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106 5. Reduction mod p and Torsion Points

For P = (1, 0, 0) and (w , x , y ) = (0, a, b) the polynomial ϕ(t) takes the

form ϕ(t) = f r (ta, tb) +· · · + f d (ta, tb), where r is the order of P on the curve
C f . Recall from 2(4.6) that L is part of the tangent cone to C f at P if and only if

r < i(P; L, C f ).For P on L where P = (at 0 , bt 0 ) we have P ∈ C f if and only

if ϕ(t 0 ) = 0.

(1.7) Proposition. With the above notations let P, P ∈ L ∩ C f where P = P and
¯ ¯ , then the
P = r p (P) = r p (P ). If the order of P on C f equals the order of Pon C ¯ f
¯ ¯ ¯
reduced line L is part of the tangent cone of C ¯ f .If P has order 1 on C ¯ f , then Lis
at P.
¯
the tangent line to C ¯ f
r
Proof. Since P is on L ∩ C f of order ≥ r, the polynomial t divides ϕ(t). Since P
r
is on L ∩ C f , the polynomial t − t 0 divides ϕ(t). Thus the product t (t − t 0 ) divides

ϕ(t). Since r p (P) = r p (P ),wehave ¯ t 0 mod p and therefore t r+1 divides ¯ϕ(t).Now
the proposition follows from the criterion in 2(4.6).
In Exercise 1 we give an example which shows that the hypothesis that the order
¯ is necessary in the proposition.
of P on C f equals the order of R on C ¯ f
Exercise
3 2
1. Let C be the conic defined by wx − p y = 0 and show that C is nonsingular over Q.
The reduction mod p is wx = 0, and show that the reduction ¯ C has a singular point at

(0, 0, 1). Find two distinct points P and P on C whose reduction is in both cases (0, 0, 1)

and such that the reduction ¯ L of the line L through P and P is different from either the
line defined by w = 0orby x = 0.
§2. Minimal Normal Forms for an Elliptic Curve
(2.1) Proposition. Let k be the field of fractions for an integral domain R, and let
E be an elliptic curve over k. Then there is a cubic equation for E in normal form
with all a i ∈ R.
Proof. Choose any normal form for E over k with coefficients ¯a i in variables ¯x and
¯ y. Let u be a common denominator for all ¯a i , in particular all u ¯a i ∈ R, and make the
3
i
2
change of variable x = u ¯x and y = u ¯y. Then the coefficients a i = u ¯a i is in R for
all i. This proves the proposition.
In the previous proposition, observe that once all a j are in R, all the related
constants b j , c j ,and , defined in 3(3.1), are also in R. Assume now that R is a
discrete valuation ring, that is, R is principal with one nonzero prime ideal Rp and
the order function ord p is a valuation denoted by v. Then R is the set of all a in K,
the field of fractions of fractions of R, such that v(a) ≥ 0.

(2.2) Definition. Let K be a field with a discrete valuation v, and let E be an elliptic
curve over K. A minimal normal form for E is a normal form with all a j in the
valuation ring R of K such that v( ) is minimal among all such equations with
coefficients a j in R.