Eisensteinkohomologie und die Konstruktion gemischter Motive

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

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