6 introduction

of the adelic points, there are nevertheless as many rational points as naively ex-

pected. Even more, E. Peyre and Y. Tschinkel [Pe/T] showed experimentally that

if

H1

(

Gal(k/k), Pic(Xk)

)

= /3 and the Brauer class does not exclude any adelic

point, then there are three times more rational points than expected. Correspond-

ingly, in E. Peyre’s [Pe95a] definition of the conjectural constant τ , there appears

an additional factor β(X) := #H1

(

Gal(k/k), Pic(X

k

)

)

.

This book is concerned with Diophantine equations from the theoretical and exper-

imental points of view. It is divided into three parts. The first part is devoted to

the various concepts of a height. In the first chapter, we start with the naive height

for -rational points on projective space. Then our goal is to deliver some insight

into the theories, which provide natural generalizations of this simple concept.

The very first generalization is the naive height for points in projective space defined

over a finite extension of . Then, following André Weil, we introduce the concept

of a height defined by an ample invertible sheaf. This is a height function, which is

defined only up to a bounded summand.

To overcome this diﬃculty, one has to work with arithmetic varieties and metrized

invertible sheaves. Arithmetic varieties are schemes projective over Spec . Actu-

ally, this leads to a beautiful geometric interpretation of the naive height.

Indeed, let X be a projective variety over , and let X be a projective model

of X over Spec . Fix a hermitian line bundle L on X . Then, according to the

valuative criterion of properness, every -rational point x on X extends uniquely

to a -valued point x: Spec → X . The height function with respect to L is

then given by

hL (x) := deg

(

x∗L

)

.

Here, deg denotes the Arakelov degree of a hermitian line bundle over Spec .

It turns out that this coincides exactly with the naive height when one works with

X =

Pn

, L = O(1), and the minimum metric, which is defined by

min

:= min

i=0, ... ,n

Xi

.

In general, h

L

admits a fundamental finiteness property as soon as L is ample.

Chapter II is devoted to some of the most popular conjectures concerning ratio-

nal points on projective algebraic varieties. We discuss Lang’s conjecture, the

conjecture of Batyrev and Manin, and, most notably, Manin’s conjecture about

the asymptotics of points of bounded height on Fano varieties (Conjecture II.7.3).

A large part of the chapter is concerned with E. Peyre’s Tamagawa type num-

ber τ(X), the coeﬃcient expected in the asymptotic formula. We discuss in detail

all factors appearing in the definition of τ(X).

In particular, we give a number of examples, for which we explicitly compute the fac-

tor α(X). We mainly consider smooth cubic surfaces of arithmetic Picard rank two.

Part B is the technical heart of the book. It deals with the concepts of a Brauer

group and its applications. The third chapter considers A. Grothendieck’s Brauer

group for arbitrary schemes. We recall the concept of a sheaf of Azumaya algebras

on a scheme and explain how such a sheaf of algebras gives rise to a class in the