2.6. Elliptic curves over finite fields 41
where all the coordinates should be regarded as congruences modulo
5. Thus, N5 = 6, which is in the range given by Hasse’s bound:
1.5278 . . . = 5 + 1 − 2
5 N5 5 + 1 + 2
5 = 10.4721 . . . .
Similarly, one can verify that N7 = 13.
The connections between the numbers Np and the group E(Q)
are numerous and of great interest. The most surprising relationship
is captured by the Birch and Swinnerton-Dyer conjecture (Conjecture
5.2.1) that relates the growth of Np (as p varies) with the rank of the
elliptic curve E/Q. We shall discuss this conjecture in Section 5.2 in
more detail. In the next proposition we describe a different connection
between Np and E(Q). We shall use the following notation: if G is
an abelian group and m ≥ 2, then the points of G of order dividing
m will be denoted by G[m].
Proposition 2.6.15 ([Sil86], Ch. VII, Prop. 3.1). Let E/Q be an
elliptic curve, p a prime number and m a natural number not divisible
by p. Suppose that E/Q has good reduction at p. Then the reduction
map modulo p,
E(Q)[m] −→ E(Fp),
is an injective homomorphism of abelian groups. In particular, the
number of elements of E(Q)[m] divides the number of elements of
The previous proposition can be very useful when calculating the
torsion subgroup of an elliptic curve. Let’s see an application:
Example 2.6.16. Let E/Q: y2 = x3 +3. In Example 2.6.14 we have
seen that N5 = 6 and N7 = 13, and E/Q has bad reduction only at
2 and 3.
If q = 5,7 is a prime number, then E(Q)[q] is trivial. Indeed,
Proposition 2.6.15 implies that |E(Q)[q]| divides N5 = 6 and also
N7 = 13. Thus, |E(Q)[q]| must divide gcd(6,13) = 1.
In the case of q = 5, we know that |E(Q)| divides N7 = 13.
Moreover, by Lagrange’s theorem from group theory, if E(Q)[p] is
non-trivial, then p divides |E(Q)[p]| (later on we will see that E(Q)[p]
is always a subgroup of Z/pZ × Z/pZ; see Exercise 3.7.5). Since 5