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 3.8
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