Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 242

Therefore, the number of points on C is less than or equal to the number of
points on E . By Hasse's theorem,

q +1+2 √ q. 1) = N ≤

2( q −

This may be rearranged to obtain
( √ q

1) 2 − ≤ 4 ,

which yields q ≤ 9. Therefore, if q ≥ 11, we must have N =0.

It remains to treat the cases q =3 , 5 , 7. First, suppose p is a square mod
q . There are no points on C 1 ,p,p with coordinates in F 3 , for example, so we
introduce denominators. Let's try

u = u 1 /q, v =1 /q, w = w 1 /q.

Then we want to solve

w 1 =1+4 q 2 ,

1 = p +2 pq 2 .

Since p is assumed to be a square mod q , Hensel's lemma implies that there
are q -adic solutions u 1 ,w 1 .
Now suppose that p is not a square mod q . Divide the second equation in
(8.15) by p to obtain

1

w 2 − v 2 =4 ,

p u 2

− v 2 =2 .

nx 2

Let n be any fixed quadratic nonresidue mod q ,andwrite1 /p ≡ (mod q ).

Letting u 1 = xu ,weobtain

w 2 v 2 =4 , v 2 =2 . − n −

1

For q =3and q =5,wemaytake n = 2 and obtain

w 2 − v 2 2 u 1 − v 2

≡ 4 , ≡ 2(mod q ) .

This has the solution ( u 1 ,v,w )=(1 , 0 , 2). As above, Hensel's lemma yields a
q -adic solution.
For q =7,take n = 3 to obtain

w 2 v 2 3 u 1 − v 2

− ≡ 4 , ≡ 2(mod .

This has the solution ( u 1 ,v,w )=(3 , 2 , 1), which yields a 7-adic solution.
Therefore, we have shown that there is a q -adic solution for all q ≤∞ .This

completes the proof of Theorem 8.28.