Functoriality and Special Values of L-Functions
Eisenstein Series and Applications
This is a semi-expository article concerning Langlands functoriality and Deligne's conjectures on the special values of L-functions. The emphasis is on symmetric power L-functions associated to a holomorphic cusp form. 2 A. RAGHURAM AND FREYDOON SHAHIDI While the integral representations of Godement and Jacquet do not seem to admit a cohomological interpretation, there is a recent work of J. Mahnkopf   which provides us with such an interpretation for certain Rankin-Selberg type
... elberg type integrals. In particular, modulo a nonvanishing assumption on local archimedean Rankin-Selberg product L-functions for forms on GL n × GL n−1 , he defines a pair of periods, which seem to be in accordance with those of Deligne  and Shimura . This work of Mahnkopf is quite remarkable and requires the use of both Rankin-Selberg and Langlands-Shahidi methods in studying the analytic (and arithmetic) properties of L-functions. His work therefore brings in the theory of Eisenstein series to play an important role. In §6 we briefly review this work of Mahnkopf. This article is an attempt to test the philosophy-to study the special values of L-functions while using functoriality-by means of recent cases of functoriality established for symmetric powers of automorphic forms on GL 2   . While a proof of the precise formulae in the conjectures of Deligne  still seem to be out of reach, we expect to be able to prove explicit connections between the special values of symmetric power L-functions twisted by Dirichlet characters and those of original symmetric power L-functions using this work of Mahnkopf. These relations are formulated in this paper as Conjecture 7.1 which seems to be compatible with the more general conjectures of Blasius  and Panchiskin  . A standard assumption made in the study of special values of L-functions is that the representations (to which are attached the L-functions) are cohomological. This is the case in Mahnkopf's work. A global representation being cohomological is entirely determined by the archimedean components. For representations which are symmetric power lifts of a cusp form on GL 2 we have the following fact. Consider a holomorphic cusp form on the upper half plane of weight k. This corresponds to a cuspidal automorphic representation, which is cohomological if k ≥ 2, and any symmetric power lift, if cuspidal, is also cohomological. (If the weight k = 1 then the representation is not cohomological, besides none of the symmetric power L-functions have any critical points.) In §5 we review representations with cohomology in the case of GL n . We recall the functorial formalism for symmetric powers in §2. We then review Deligne's conjecture for the special values of symmetric power L-functions in §3 and give a brief survey as to which cases are known so far. In §4 we sketch a proof of the conjecture for dihedral representations; the details will appear elsewhere . Acknowledgements: This paper is based on a talk given by the second author during the workshop on Eisenstein series and Applications at the American Institute of Mathematics (AIMS) in August of 2005; he would like to thank the organizers Wee Teck Gan, Stephen Kudla and Yuri Tschinkel as well as Brian Conrey of AIMS for a most productive meeting.