Preface

In this book, we study existence and asymptotics of rational points on algebraic

varieties of Fano and intermediate type. The book consists of three parts. In the

first part, we discuss to some extent the concept of a height and formulate Manin’s

conjecture on the asymptotics of rational points on Fano varieties.

In the second part, we study the various versions of the Brauer group. We explain

why a Brauer class may serve as an obstruction to weak approximation or even to

the Hasse principle. This includes two sections devoted to explicit computations of

the Brauer–Manin obstruction for particular types of cubic surfaces.

The final part describes numerical experiments related to the Manin conjecture

that were carried out by the author together with Andreas-Stephan Elsenhans.

Prerequisites. We assume that the reader is familiar with some basic mathemat-

ics, including measure theory and the content of a standard course in algebra.

In addition, a certain background in some more advanced fields shall be necessary.

This essentially concerns three areas.

a) We will make use of standard results from algebraic number theory and class

field theory, as well as such concerning the cohomology of groups. The content

of articles [Cas67, Se67, Ta67, A/W, Gru] in the famous collection edited by

J. W. S. Cassels and A. Fröhlich shall be more than suﬃcient.

Here, the most important results that we shall use are the existence of the global

Artin map and, related to this, the computation of the Brauer group of a number

field [Ta67, 11.2].

b) We will use the language of modern algebraic geometry as described in the

textbook of R. Hartshorne [Ha77, Chapter II]. Cohomology of coherent sheaves

[Ha77, Chapter III] will be used occasionally.

c) In Chapter III, we will make substantial use of étale cohomology. This is probably

the deepest prerequisite that we expect from the reader. For this reason, we will

formulate its main principles, as they appear to be of importance for us, at the

beginning of the chapter. It seems to us that, in order to follow the arguments, an

understanding of Chapters II and III of J. Milne’s textbook [Mi] should be suﬃcient.

At a few points, some other background material may be helpful. This concerns,

for example, Artin L-functions. Here, [Hei] may serve as a general reference. Pre-

cise citations shall, of course, be given wherever the necessity occurs.

A suggestion that might be helpful for the reader. Part C of this book describes

experiments concerning the Manin conjecture. Clearly, the particular samples are

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