The aim of this book is to show that Shimura varieties provide a tool to
construct certain interesting objects in arithmetic algebraic geometry.
These objects are the so-called mixed motives: these are of great
arithmetic interest. They can be viewed as quasiprojective algebraic
varieties over Q which have some controlled ramification and where we
know what we have to add at infinity to compactify them. The existence
of certain of these mixed motives is related to zeroes of L-functions
attached to certain pure motives. This is the content of the
Beilinson-Deligne conjectures which are explained in some detail in the
first chapter of the book. The rest of the book is devoted to the
description of the general principles of construction (Chapter II) and
the discussion of several examples in Chapter II-IV. In an appendix we
explain how the (topological) trace formula can be used to get some
understanding of the problems discussed in the book. Only some of this
material is really proved: the book also contains speculative
considerations, which give some hints as to how the problems could be
tackled. Hence the book should be viewed as the outline of a programme
and it offers some interesting problems which are of importance and can
be pursued by the reader. In the widest sense the subject of the paper
is number theory and belongs to what is called arithmetic algebraic
geometry. Thus the reader should be familiar with some algebraic
geometry, number theory, the theory of Liegroups and their arithmetic
subgroups. Some problems mentioned require only part of this background
knowledge.
