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§4. The Kernel of Reduction mod p and the p-Adic Filtration 111

3
2
d) y − y = x − 7.
2
2
3
e) y + xy + y = x − x .
2 3 2
f) y + xy = x − x − 5.
4. Which of the following curves have good reduction at p = 2, p = 3, p = 5, and p = 7?
3
2
a) y + xy + y = x − x.
2
2
3
b) y + xy = x − x − 2x − 1.
2 3 2
c) y + xy + y = x − x − 3x + 3.
3
2
2
d) y + xy = x + x − 2x − 7.
2
2
3
e) y + xy + y = x + x − 4x + 5.
2
2
2
3
3
2
5. The curves y + y = x − x and y + y = x + x have good reduction at 2 and the
same reduced curve mod 2. Determine r 2 on the subgroup generated by (0, 0) with values
in the points over F 2 of the reduced curve in each case, see 1(2.3) and 1(2.4).
3
2
6. Find the discriminant of y + y = x − 7x + 6 and show that it is a prime.
§4. The Kernel of Reduction mod p and the p-Adic Filtration
Following (1.1), we use the notations of the previous section where R is a facto-
rial ring, p is an irreducible, and k is its field of fractions. Recall from (3.5) that
¯
the inverse image of (0, 0, 1) under the reduction mapping r p : E(k) → E(k(p))
consists of all (w, x, 1) ∈ E(k) such that ord p (w), ord p (x)> 0. These conditions
are analyzed further in the next two propositions using the normal form of the cubic
equation.
(4.1) Proposition. Let E be an elliptic curve over k, and (w, x, 1) ∈ E(k).If
ord p (w) > 0, then we have ord p (x)> 0, and the relation ord p (w) = 3ord p (x)
holds.
Proof. For y = 1, the projective normal form of the cubic equation for E has the
form
2
3
2
3
2
w + a 1 wx + a 3 w = x + a 2 wx + a 4 w x + a 6 w .
We assume that ord p (w) > 0 and ord p (x) ≤ 0, and we derive a contradiction. If R
is the right-hand side of the normal cubic, then
3
ord p (R) = ord p (x ) = 3 ord p (x) ≤ 0,
and if L is the left-hand side of the normal cubic, then
ord p (L) = min{ord p (w), ord p (x) + ord p (w) + ord p (a 1 )}.
Since ord p (w) > 0, we would derive the relation
3 ord p (x) ≥ ord p (x) + ord p (w) or 0 ≥ 2 ord p (x) ≥ ord p (w).
This is a contradiction.
2
Next observe that ord p (w) = ord p (w + a 1 wx + a 3 w ) since ord p (w) <
2
min{ord p (a 1 wx), ord p (a 3 w )}. Thus we obtain the second assertion