Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 335

then find a new set of τ k corresponding to the discriminant − 4 · 171, a new

j -invariant mod p ,andanew E .

If we had used R = Z 1+ √ − 1 2 , then we would have obtained an elliptic
curve with group isomorphic to Z 6 ⊕ 1(mod6 R )inthis

Z 42 ,since φ p ≡

case.
This technique has many uses. For example, in [100], the curve E defined

by

y 2 = x 3 +3 x − 31846

(mod 158209)

was dedicated to Arjen Lenstra on the occasion of his thesis defense on May
16, 1984. The curve satisfies

E ( F 158209 ) Z 5 ⊕ Z 16 ⊕ Z 1984 .

(If the defense had been one month later, such a dedication would have been
impossible.) Finding elliptic curves with groups that are cyclic of large prime
order is very useful in cryptography (see Chapter 6). Finding elliptic curves of
a given order is also useful in primality proving (see Section 7.2). A detailed
discussion of the problem, with improvements on the method presented here,
is given in [73]. See also [7], [8].

10.5 Kronecker's Jugendtraum

The Kronecker-Weber theorem says that if K/ Q is a finite Galois extension
with abelian Galois group, then

Q ( e 2 πi/n )

K ⊆

for some integer n . This can be viewed as saying that the abelian extensions of
Q are generated by the values of an analytic function, namely e 2 πiz ,atrational
numbers. Kronecker's Jugendtraum (youthful dream) is that the abelian
extensions of an arbitrary number field might similarly be generated by special
values of a naturally occurring function. This has been accomplished for
imaginary quadratic fields. Some progress has also been made for certain other
fields by Shimura using complex multiplication of abelian varieties (higher
dimensional analogues of elliptic curves).
If E is an elliptic curve given by y 2 = x 3 + Ax + B ,thenits j -invariant is
given by j = 6912 A 3 / (4 A 3 +27 B 2 ). Therefore, if E is defined over a field L ,
then the j -invariant of E is contained in L . Conversely, if j =0 , 1728 lies in
some field L , then the elliptic curve

3 j
1728 − j x +

2 j
1728 − j

y 2 = x 3 +