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

E(Fp).

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)[5]| 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