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112 5. Reduction mod p and Torsion Points

3 2 2 3
ord p (w) = ord p x + a 2 wx + a 4 w x + a 6 w = 3 ord p (x)
where we have checked the four possible minima in

$ %
3
2
3
2
ord p (w) ≥ min ord p (x ), ord p (a 2 wx ), ord p (a 4 w x), ord p (a 6 w ) .
This proves the proposition.
The converse to (4.1) does not hold; it is possible to have ord p (x)> 0and
ord p (w) ≤ 0. For example, if all a i = 0 except a 6 in the normal form, then the cubic
reduces to
3
3
w = x + a 6 w ,
and we would have only the relation 0 = ord p (a 6 ) + 2 ord p (w).
The next definition leads to a sequence of subgroups of E(k) which are used to
¯
analyse ker(r p : E(k) → E(k(p))).
(4.2) Definition. Let E be a elliptic curve over k defined by a cubic in normal form.
The p-adic filtration on E is the sequence of subgroups E (n) (k) defined by the condi-
tion that w : x :1 ∈ E (n) (k) provided ord p (w) > 0 and ord p (x) ≥ n, or equivalently
provided ord p (w) ≥ 3n for n ≥ 1.
The equivalence of the two conditions for a point to be in E (n) (k) follows from
(4.1). Observe that E (1) (k) = r −1 (0), where 0 = 0:0:1 in E ¯ (n) (k(p)). We extend
p
(E(k(p)) ns ). The curve E has good
the definition to one other term E (0) (k) = r −1 ¯
p
reduction at p if and only if E(k) = E (0) (k). The subset E (0) (k) is a subgroup of
E(k) since a line L intersecting E(k) at one point of E(k) − E (0) (k) must intersect
it twice or at another point. We leave it to the reader to check in Exercise 3 that the
restriction r p : E (0) (k) → E(k) ns is a group morphism.
¯
(4.3) Proposition. Let P = w : x :1,P = w : x :1, and P = w : x :1 be

three points on the intersection E ∩ L over k, where E is an elliptic curve defined by

a cubic in normal form and L is a line. If P, P ∈ E (n) (k) for n ≥ 1, then we have

ord p (x + x + x ) ≥ 2n and ord p (w ) = 3 ord p (w ) ≥ 3n.

Proof. Let w = cx + b be the equation of the line L through the three points P, P ,

and P . Now we calculate c and ord p (c) using the equation of the cubic.

Case 1. P = P . Then c = (w−w )/(x −x ) is the slope of the line. Subtracting
the normal form of the cubic
2
3
2
2
w + a 1 wx + a 3 w = x + a 2 wx + a 4 w x + a 6 w 3
for P from the equation for P on the curve, we get

2
2
(w − w ) +· · · = (x − x )(x + xx + x ) +· · · .