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§6. The General Notion of Height on Projective Space 135

§6. The General Notion of Height on Projective Space

A height on projective space is a proper, positive real valued function with a certain
behavior when composed with algebraic maps of projective space onto itself.
(6.1) Definition. Let k be a field. A k-morphism f : P m (k) → P m (k) of degree d is
a function of the form
f (y 0 : ··· : y m ) = f 0 (y 0 ,..., y m ) : ··· : f m (y 0 ,..., y m ),

where each f i (y 0 ,..., y m ) ∈ k[y o ,..., y m ] is homogeneous of degree d and not all
¯
¯
are equal to zero at any y 0 : ··· : y m ∈ P m (k).Here k denotes an algebraic closure
of k.
(6.2) Definition. A height h on P m (k) is a proper function h : P m (k) → R such that
for any k-morphism f : P m (k) → P m (k) of degree d the composite hf is equivalent
to d · h, that is, there is a constant c with |h( f (y)) − d · h(y)|≤ c for all y ∈ P m (k).
There is a canonical height function on projective space over a global field which
is basic in many considerations in diophantine geometry. We will consider some
special cases which are used to construct a norm on the rational points of an elliptic
curve over the rational numbers.

(6.3) Notations. For a point in P m (Q) we choose a Z-reduced representative
y 0 : ··· : y m and denote by

H(y 0 : ··· : y m ) = max{|y 0 |,..., |y m |} and h(P) = log H(P),
where P = y 0 : ··· : y m . Recall that a Z-reduced representative of P is integral and
without common divisor, and so unique up to sign. This h(P) is called the canonical
height on P m (Q).
In the one-dimensional case there is a bijection u : Q ∪ {+∞} → P 1 (Q) defined
by u(m/n) = n : m and u(∞) = 0 : 1. The composite hu restricted to Q is given by
h(m/n) = log max{|m|, |n|}, where m/n is reduced to lowest terms.

(6.4) Remark. Since h(y 0 ,..., y m ) = log(max{|y 0 |,... , |y m |}) is a proper map of
R m+1 −{0}→ R and since Z m+1 is a discrete subset of R m+1 , the map h : P m (Q) →
R is proper where P m (Q) has the discrete topology.
To obtain a norm on an elliptic curve from a height, we will make use of the

following function s : P 1 (k) × P 1 (k) → P 2 (k) given by s(w : x,w : x ) = ww :

(xw + x w) : xx . It has the following elementary property related to heights.
(6.5) Proposition. For s : P 1 (Q) × P 1 (Q) → P 2 (Q) the difference h(s(w : x,w :

x )) − h(w : x) − h(w : x ) has absolute value less than log 2 on the product

P 1 (Q) × P 1 (Q).