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104 5. Reduction mod p and Torsion Points
u
n
a = p · ,
v
where p does not divide either u or v and n is an integer uniquely determined by a.
Let ord p (a) = n denote the order function associated with p. Let r p (a) =¯a denote
the canonical reduction mod p defined R → k(p).If R (p) denotes the subring of
all a in k with ord p (a) ≥ 0, then the mod p reduction function is well defined on
R (p) → k(p).
The order function satisfies the following properties:
(V1) ord p (ab) = ord p (a) + ord p (b)
and
(V2) ord p (a + b) ≥ min{ord p (a), ord p (b)}.
The second property can be refined in the case where ord p (a)< ord p (b) in which
case ord p (a) = ord p (a + b). This refinement can be deduced directly from the
second property of ord p (a) using an elementary argument. The function is a special
case of a discrete valuation (of rank 1) and R (p) is the valuation ring associated to
ord p . It is a principal ring with one irreducible p and one maximal ideal R (p) p where
r p induces an isomorphism R (p) /R (p) p → k(p).
There are some cases, especially over a number field, where it is confusing to use
p for the irreducible or local uniformizing element of R, in those cases we will tend
to use t or π.
The reduction mod p function r p : R (p) → k(p) can be defined on affine space
by taking products
n
n
k ⊃ R n → k(p) ,
(p)
but it is only defined on the points x = (x 1 ,... , x n ) all of whose coordinates x j
have positive ord p . Then the formula is r p (x 1 ,... , x n ) = (¯x 1 ,... , ¯x n ). On the other
hand, reduction mod p is defined on the entire projective space.
(1.2) Definition. The reduction mod p function r p : P n (k) → P n (k(p)) is defined
by the relation
r p (y 0 : ··· : y n ) = ( ¯y 0 : ··· : ¯y n ),

where (y 0 : ··· : y n ) is the homogeneous coordinates of a point in P n (k) with all
y i in R and without a common irreducible factor. Such a representative of a point is
called reduced.

Observe that r p (y 0 : ··· : y n ) = ( ¯y 0 : ··· : ¯y n ) is defined when each ord p (y i ) ≥
0 and some ord p (y j ) = 0 so that ¯y j = 0. Such a representative (y 0 : ··· : y n )
of a point in P n (k) is called p-reduced. The reduced representatives are unique up
to multiplication by a unit in R, and the p-reduced representatives are unique up to
multiplication by a unit in R (p) .
We saw in Chapter 2 that a good intersection theory had to be formulated in pro-
jective space. The above construction of mod p reduction shows another advantage
of projective space over affine space which is arithmetic instead of geometric.