Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 316

Then γx K . It follows that I 1 and I 2 are equivalent ideals.

Therefore, we have a bijection

∈ I 2 ⊂ K ,so γ ∈

Homothety classes of lattices L
with RL

Equivalence classes of
nonzero ideals of R

←→

⊆ L

It can be shown that the set of equivalence classes of ideals is finite (when
R = O K , this is just the finiteness of the class number). Therefore, the set of
homothety classes is finite. This observation has the following consequence.

PROPOSITION 10.4
Let R be an order inanimaginary quadraticfieldand et L be a lattice su ch
that RL ⊆ L .Then j ( L ) isalgebraic over Q .

PROOF Let E be the elliptic curve corresponding to L . We may assume
that E is given by an equation y 2 =4 x 3 −g 2 x−g 3 .Let σ be an automorphism

of C .Let E σ be the curve y 2 =4 x 3

−σ ( g 2 ) x−σ ( g 3 ). If α is an endomorphism
of E ,then α σ is an endomorphism of E σ ,where α σ means applying σ to all
of the coe cients of the rational functions describing α .Thisimpliesthat

End( E σ ) . End( E )

Therefore, the lattice corresponding to E σ belongs to one of the finitely many
homothety classes of lattices containing R in their endomorphism rings (there
is a technicality here; see Exercise 10.2). Since σ ( j ( L )) is the j -invariant
of E σ , we conclude that j ( L ) has only finitely many possible images under
automorphisms of C . This implies (see Appendix C) that j ( L ) is algebraic
over Q .

In Section 10.3, we'll prove the stronger result that j ( L ) is an algebraic
integer.

COROLLARY 10.5
Let K be an imaginary quadraticfie d.

1. Let τ ∈H .Then C / ( Z τ + Z ) has com plex m ultiplication by som e order
in K ifand onlyif τ ∈ K .

2. If τ

∈H iscontained in K ,then j ( τ ) isalgebraic.

PROOF We have already shown (see (10.1)) that if there is complex mul-
tiplication by an order in K then τ ∈ K . Conversely, suppose τ ∈ K .Then

τ satisfies a relation

aτ 2 + bτ + c,