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§5. Torsion in Elliptic Curves over Q: Nagell–Lutz Theorem 115

2n
x(P) + x(Q) + x(T ) ≡ 0mod p R,
and this means that
2n
x(P + Q) + x(T ) ≡ x(P) + x(Q) mod p R.
2n
For the last statement note that x(P) ∈ p R if and only if P ∈ E (2n) (k). This
proves the theorem.

(4.6) Remark. The above result can be modified when a 1 = 0togiveaninjection
n
E (n) (k) p R
→ .
3n
E (3n) (k) p R
In the next section we will see one case where this modification is useful.

Exercises

1. Carry out the proofs of the modifications indicated in (4.4) and (4.6).
2. Using the notations in (2.1), show that R is a valuation ring with maximal ideal Rp if and
only if R (p) = R. Then show that Rp is the unique maximal ideal in R. Show that the
following is a filtration by subgroups of the multiplicative group k :
∗
n
2
∗
∗
k ⊃ R ⊃ 1 + Rp ⊃ 1 + Rp ⊃ ··· ⊃ 1 + Rp ⊃ ··· .
∗
∗
In this case show that ord p induces an isomorphism ord p : k /R → Z, reduction mod p
n
∗
∗
∗
induces an isomorphism R /(1 + Rp) → k(p) = (R/Rp) , and 1 + ap → a induces
n
an isomorphism (1 + Rp )/(1 + Rp n+1 ) → k(p) = R/Rp as additive groups.
3. Let ¯ E(k) ns denote the subgroup of nonsingular points of ¯ E(k). Show that E (0) (k) =
−1
r p ( ¯ E(k) ns ) is a subgroup of E(k) with zero equal to 0 : 0 : 1. Show that the restriction
r p : E (0) (k) → ¯ E(k(p)) ns is a group homomorphism.
§5. Torsion in Elliptic Curves over Q: Nagell–Lutz Theorem
Let Tors(A) or A tors denote the torsion subgroup of an abelian group A. In this sec-
tion we will see that E(Q) tors is a group that can be determined effectively from the
equation of the curve E, and is in particular a finite group.
(5.1) Theorem (Nagell–Lutz). Let E be an elliptic curve over the rational numbers
Q.
(1) The subgroupE(Q) tors ∩ E (1) (Q) is zero for each odd prime p and

E(Q) tors ∩ E (2) (Q)

is zero for p = 2.