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§3. Good Reduction of Elliptic Curves 109

§3. Good Reduction of Elliptic Curves

Continuing with the notations (1.1), we have for an irreducible p a canonical reduc-
tion homomorphism r p : R (p) → k(p) denoted by r p (a) =¯a.
2
(3.1) Definition. Let E be an elliptic curve over k with minimal normal form y +
3
2
a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6 . The reduction E of E modulo p is given by
¯
2
2
3
y +¯a 1 xy +¯a 3 y = x +¯a 2 x +¯a 4 x +¯a 6 .
It is a plane cubic curve over k(p). The curve E is also denoted E (p) .
¯
Note that the normal form of the equation for E only has to be minimal at p for
this definition. Observe that an admissible change of variable between two minimal
2 3 2
normal forms of E at p given by x = u x + r and y = u y + su x + t over R (p)
2
2
reduces to x =¯u x +¯r and y =¯u y +¯s ¯u x + ¯ t an admissible change of variable
3
for E over k(p). Hence the reduction E is well defined up to isomorphism.
¯
¯
(3.2) Remarks. With the above notations the discriminant of the reduced curve E ¯
is ¯ , the reduction mod p of the discriminant of E. Clearly E is nonsingular if
¯
and only if ¯ = 0, or equivalently, if and only if ord p ( ) = 0.
(3.3) Definition. An elliptic curve E defined over k has a good reduction at p pro-
¯
vided E, the reduced curve at p, is nonsingular. When E is singular, we say E has
¯
bad reduction at p.
In general the reduction function r p : P 2 (k) → P 2 (k(p)) restricts to r p : E(k)
→ E(k(p)), and in the case of good reduction we have the following result.
¯
(3.4) Proposition. Let E be an elliptic curve over k with good reduction at p. Then
the reduction function r p : E(k) → E(k(p)) is a groupmorphism.
¯
Proof. Clearly r p (0:0:1) = 0 : 0 : 1 so that zero is preserved. For P, Q ∈ E(k)
let L be the line through P and Q when P = Q and the tangent line to E at P
¯
¯
when P = Q. Then L reduces to L, the line through r p (P) and r p (Q).Again L is
¯
tangent to E at r p (P) when r p (P) = r p (Q) by (1.7). If PQ denotes, as usual, the
third intersection point of L with E,thenwehave r p (PQ) = r p (P)r p (Q). Since the
¯
third intersection point of L with E for the x-coordinate is given by reduction mod
¯
¯
p of the equation giving the x-coordinate of PQ. Thus we calculate
r p (P + Q) = r p ((PQ)0) = (r p (P)r p (Q))r p (0) = r p (P) + r p (Q),
and thus r p is a group morphism. This proves the proposition.
(3.5) Remark. Since 0 = 0 : 0 : 1 on both E and the reduced curve over k(p),we
seethatthe p-reduced w : x : y on E(k) is in ker(r p ) if and only if ord p (y) = 0,
ord p (w) > 0, and ord p (x)> 0. In fact, we can divide by y and assume that the point
is of the form w : x : 1, where w and x have strictly positive ordinal at p.