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
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
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
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 =
, L = O(1), and the minimum metric, which is defined by
:= min
i=0, ... ,n
In general, h
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
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