Spectral Decomposition and Eisenstein Series

Mœglin, Colette; Waldspurger, Jean-Loup

doi:10.1017/cbo9780511470905

Cited by 341 publications

(415 citation statements)

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“…However, it never appeared in a journal as I was unable to generalise the boudary conditions required for Faddeev's method, while Jacquet's elegant simplification of Selberg's original proof, as exposed in ch. IV of [19], made the search for alternative methods pointless. Nevertheless, I include that kernel estimate here as it might still have other applications and its proof is a byproduct of the arguments we have to expound anyway.…”

Section: The Resultsmentioning

confidence: 99%

Geometric estimates for the trace formula

Hoffmann

2008

Ann Glob Anal Geom

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In order to study the asymptotic distribution of geometric or spectral data associated with quotients of a reductive group by a lattice, one needs a trace formula for test functions on that group with noncompact support. Arthur has proved a trace formula for compactly supported test functions on reductive groups of arbitrary rank. We show that the coarse geometric expansion in his formula converges for rapidly decreasing functions.Mathematics Subject Classification. Primary: 11F72; Secondary: 22E55.

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Section: The Resultsmentioning

confidence: 99%

Geometric estimates for the trace formula

Hoffmann

2008

Ann Glob Anal Geom

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“…As before, we rely on Langlands' theory of Eisenstein series ([L2], [L3], [HC], [MW1]). Our main references are Arthur's truncation ( [A2], [A3]), Langlands' inner product formula ([A3], [L2], [L3]), Harish-Chandra [HC], and Muller [Mu] (see our Introduction).…”

Section: Analytic Behaviour Of Fourier Coefficients Of Cuspidalmentioning

confidence: 99%

“…As in [MW1] one can simply defineτ P to be the characteristic function of the sum of z, the real Lie algebra of the center of G, and the interior of the cone generated by α 1 . We then need to project T on a 1 .…”

Section: Analytic Behaviour Of Fourier Coefficients Of Cuspidalmentioning

confidence: 99%

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Boundedness of automorphic 𝐿–functions in vertical strips

Gelbart

Shahidi

2000

J. Amer. Math. Soc.

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In this paper we prove the boundedness in vertical strips of finite width for all the L L –functions that appear in constant terms of Eisenstein series, under a certain natural assumption on local normalized intertwining operators. This result has been quite important in establishing recent cases of Langlands functoriality by Cogdell, Kim, Piatetski-Shapiro, and Shahidi, using converse theorems.

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“…References include [3], [7], [44], [56], [118], [166], [168]. Our purpose here is to describe the general method of obtaining analytic properties of L-functions from Eisenstein series, generalizing Selberg's method for ζ(s) in Section 4.1.…”

Section: Langlands-shahidi (1967-)mentioning

confidence: 99%

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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Spectral Decomposition and Eisenstein Series

Cited by 341 publications

References 0 publications

Geometric estimates for the trace formula

Geometric estimates for the trace formula

Boundedness of automorphic 𝐿–functions in vertical strips

Riemann’s zeta function and beyond

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