Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Henniart, Guy

doi:10.1007/s002220050012

Cited by 360 publications

(349 citation statements)

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“…The full, completed L-function involves extra factors for the places in S, whose definition is technical and in general difficult. This is connected to the local Langlands correspondence, proven recently by Harris and Taylor for GL(n) and by Jiang and Soudry for SO(2n + 1) (see [14], [57], [59], [62], [63], [65], [78], [101]). When ρ is the standard representation of L GL(n) = GL(n) and F = Q, the Euler factors in (8.3) agree with those in (7.23); in general the degree of L S (s, π, ρ) equals the dimension of ρ.…”

Section: Langlands L-functionsmentioning

confidence: 59%

Riemann’s zeta function and beyond

Gelbart

Miller

2003

Bull. Amer. Math. Soc.

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Abstract. 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 technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of L-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of L-functions.

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Section: Langlands L-functionsmentioning

confidence: 59%

Riemann’s zeta function and beyond

Gelbart

Miller

2003

Bull. Amer. Math. Soc.

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“…Then ξ • ϕ is a Langlands parameter for G invariant underθ . Let π be the representation of G(F) assigned to ξ • ϕ by the local Langlands correspondence [Harris and Taylor 2001;Henniart 2000]. We have π ∼ = π • θ .…”

Section: Tempered Representations and Their Contragredientmentioning

confidence: 99%

“…Furthermore, many cases are known when F is a finite extension of the field ‫ޑ‬ p of p-adic numbers. Most notably, the correspondence over p-adic fields is known when the reductive group is GL n by work of Harris and Taylor [2001] and Henniart [2000], and has very recently been obtained for quasisplit symplectic and orthogonal groups by Arthur [2013]. Other cases include the group U 3 by work of Rogawski, Sp 4 and GSp 4 by work of Gan-Takeda.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2013

Algebra Number Theory

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This paper contains a complete proof of Fukaya and Kato's -isomorphism conjecture for invertible -modules (the case of V = V 0 (r ), where V 0 is unramified of dimension 1). Our results rely heavily on Kato's proof, in an unpublished set of lecture notes, of (commutative) -isomorphisms for one-dimensional representations of G ‫ޑ‬ p , but apart from fixing some sign ambiguities in Kato's notes, we use the theory of (φ, )-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of Fukuya and Kato and extend it to the (slightly noncommutative) semiglobal setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves E over the extension of ‫ޑ‬ p which trivialises the p-power division points E( p) of E. In this sense the paper is complimentary to our work with Bouganis (Asian J. Math. 14:3 (2010), 385-416) on noncommutative Main Conjectures for CM elliptic curves.

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“…Specific information about the Hecke algebra H(G, λ) is employed to give a purely local proof of Shahidi's theorem on the reducibility of parabolic induction [20], which can also be described in terms of the poles of L-functions. On the other hand, recent work of Henniart [14], building on work of Harris and Taylor [11] and Henniart [13], relates this to the classification of local galois representations by their images (e.g. symplectic, orthogonal).…”

Section: Introductionmentioning

confidence: 99%