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The Beilinson conjectures

1991; Cambridge University Press; Linguagem: Inglês

10.1017/cbo9780511526053.007

Christopher Deninger, A. J. Scholl,

Homotopy and Cohomology in Algebraic Topology

The Beilinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators. We begin with a short but almost selfcontained introduction to this circle of ideas. This is possible by using Bloch's description of Beilinson's motivic cohomology and regulator map in terms of higher Chow groups and generalized cycle maps. Here we follow [Bl3] rather closely. We will then sketch how much of the known evidence in favour of these conjectures — to the left of the central point — can be obtained in a uniform way. The basic construction is Beilinson's Eisenstein symbol which will be explained in some detail. Finally in an appendix a map is constructed from higher Chow theory to a suitable Ext-group in the category of mixed motives as defined by Deligne and Jannsen. This smooths the way towards an interpretation of Beilinson's conjectures in terms of a Deligne conjecture for critical mixed motives [Sc2]. It also explains how work of Harder [Ha2] and Anderson fits into the picture.