18 the concept of a height [Chap. I

2.4. Definition. Let K be a number field of degree d. Then, for

(x0 : . . . : xn) ∈ Pn(K), one puts

Hnaive(x0 : . . . : xn) :=

ν∈Val(K)

max

i=0,...,n

xi

ν

1

d

.

This height function is the number field version of the naive height on Pn. It is

usually called the absolute height.

2.5. Lemma. Let K be a number field, and let (x0 : . . . : xn) ∈

Pn(K).

Further, let L ⊃ K be a finite extension.

Then the absolute height Hnaive(x0 : . . . : xn) remains unchanged when

(x0 : . . . : xn) is considered as an L-rational point.

Proof. Put d := [L : K]. Then, by the properties of the valuations, we have

Hnaive(x0 : . . . : xn) =

ν∈Val(K)

max

i=0,...,n

xi

ν

1

d

=

ν∈Val(K)

μ∈Val(L)

μ above ν

max

i=0,...,n

xi

μ

1

dd

=

μ∈Val(L)

max

i=0,...,n

xi

μ

1

dd

,

which is exactly the formula for Hnaive(x0 : . . . : xn) considered as an

L-rational point.

2.6. Proposition (D. G. Northcott). Let B, D ∈ . Then

{x ∈

Pn(

) | x ∈

Pn(K)

for [K : ] D and Hnaive(x) B}

is a finite set.

Proof. We may work with the number fields K of a fixed degree d.

For x, we choose homogeneous coordinates such that some coordinate equals 1.

Then it is clear that, for every valuation .

ν

and every index i, we have

max{x0 ν, . . . , xn ν} ≥ max{1, xi ν} .

Multiplying over all ν and taking the d-th root therefore shows

Hnaive(x0 : . . . : xn) ≥ Hnaive(1 : xi) .

Hence, it suﬃces to verify that the set

{(1 : x) ∈

P1(

) | (1 : x) ∈

P1(K)

for [K : ] = d and Hnaive(1 : x) B}

is finite.