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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.
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.