Asymptotique des nombres de Betti des variétés arithmétiques

Cossutta, Mathieu

doi:10.1215/00127094-2009-057

Cited by 4 publications

(24 citation statements)

References 25 publications

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“…Part 4 is devoted to applications. Apart from those already mentioned, we deduce from our results and recent results of Cossutta [16] and Cossutta-Marshall [17] an estimate on the growth of the small degree Betti numbers in congruence covers of hyperbolic manifolds of simple type. We also deduce from our results an application to the non-vanishing of certain periods of automorphic forms.…”

Section: 19supporting

confidence: 67%

“…In the general case where G = SO(p, q) we can proceed as in the preceeding section to deduce Corollary 1.15 from Theorem 1.14 and Proposition 5. 16. Note however that in this general case we have to use both totally geodesic cycles associated to SO(p − n, q) (related to cohomological representations associated to a Levi of the form L = C × SO 0 (p − 2n, q)) and totally geodesic cycles associated to SO(p, q − n) (related to cohomological representations associated to a Levi of the form L = C × SO 0 (p, q − 2n)).…”

Section: According To the Hard Lefschetz Theorem The Mapmentioning

confidence: 99%

“…In fact the inequality is an equality but we do not need it here. 16.11. Proposition 16.10 implies Proposition 16.2.…”

Section: 7mentioning

confidence: 99%

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Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

Bergeron

Millson

Mœglin

2016

Int Math Res Notices

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Abstract. We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds generate the cohomology groups of degree n of compact congruence p-dimensional hyperbolic manifolds "of simple type" as long as n is strictly smaller than p 3 . We also prove that for connected Shimura varieties associated to O(p, 2) the Hodge conjecture is true for classes of degree < p+13 . The proof of our general theorem makes use of the recent endoscopic classification of automorphic representations of orthogonal groups by [5]. As such our results are conditional on the hypothesis made in this book, whose proofs have only appear on preprint form so far; see the second paragraph of subsection 1.19 below.

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Section: 19supporting

confidence: 67%

Section: According To the Hard Lefschetz Theorem The Mapmentioning

confidence: 99%

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Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

Bergeron

Millson

Mœglin

2016

Int Math Res Notices

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“…Both π and ρ determines real linear forms Λ π and Λ ρ -the exponents -on a M ; it measures the failure of the representation to be tempered. 12 The use of the stable twisted trace formula being replaced by the (untwisted) stable trace formula. 13 We will mainly deal with the situation where v is Archimedean -so that H(G) = C ∞ c (G) -except in Appendice A where we have to deal with all places.…”

Section: Representations Induced From Square Integrable Automorphic Rmentioning

confidence: 99%

“…The restriction to C × of the Langlands' parameter of a cohomological representation A q is explicitly described in [7, §5.3]. We use here the following alternate description given by Cossutta [12]:…”

Section: Infinitesimal Charactermentioning

confidence: 99%

The Hodge conjecture and arithmetic quotients of complex balls

2016

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In memory of Raquel Maritza Gilbert beloved wife of the second author.Abstract. Let S be a closed Shimura variety uniformized by the complex n-ball. The Hodge conjecture predicts that every Hodge class in H 2k (S, Q), k = 0, . . . , n, is algebraic. We show that this holds for all degrees k away from the neighborhood ]n/3, 2n/3[ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension c of the subvariety) centered at the middle dimension of S. Finally we extend most of these results to Shimura varieties associated to unitary groups of any signature. The proofs make use of the recent endoscopic classification of automorphic representations of classical groups by [1,58]. As such our results are conditional on the stabilization of the trace formula for the (disconnected) groups GL(N ) ⋊ θ associated to base change. At present the stabilization of the trace formula has been proved only for the case of connected groups. But the extension needed is now announced, see §1.13 for more details.

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Theta Lifting and Cohomology Growth in p-adic Towers

Cossutta¹,

Marshall

2012

International Mathematics Research Notices

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We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called 'p-adic congruence tower' we obtain sharp asymptotics for the number of representations which occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.The second author is supported by NSF grant this ensures that the metaplectic cover of Sp(W ) splits on restriction to G × G ′ by a result of Kudla [6]. Let L ⊂ V be an O E lattice, whose completion in V v we shall assume to be self dual whenever V is unramified, and Γ ⊂ G(R) the arithmetic group stabilising L. Let r be the dimension of the maximal isotropic subspace of V , and define

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Asymptotique des nombres de Betti des variétés arithmétiques

Cited by 4 publications

References 25 publications

Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

Hodge Type Theorems for Arithmetic Manifolds Associated to Orthogonal Groups

The Hodge conjecture and arithmetic quotients of complex balls

Theta Lifting and Cohomology Growth in p-adic Towers

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