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§4. The Kernel of Reduction mod p and the p-Adic Filtration 113

Each term in the previous relation is of the form

a
a b
a b
a
b
b
b
a
w x − w x = (w − w )x + w (x − x ).
Hence the difference of the two normal forms has the form
2
2

(w − w )(1 + u) = (x − x )(x + xx + x + v), for u,v ∈ k,
where ord p (u)> 0 so that ord p (1 + u) = 0. Further, each term of v is divisible

by some w or w so that ord p (v) ≥ 3n. Since ord p (x) and ord p (x ) ≥ n,wehave
2
2

ord p (x + xx + x + v) ≥ 2n. Thus we obtain the following inequality:
w − w 2 2

ord p (c) = ord p ≥ ord p (x + xx + x − v) − ord p (1 + u)
x − x
≥ 2n.

Case 2. P = P . Then c = dw/dx. Differentiate the normal form of the cubic
implicitly
dw dw dw
+ a 1 w + x + 2a 3 w
dx dx dx
dw 2 dw 2 dw

2
2
= 3x + a 2 2wx + x + a 4 w + 2wx + 3a 6 w ,
dx dx dx
and collecting the terms we obtain
dw
2 2
1 + a 1 x + 2a 3 w − a 2 x − 2a 4 wx − 3a 6 w
dx
2
2
= 3x + 2a 2 xw + a 4 w − a 1 w.
The coefficient of dw/dx has the form 1 + u where ord p (u)> 0 and this means that
2 2
ord p (1 + u) = 0. The right-hand side has the form 3x + v where ord p (3x + v) ≥
2
ord p (3x ) ≥ 2ord p (x) since ord p (w) = 3n, and thus we have the inequality

dw 2
ord p (c) = ord p = ord p (3x + v) ≥ 2n.
dx
Therefore in both cases ord p (c) ≥ 2n. As for the coefficient b in the equation of

the line through P and P ,weuse b = w − cx to obtain the inequality
ord p (b) ≥ min{ord p (w), ord p (c) + ord p (x)}≥ 3n.

In order to estimate ord p (x + x + x ), we substitute the equation for the line

w = cx + b through P and P in the normal form for the equation of the cubic to

obtain an equation in x
(cx + b) + a 1 (cx + b)x + a 3 (cx + b) 2