A Proof of Langlands' Conjecture on Plancherel Measures; Complementary Series of -adic groups

Shahidi, Freydoon

doi:10.2307/1971524

Cited by 500 publications

(733 citation statements)

References 27 publications

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“…If m = 1, granting the functional equation (2.7), Assumption 2.1, and analytic properties of matrix coefficients of normalized operators (cf. [A1], [Si], [Sh1]), this immediately implies the boundedness in each T ε,I .…”

Section: Introductionmentioning

confidence: 77%

“…Let χ be a non-degenerate character of U(F )\U , defined by ψ and compatible withw (cf. [Sh1]). If U 0 = U ∩ M, we will use χ to also denote χ|U 0 .…”

Section: L-functions and Intertwining Operatorsmentioning

confidence: 99%

“…We start by recalling the general induction ([Sh1], [Sh2]) which is needed in order to answer different questions about each individual L(s, π, r i ) using this approach. As observed by Arthur [A4], our induction is endoscopic.…”

Section: Induction and Finiteness Of Polesmentioning

confidence: 99%

“…As observed by Arthur [A4], our induction is endoscopic. For the sake of completeness, we include a result which captures the main induction step of [Sh1] (cf. Propositions 4.1, 4.2, 5.1, Theorem 3.5 of [Sh1]).…”

Section: Induction and Finiteness Of Polesmentioning

confidence: 99%

“…Decompose the adjoint action r of L M , the L-group of M, on the complex Lie algebra L n of L N , the L-group of N in the sense of [B], as r = m i=1 r i , ordered as in Section 1. Let L(s, π, r i ), s ∈ C, be the completed automorphic L-function attached to π and r i , where local factors at ramified places are defined as in [Sh1]. We use the same local L-functions (and root numbers) to normalize the local standard intertwining operators (Section 2).…”

Section: Introductionmentioning

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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Section: Introductionmentioning

confidence: 77%

“…Let χ be a non-degenerate character of U(F )\U , defined by ψ and compatible withw (cf. [Sh1]). If U 0 = U ∩ M, we will use χ to also denote χ|U 0 .…”

Section: L-functions and Intertwining Operatorsmentioning

confidence: 99%

Section: Induction and Finiteness Of Polesmentioning

confidence: 99%

Section: Induction and Finiteness Of Polesmentioning

confidence: 99%

Section: Introductionmentioning

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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Zeros of Automorphic L-Functions and Noncyclic Base Change

Liu

Number Theory

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Abstract.Let π be an automorphic irreducible cuspidal representation of GL m over a Galois (not necessarily cyclic) extension E of Q of degree . Assume that π is invariant under the action of the Galois group Gal E/Q . We computed the n-level correlation of normalized nontrivial zeros of L(s, π) and proved that it is equal to the n-level correlation of normalized nontrivial zeros of a product of distinct L-functions L(s, π 1 ) · · · L(s, π ) attached to cuspidal representations π 1 , . . . , π of GL m over Q. This is done unconditionally for m = 1, 2 and for m = 3, 4 with the degree having no prime factor ≤ (m 2 + 1)/2. In other cases, the computation is under a conjecture of bounds toward the Ramanujan conjecture over E, and a conjecture on convergence of certain series over prime powers (Hypothesis H over E and Q). The results provide an evidence that π should be (noncyclic) base change of distinct cuspidal representations π 1 , . . . , π of GL m (Q A ), if it is invariant under the Galois action. A technique used in this article is a version of Selberg orthogonality for automorphic L-functions (Lemma 6.2 and Theorem 6.4), which is proved unconditionally, without assuming π and π 1 , . . . , π being self-contragredient. Introduction.According to Langlands' functoriality conjecture, the L-function attached to an automorphic irreducible cuspidal representation π of GL m over a number field E should equal a product of L(s, π j ) for certain cuspidal representations π j of GL m j over Q. Arthur and Clozel [ArtClo] proved that this is indeed the case when E is a cyclic Galois extension of Q and π is stable under Gal E/Q . In fact in this case, π is the base change of exactly nonequivalent cuspidal representations π Q , π Q ⊗ η E/Q , . . . , π Q ⊗ η

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Residues of Eisenstein Series and Related Problems

Jiang

Eisenstein Series and Applications

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A Proof of Langlands' Conjecture on Plancherel Measures; Complementary Series of -adic groups

Cited by 500 publications

References 27 publications

Boundedness of automorphic 𝐿–functions in vertical strips

Boundedness of automorphic 𝐿–functions in vertical strips

Zeros of Automorphic L-Functions and Noncyclic Base Change

Residues of Eisenstein Series and Related Problems

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