A man hears what he wants to hear and disregards the rest.– Paul Simon and Art Garfunkel, The Boxer
This article is an expanded version of a lecture given at the conference on Special Values of Rankin L-Series at MSRI in December of 2001. I have tried to retain some of the tone of an informal lecture. In particular, I have attempted to outline, in very broad terms, a program involving relations among
(i) algebraic cycles,
(ii) Eisenstein series and their derivatives, and
(iii) special values of Rankin–Selberg L-functions and their derivatives,
ignoring many important details and serious technical problems in the process. I apologize at the outset for the very speculative nature of the picture given here. I hope that, in spite of many imprecisions, the sketch will provide a context for a variety of particular cases where precise results have been obtained. Recent results on one of these, part of an ongoing joint project with Michael Rapoport and Tonghai Yang on which much of the conjectural picture is based, are described in Yang's article. A less speculative discussion of some of this material can be found in.
I thank my collaborators B. Gross, M. Harris, J. Millson, S. Rallis, M. Rapoport and T. Yang for generously sharing their mathematical ideas and for their support over many years. I also thank R. Borcherds, J.-B. Bost, J. Cogdell, J. Funke, R. Howe, D. Kazhdan, K. Keating, J. Kramer, U. Kühn, J.-S. Li, J. Schwermer, and D. Zagier for helpful discussions, comments and suggestions.