Page 139 -

116 5. Reduction mod p and Torsion Points

(2) The restriction of the reduction homomorphism r p |E(Q) tors : E(Q) tors →
E (p) (F p ) is injective for any odd prime p where E has good reduction and
r 2 |E(Q) tors : E(Q) tors → E (2) (F 2 ) has kernel at most Z/2Z when E has good
reduction at 2.
Proof. The function T → x(T ) defines a monomorphism E (n) (Q)/E (2n) (Q) →
n
= Z/Zp by (4.5), and this implies that there is no torsion prime to p in
Zp /Zp 2n ∼
E (1) (Q) prime to p. Assume that pT = 0 where T ∈ E (r) (Q) − E (r+1) (Q) and
r ≥ 1. If p is odd, then we can use (4.5) and (4.6) to show that
3r
0 = x(pT ) ≡ px(T ) mod p .
Hence x(T ) ∈ p 3r−1 Z, and this means that T ∈ E (3r−1) (Q). This implies that
r ≤ 3r −1or2r ≤ 1sothat r = 0. If p = 2, then we can only use (4.5) to show that

2r
0 = x(2T ) ≡ 2x(T ) mod 2 .
Hence x(T ) ∈ 2 2r−1 Z, and this means that T ∈ E (2r−1) (Q). This implies that
r = 2r − 1 r = 1. Hence E(Q) tors has zero intersection with E (1) (Q) for p odd and
with E (2) (Q) for p = 2.
For the second assertion recall that ker(r p ) = E (1) (Q) at any prime p. The first
assertion implies that for good reduction at p the restriction r p |E(Q) tors has zero
kernel for p odd and kernel isomorphic to E(Q) tors ∩ E (1) (Q)/E (2) (Q) for p = 2.
The group E(Q) tors ∩ E (1) (Q)/E (2) (Q) injects into 2Z/4Z = Z/2Z by (4.5). This
proves the second assertion and the theorem.

(5.2) Remark. If C is a cubic curve defined by an equation over F q in normal form,
then for each x in F q we have at most two possible (x, y) on the curve C(F q ) and so
the cardinality #C(F q ) ≤ 2q + 1.
(5.3) Corollary. Let E be an elliptic curve over Q. If E has good reduction at an odd
prime p, then the cardinality of the torsion subgroup satisfies #E(Q) tors ≤ 2p + 1.
If E has good reduction at 2, then #E(Q) tors ≤ 10.

(5.4) Corollary. For an elliptic curve E over Q, the torsion subgroupE(Q) tors of
E(Q) is finite and is either cyclic or cyclic direct sum with Z/2Z.

Proof. Every elliptic curve has good reduction at some p giving the finiteness asser-
tion. Since E(Q) tors is a finite subgroup of E(R), we can apply (7.2) of the Intro-
duction and the structure of such finite subgroups of the cicle or the circle direct sum
with Z/2Z.
In the first few sections of Chapter 1, we introduced some elliptic curves in order
to see how the group law works, and now we return to these curves to illustrate the
use of Theorem (5.1) and some other general ideas.
3
2
(5.5) Example. The curve E : y + y −xy = x was considered in 1(1.6) where we
saw that (1, 1) generated a subgroup of order 6 in E(Q) tors . Reducing this curve mod
2
3
2, we apply the ideas in 3(6.4) to see that the curve is isomorphic to y +xy = x +x 2