Compatibility of local and global Langlands correspondences

Taylor, Richard; Yoshida, Teruyoshi

doi:10.1090/s0894-0347-06-00542-x

Cited by 99 publications

(104 citation statements)

References 14 publications

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“…• Π is square integrable at some finite place, then Conjecture 1.1 is known by a series of works [Kot92a], [Clo91], [HT01] and [TY07] for every y l. More precisely, the assertions of Theorem 1.2 below, without any condition on Π ∞ when m is even, are known under the additional assumption on Π as above. (Although the assertion (vi) is not explicitly recorded, it follows easily from the contents of [TY07].…”

Section: If We Further Assume Thatmentioning

confidence: 99%

“…(Although the assertion (vi) is not explicitly recorded, it follows easily from the contents of [TY07]. )…”

Section: If We Further Assume Thatmentioning

confidence: 99%

“…In Corollary 7.9 we prove relevant cases of the Ramanujan-Petersson conjecture. Finally in Section 7.3, we imitate the argument of [TY07] to prove a stronger result on the local-global compatibility and the last assertion of Theorem 1.2.…”

Section: If We Further Assume Thatmentioning

confidence: 99%

“…I am very grateful and indebted to Richard Taylor and Michael Harris for generously sharing their knowledge on the automorphic representations of unitary (similitude) groups and the construction of Galois representations. Taylor also explained to me important points in adapting [TY07] to my setting and pointed out several errors in the latest version of the article. Harris envisaged a program for obtaining Theorem 1.2 and more as part of the "Paris book project", which greatly influenced and inspired our work.…”

Section: If We Further Assume Thatmentioning

confidence: 99%

“…(See [TY07,§1] for instance, to review these notions.) The notation L m,Fy (Π y ) means the local Langlands image of Π y , where the geometric normalization is used ( §2.3).…”

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confidence: 99%

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Galois representations arising from some compact Shimura varieties

2011

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Our aim is to establish some new cases of the global Langlands correspondence for GLm. Along the way we obtain a new result on the description of the cohomology of some compact Shimura varieties. Let F be a CM field with complex conjugation c and Π be a cuspidal automorphic representation of GLm(AF ). Suppose that Π ∨ Π • c and that Π∞ is cohomological. A very mild condition on Π∞ is imposed if m is even. We prove that for each prime l there exists a continuous semisimple representation R l (Π) : Gal(F /F ) → GLm(Q l ) such that Π and R l (Π) correspond via the local Langlands correspondence (established by Harris-Taylor and Henniart) at every finite place w l of F ("local-global compatibility"). We also obtain several additional properties of R l (Π) and prove the RamanujanPetersson conjecture for Π. This improves the previous results obtained by Clozel, Kottwitz, Harris-Taylor and Taylor-Yoshida, where it was assumed in addition that Π is square integrable at a finite place. It is worth noting that the mild condition on Π∞ in our theorem is removed by a p-adic deformation argument, thanks to Chenevier-Harris.Our approach generalizes that of Harris-Taylor, which constructs Galois representations by studying the l-adic cohomology and bad reduction of certain compact Shimura varieties attached to unitary similitude groups. The central part of our work is the computation of the cohomology of the so-called Igusa varieties. Some of the main tools are the stabilized counting point formula for Igusa varieties and techniques in the stable and twisted trace formulas.Recently there have been results by Morel and Clozel-Harris-Labesse in a similar direction as ours. Our result is stronger in a few aspects. Most notably, we obtain information about R l (Π) at ramified places.

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Section: If We Further Assume Thatmentioning

confidence: 99%

“…(Although the assertion (vi) is not explicitly recorded, it follows easily from the contents of [TY07]. )…”

Section: If We Further Assume Thatmentioning

confidence: 99%

Section: If We Further Assume Thatmentioning

confidence: 99%

Section: If We Further Assume Thatmentioning

confidence: 99%

“…(See [TY07,§1] for instance, to review these notions.) The notation L m,Fy (Π y ) means the local Langlands image of Π y , where the geometric normalization is used ( §2.3).…”

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confidence: 99%

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Galois representations arising from some compact Shimura varieties

2011

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Bigness in Compatible Systems

Snowden

Wiles

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if ρ is an ℓ-adic representation of the absolute Galois group of a number field for which the residual representation ρ comes from a modular form then so does ρ. This theorem has numerous hypotheses; a crucial one is that the image of ρ must be "big," a technical condition on subgroups of GLn. In this paper we investigate this condition in compatible systems. Our main result is that in a sufficiently irreducible compatible system the residual images are big at a density one set of primes. This result should make some of the work of Clozel, Harris and Taylor easier to apply in the setting of compatible systems. Elementary properties of bignessIn this section we establish some elementary properties of bigness. Throughout this section, k denotes a finite field of characteristic ℓ, V a finite dimensional vector space over k and G a subgroup of GL(V ).

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On the $$\mathrm {GL}_n$$ GL n -eigenvariety and a conjecture of Venkatesh

Hansen

Thorne

2017

Sel. Math. New Ser.

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Let π be a cuspidal, cohomological automorphic representation of GL n (A). Venkatesh has suggested that there should exist a natural action of the exterior algebra of a certain motivic cohomology group on the π -part of the Betti cohomology (with rational coefficients) of the GL n (Q)-arithmetic locally symmetric space. Venkatesh has given evidence for this conjecture by showing that its 'l-adic realization' is a consequence of the Taylor-Wiles formalism. We show that its ' p-adic realization' is related to the properties of eigenvarieties.

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Compatibility of local and global Langlands correspondences

Cited by 99 publications

References 14 publications

Galois representations arising from some compact Shimura varieties

Galois representations arising from some compact Shimura varieties

Bigness in Compatible Systems

On the $$\mathrm {GL}_n$$ GL n -eigenvariety and a conjecture of Venkatesh

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