The concept of a height
Equations are just the boring part of mathematics. I attempt to see
things in terms of geometry.
Stephen Hawking (A Biography (2005) by Kristine Larsen, p. 43)
1. The naive height on the projective space over
1.1. Heights have been studied by number theorists for a very long time.
A height is a function measuring the size or, more precisely, the arithmetic com-
plexity of certain objects. These objects are classically solutions of Diophantine
equations or rational points on an algebraic variety. A height then might answer
the question, How many bits would one need in order to store the solution or the
point on a computer?
More recently, starting with G. Faltings’ ideas of heights on moduli spaces, it be-
came more common to consider heights for more complicated objects, such as cycles.
1.2. Definition. For (x0 : . . . : xn) ∈
Hnaive(x0 : . . . : xn) := max
(x0 : . . . : xn) = (x0 : . . . : xn),
such that all xi are integers and gcd(x0, . . . , xn) = 1.
The function Hnaive : Pn ( ) → is called the naive height.
1.3. Fact (Fundamental finiteness). For every B ∈ , there are only
finitely many points x ∈ Pn ( ) such that
Hnaive(x) B .
Proof. For each component of (x0, . . . , xn), we have −B xi B. Thus, there
are only finitely many choices.
1.4. The naive height is probably the simplest function one might think of
that fulfills the fundamental finiteness property. For more general height functions,
the fundamental finiteness property will always be required.