Tate’s thesis, Fourier analysis in number fields and Hecke’s zeta-functions, Princeton, 1950, first appeared in print as Chapter XV of the conference proceedings Algebraic Number Theory, edited by Cassels and Frolich, published by the Thompson Book Co., Washington, D.C., in 1967. In it, Tate provides an elegant and unified treatment of the analytic continuation and functional equation of the L-...

All known ways to analytically continue automorphic L-functions involve integral representations using the corresponding automorphic forms. The simplest cases, extending Hecke’s treatment of GL2, need no further analytic devices and very little manipulation beyond Fourier-Whittaker expansions. [1] Poisson summation is a sufficient device for several accessible classes of examples, as in Riemann...

Abstract The main aim of this article is to show that normalised standard intertwining operator between induced representations p -adic groups, at a very specific point evaluation, has an arithmetic origin. This result applications Eisenstein cohomology and the special values automorphic L -functions.

This essay contains somewhat drymaterial, mostly useful inmotivating eventually a certain crucial but at first sight somewhat technical transition from representations of groups to representations of Lie algebras. Parts of it will also be used in the theory of automorphic forms. I have made some effort to reduce everything to well known facts inmeasure theory and topology. The standard referenc...

We describe a new method to estimate the trilinear period on automorphic representations of PGL2(R). Such a period gives rise to a special value of the triple Lfunction. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value. Our method is based on the study of the analytic structure of the corresponding unique trilinear functional on un...

We establish some new cases of Artin’s conjecture. Our results apply to Galois representations over Q with image S5 satisfying certain local hypotheses, the most important of which is that complex conjugation is conjugate to (12)(34). In fact, we prove the stronger claim conjectured by Langlands that these representations are automorphic. For the irreducible representations of dimensions 4 and ...

We provide a family of representations of GL2n over a p-adic field that admit a non-vanishing linear functional invariant under the symplectic group (i.e. representations that are Sp2ndistinguished). While our result generalizes a result of M. Heumos and S. Rallis our methods, unlike their purely local technique, relies on the theory of automorphic forms. The results of this paper together with...

In this paper, we decompose the space of nearly holomorphic Hilbert-Siegel automorphic forms as representations adele group under certain assumptions. We also give an application for classical modular forms. particular, show surjectivity global Siegel operator congruence subgroups with large weights.