On the $p$-adic cohomology of some $p$-adically uniformized varieties

Grosse-Klönne, Elmar

doi:10.1090/s1056-3911-10-00541-2

Cited by 3 publications

(2 citation statements)

References 28 publications

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“…In [AdS03] Alon and de Shalit prove a version of the above corollary for the cohomology of cocompact, discrete subgroups of P GL d (F p ). See also the article [GK11] of Große-Klönne for the case of non-trivial coefficient systems.…”

Section: Automorphic L-invariantsmentioning

confidence: 99%

Automorphic L-invariants for reductive groups

Gehrmann

2019

Preprint

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Let G be a reductive group over a number field F , which is split at a finite place p of F , and let π be a cuspidal automorphic representation of G, which is cohomological with respect to the trivial coefficient system and Steinberg at p. We use the cohomology of p-arithmetic subgroups of G to attach automorphic L-invariants to π. This generalizes a construction of Darmon (respectively Spieß), who considered the case G = GL 2 over the rationals (respectively over a totally real number field). These L-invariants depend a priori on a choice of degree of cohomology, in which the representation π occurs. We show that they are independent of this choice provided that the π-isotypical part of cohomology is cyclic over Venkatesh's derived Hecke algebra. Further, we show that automorphic L-invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic L-invariants are equal to the Fontaine-Mazur L-invariants of the associated Galois representation.

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Section: Automorphic L-invariantsmentioning

confidence: 99%

Automorphic L-invariants for reductive groups

Gehrmann

2019

Preprint

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“…[16]). Finally, in [17], Große-Klönne used global methods to prove part (ii) of Conjecture 4.1 under the assumption that Γ is of arithmetic type in the sense of [17], §4.…”

Section: Applications To P-adic Symmetric Spacesmentioning

confidence: 99%

Homological vanishing theorems for locally analytic representations

Kohlhaase

Schraen

2011

Math. Ann.

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Let G be the group of rational points of a split connected reductive group over a p-adic local field, and let Γ be a discrete and cocompact subgroup of G. Motivated by questions on the cohomology of p-adic symmetric spaces, we investigate the homology of Γ with coefficients in locally analytic principal series and related representations of G. The vanishing and finiteness results that we find partially rely on the compactness of certain Banach-Hecke operators. We also give a new construction of P. Schneider's reduced Hodge-de Rham spectral sequence and show that the induced filtration is the Hodge-de Rham filtration. In a previously unknown case, our vanishing theorems then also imply two other of P. Schneider's conjectures.Mathematics Subject Classification (1991) 22E50 · 20G10 · 11F70

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Automorphic ℒ‐invariants for reductive groups

Gehrmann

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Let G be a reductive group over a number field F, which is split at a finite place 𝔭 {\mathfrak{p}} of F, and let π be a cuspidal automorphic representation of G, which is cohomological with respect to the trivial coefficient system and Steinberg at 𝔭 {\mathfrak{p}} . We use the cohomology of 𝔭 {\mathfrak{p}} -arithmetic subgroups of G to attach automorphic ℒ {\mathcal{L}} -invariants to π. This generalizes a construction of Darmon (respectively Spieß), who considered the case G = GL 2 {G={\mathrm{GL}}_{2}} over the rationals (respectively over a totally real number field). These ℒ {\mathcal{L}} -invariants depend a priori on a choice of degree of cohomology, in which the representation π occurs. We show that they are independent of this choice provided that the π-isotypic part of cohomology is cyclic over Venkatesh’s derived Hecke algebra. Further, we show that automorphic ℒ {\mathcal{L}} -invariants can be detected by completed cohomology. Combined with a local-global compatibility result of Ding it follows that for certain representations of definite unitary groups the automorphic ℒ {\mathcal{L}} -invariants are equal to the Fontaine–Mazur ℒ {\mathcal{L}} -invariants of the associated Galois representation.

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On the $p$-adic cohomology of some $p$-adically uniformized varieties

Cited by 3 publications

References 28 publications

Automorphic L-invariants for reductive groups

Automorphic L-invariants for reductive groups

Homological vanishing theorems for locally analytic representations

Automorphic ℒ‐invariants for reductive groups

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