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§and their arithmetic subgroups. Some problems mentioned§require only part of this background knowledge.