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Eisenstein Cohomology and the Construction of Variable Motives (English, German, Paperback, 1993 ed.)
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Eisenstein Cohomology and the Construction of Variable Motives (English, German, Paperback, 1993 ed.)
Series: Lecture Notes in Mathematics, 1562
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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.
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