Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 336

is defined over L and has j -invariant equal to j L . Therefore, for any j there

is an elliptic curve with j -invariant equal to j defined over the field generated
by j .

∈

THEOREM 1 0.16

Let K = Q ( √ −

D ) be an imaginary quadraticfie d, let O K be the ring of
algebraicintegers in K ,and let j = j (

O K ) ,where O K is regarded as a lattice
in C .Let E be an elliptic curve defined over K ( j ) with j -invariant equalto
j .

1. A ssu m e K = Q ( i ) , Q ( e 2 πi/ 3 ) .Let F be the fiel d generated over K ( j ) by
the x -co o rd inates of the torsion pointsin E ( Q ) .Then F/K has abelian
Ga ois group, and every extension of K withabe ian G aloisgroupis
con tained in F .

2. If K = Q ( i ) ,the result of (1) holds w hen F isthe extension generated
by the squares of the x -co o rd inates of the torsion points.

3. If K = Q ( e 2 πi/ 3 ) ,the result of (1) holds w hen F isthe extension gen-
erated by the cubes of the x -co o rd inates of the torsion points.

O K )
is algebraic, by Proposition 10.4. The j -invariant determines the lattice for
the elliptic curve up to homothety (Corollary 9.20), so an elliptic curve with
invariant j ( O K ) automatically has complex multiplication by O K .
The x -coordinates of the torsion points are of the form

For a proof, see, for example, [111, p. 135] or [103]. Note that j (

℘ ( r 1 ω 1 + r 2 ω 2 ) , 1 ,r 2 ∈ Q ,

where ℘ is the Weierstrass ℘ -function for the lattice for E . Therefore, the
abelian extensions of K are generated by j ( O K ) and special values of the
function ℘ . This is very much the analogue of the Kronecker-Weber theorem.
There is much more that can be said on this subject. See, for example,
[111] and [70].

Exercises
10.1 Let K = Q ( √ d )and K = Q ( √ d ) be quadratic fields. Let β ∈ K
and β ∈ K and assume β, β ∈ Q . Suppose that β + β lies in a
quadratic fie ld . Show that K = K .( Hint: It suces to consider the

case β = a √ d and β = b √ d .Let α = β + β . Show that if α is a root
of a quadratic polyno mia l with coeci e nts in Q ,thenwecansolvefor

√ d ,say,intermsof √ d and obtain √ d ∈ K .)