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114 5. Reduction mod p and Torsion Points

2
3
2
3
= x + a 2 (cx + b)x + a 4 (cx + b) x + a 6 (cx + b) .
Collecting coefficients of powers of x, we have the following first two terms:

3 2 3
0 = x 1 + a 2 c + a 4 c + a 6 c

2 2 2
+ x a 2 b + 2a 4 bc + 3a 6 bc − a 1 c − a 3 c +· · · .

The sum of the roots of this cubic equation in x is x + x + x the x-coordinates of

P, P ,and P and is given by the following expression:

2
a 2 b + 2a 4 bc + 3a 6 bc − a 1 c − a 3 c 2

x + x + x =− .
2
1 + a 2 c + a 4 c + a 6 c 3
Since the denominator is of the form 1 + u where ord p (u)> 0, we see that
3
2
ord p (1 + a 2 c + a 4 c + a 6 c ) = 0.

From the relations ord p (b) ≥ 3n and ord p (c) ≥ 2n, we deduce that ord p (x + x +

x ) ≥ 2n.
For the last statement of the proposition observe that

ord p (x ) ≥ min ord p (x + x + x ), ord p (−x), ord p (−x ) ≥ n.

Since w = cx + b, or from (4.1), it is now clear that ord p (w ) ≥ 3n and the point
(w , x , 1) is in E (n) (k). This proves the proposition.

(4.4) Remark. At the end of the proof of the previous proposition, we see that

ord p (x + x + x ) ≥ 3n whenever a 1 = 0, so that the term a 1 c is not in the nu-

merator of the expression for x + x + x .
(4.5) Theorem. Let E be an elliptic curve over k defined by a cubic in normal
form with p-adic filtration E (n) (k) on E(k). The subsets E (n) (k) are subgroups.
n
The function P = (w, x, 1) → x(P) = x defined E (n) (k) → p R composed with
n
2n
n
the canonical quotient morphism p R → p R/p R defines a groupmorphism
n
2n
E (n) (k) → p R/p R with kernel in E (2n) (k) and it induces a monomorphism
n
2n
E (n) (R)/E (2n) (k) → p R/p R for n ≥ 1.
Proof. For the first statement observe that the condition on P = (w, x, 1) where
ord p (w) ≥ 3n and ord p (x) ≥ n is equivalent for the coordinates P(1, x, y) to the
condition ord p (x) ≤−2n and ord p (y) ≤−3n. Since
−(1, x, y) = (1, x, −y − a 1 x − a 3 ),
it follows that the sets E (n) (k) are stable under taking inverse of elements. This to-
gether with (4.3) implies that E (n) (k) is a subgroup for n ≥ 1.
For the second statement, consider PQ = T . By (4.3) we have