Suppose that G and H are connected reductive groups over a number field F and that an L-homomorphism ρ : LG −→ LH is given. The Langlands functoriality conjecture predicts the existence of a map from the automorphic representations of G(A) to those of H(A). If the adelic points of the algebraic groups G, H are replaced by their metaplectic covers, one may hope to specify an analogue of the Lgro...

The object of this work is the spinor L-function of degree 3 and certain degeneration related to the functoriality principle. We study liftings of automorphic forms on the pair of symplectic groups (GSp(2), GSp(4)) to GSp(6). We prove cuspidality and demonstrate the compatibility with conjectures of Andrianov, Panchishkin, Deligne and Yoshida. This is done on a motivic and analytic level. We di...

Let X be a smooth projective curve over an algebraically closed field of characteristic > 2. Consider the dual pair H = SO2m, G = Sp2n over X with H split. Write BunG and BunH for the stacks of G-torsors and H-torsors on X . The theta-kernel AutG,H on BunG ×BunH yields the theta-lifting functors FG : D(BunH) → D(BunG) and FH : D(BunG) → D(BunH) between the corresponding derived categories. Assu...

When I was asked to suggest a Bulletin article worthy of reprinting, I knew the answer right away: An Elementary Introduction to the Langlands Program by Steven Gelbart, published by the Bulletin in 1984, remains one of the best reviews of the Langlands Program, even though this subject has expanded tremendously in the intervening years. This paper has greatly influenced my own research and is ...

In recent years L-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of L-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to te...

We prove an algebraicity result for the central critical value of certain RankinSelberg L-functions for GLn×GLn−1. This is a generalization and refinement of the results of Harder [15], Kazhdan, Mazur and Schmidt [23], Mahnkopf [29], and Kasten and Schmidt [22]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric pow...

An excellent introduction to the theory of automorphic representations and the relations with number theory is the Corvallis proceedings, Automorphic Forms, Representations and L-Functions, Parts 1 and 2, Proc. Sympos. Pure Math., vol. 33, 1979. Although composed mainly of survey articles, the proceedings are already rather formidable. They are a measure of the breadth of the field. They will b...

Following the explicit instructions of the organizers, I have tried to write an article that is suitable for a general mathematical audience. It contains some analogies and metaphors that might even be put to nonmathematicians. I hope that experts will be tolerant of the inevitable simplifications. The principle of functoriality is one of the central questions of present day mathematics. It is ...

Let π be a cuspidal automorphic representation of PGL3(A) (where A is the adele ring of F ). The functoriality conjecture of Langlands predicts that, via the inclusion γ of dual groups, there is a global L-packet L(π) of G2 associated to π. Indeed, for each place v of F , there should be a local L-packet L(πv) of G2(Fv). It is expected that L(πv) contains a unique generic representation σ(πv). ...

According to the Langlands functoriality conjecture, broadened setting of spherical varieties (of which reductive groups are special cases), a map between L -groups should give rise functorial transfer their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this will be realized as comparison limiting forms (relative) trace formulas these spaces. In paper, we establis...