadratic imaginary field . Let be the set of -isomorphism classes of elliptic curves with . By the above results we may also view as the set of lattices with .

If is a fractional ideal, then is a lattice in . For the elliptic curve we have

because is an -module by definition. Since rescaling a lattice produces an isomorphic elliptic curve, for any nonzero the fractional ideals and define the same elements of .

The class group is the group of fractional ideals modulo principal fractional ideals. If is a fractional ideal, denote by its ideal class in the class group of . We have a natural map

which sends to .

Theorem 3.7   Fix a quadratic imaginary field , and let be a lattice in such that . Let and be nonzero fractional -ideals. Then
1. is a lattice in ,
2. We have .
3. We have if and only if .
Thus there is a well-defined action of on given by

Theorem 3.8   The action of on is simply transitive.

Example 3.9   Let . Then the class number is . An elliptic curve with CM by is , and one can obtain the other two elements of by multiplying the lattice by two representative ideal classes for .

William 2007-05-25