CHAPTER I 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) ∈ Pn( ), put Hnaive(x0 : . . . : xn) := max i=0,...,n |x i | . Here (x0 : . . . : xn) = (x0 : . . . : xn), such that all x i are integers and gcd(x 0 , . . . , x n ) = 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. 15 http://dx.doi.org/10.1090/surv/198/02

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