Edited by J. Coates
Edited by M. J. Taylor
Publisher: Cambridge University Press
Print Publication Year: 1991
Online Publication Date:December 2009
Online ISBN:9780511526053
Paperback ISBN:9780521386197
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511526053.016
Subjects: Number theory
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INTRODUCTION
This article does not represent precisely a talk given at the symposium, but is complementary to [DenS]. Its purpose is to explain a setting in which the various conjectures on special values of L-functions admit a unified formulation. At critical points, Deligne's conjecture [Del2] relates the value of an L-function to a certain period, and at non-critical points, the conjectures of Beilinson [Bel] give an interpretation in terms of regulators. Finally, at the point of symmetry of the functional equation, there is the conjecture of Birch and Swinnerton-Dyer, generalised by Bloch [Bl2] and Beilinson [Be2], in which the determinant of the height pairing on cycles appears.
Both the periods and the regulators are constructed globally, and their definitions are in some sense archimedean. The height pairing, on the other hand, is defined as a sum of local terms. Our aim is to show how all of these objects—periods, regulators, and heights—may be interpreted as ‘periods of mixed motives’.
That such a reformulation is possible in the case of regulators is clearly indicated in the letter of Deligne to Soulé [Del3], Perhaps the only novel feature of our account is to regard the mixed motives as primary objects, rather than the Ext groups. It is appropriate to mention in this connection work of Anderson and of Harder [H], in which certain particular mixed motives arising in the study of the cohomology of Shimura varieties are investigated.
pp. i-iv
pp. v-vi
pp. vii-vii
pp. viii-viii
Lectures on automorphic L-functions : Read PDF
pp. 1-60
Gauss sums and local constants for GL(N) : Read PDF
pp. 61-74
L-functions and Galois modules : Read PDF
pp. 75-140
Motivic p-adic L-functions : Read PDF
pp. 141-172
The Beilinson conjectures : Read PDF
pp. 173-210
Iwasawa theory for motives : Read PDF
pp. 211-234
Kolyvagin's work for modular elliptic curves : Read PDF
pp. 235-256
Index theory, potential theory, and the Riemann hypothesis : Read PDF
pp. 257-270
Katz p-adic L-functions, congruence modules and deformation of Galois representations : Read PDF
pp. 271-294
Kolyvagin's work on Shafarevich-Tate groups : Read PDF
pp. 295-316
Arithmetic of diagonal quartic surfaces I : Read PDF
pp. 317-338
On certain Artin L-Series : Read PDF
pp. 339-352
The one-variable main conjecture for elliptic curves with complex multiplication : Read PDF
pp. 353-372
Remarks on special values of L-functions : Read PDF
pp. 373-392