Simple ℒ-invariants for GL_{𝓃}

Ding, Yiwen

doi:10.1090/tran/7859

Cited by 14 publications

(28 citation statements)

References 31 publications

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“…Remark 2.6. In fact, one can identify L 1 and L 2 with Fontaine-Mazur L -invariants of the corresponding Galois representation via local-global compatibility, according to Remark 3.1 of [Ding19]. This is the reason for the appearance of a sign in (2.23).…”

Section: P-adic Logarithm and Dilogarithmmentioning

confidence: 99%

Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})

Qian

2021

Represent. Theory

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p. via the construction of some interesting locally analytic representations. Let E be a sufficiently large finite extension of Q p and ρ p be a p-adic semi-stable representation Gal(Q p /Q p ) → GL 3 (E) such that the associated Weil-Deligne representation WD(ρ p ) has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one (ϕ, Γ)-modules shows that the Hodge filtration of ρ p depends on three invariants in E. We construct a family of locally analytic representations Σ min (λ, L 1 , L 2 , L 3 ) of GL 3 (Q p ) depending on three invariants L 1 , L 2 , L 3 ∈ E, such that each representation in the family contains the locally algebraic representation Alg ⊗ Steinberg determined by WD(ρ p ) (via classical local Langlands correspondence for GL 3 (Q p )) and the Hodge-Tate weights of ρ p . When ρ p comes from an automorphic representation π of a unitary group over Q which is compact at infinity, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the Hecke eigenspace (associated with π) in the completed cohomology. We note that [Amer. J. Math. 141 (2019), pp. 611-703] constructs a family of locally analytic representations depending on four invariants ( cf. (4) in that publication ) and proves that there is a unique representation in this family that embeds into the Hecke eigenspace above. We prove that if a representation Π in Breuil's family embeds into the Hecke eigenspace above, the embedding of Π extends uniquely to an embedding of a Σ min (λ, L 1 , L 2 , L 3 ) into the Hecke eigenspace, for certain L 1 , L 2 , L 3 ∈ E uniquely determined by Π. This gives a purely representation theoretical necessary condition for Π to embed into completed cohomology. Moreover, certain natural subquotients of Σ min (λ, L 1 , L 2 , L 3 ) give an explicit complex of locally analytic representations that realizes the derived object Σ(λ, L ) in (1.14) of [Ann. Sci.Éc. Norm. Supér. 44 (2011), pp. 43-145]. Consequently, the locally analytic representation Σ min (λ, L 1 , L 2 , L 3 ) gives a relation between the higher L -invariants studied in [Amer. J. Math. 141 (2019), pp. 611-703] as well as the work of Breuil and Ding and the p-adic dilogarithm function which appears in the construction of Σ(λ, L ) in [Ann. Sci.Éc. Norm. Supér. 44 (2011), pp. 43-145].

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Section: P-adic Logarithm and Dilogarithmmentioning

confidence: 99%

Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})

Qian

2021

Represent. Theory

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“…Remark 2.4. A locally analytic version of part (2) of Theorem 2.3 (when D = F and char(F ) = 0) is established in the work of Ding [18] and generalized to split reductive groups by Gehrmann [45]. Higher Ext groups are computed by Orlik and Strauch [55], for split reductive groups and in a suitable category of locally analytic representations (but not in the category of admissible locally analytic representations).…”

Section: 2mentioning

confidence: 99%

“…between the sets of isomorphism classes of the corresponding objects 18 . Every b ∈ G(F ) yields therefore an exact, faithful, F -linear tensor functor…”

Section: 12mentioning

confidence: 99%

“…If the group G is connected, as it is the case in this paper, this assumption is not necessary by the vanishing theorem of Steinberg [17, Lemma 9.1.5]. 18 Despite its innocuous-looking character, this is one of the most difficult results in the book of Fargues and Fontaine [24]. 19 There is a natural equivalence between the category of G-bundles on X and the category of G-torsors on X locally trivial for the étale topology: if Y is G-torsor étale locally trivial, we obtain a G-bundle by sending (V, ρ) ∈ Rep F (G) to Y ×G,ρ V ; conversely, each G-bundle ω yields a locally trivial G-torsor Isom ⊗ (ωcan, ω), where ωcan(V, ρ) = V ⊗F OX .…”

Section: The Structure Of B(g)mentioning

confidence: 99%

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p-adic étale cohomology of period domains

et al. 2021

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HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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“…These can be viewed as multiplicative refinements of the extensions studied in [15], Section 2.5, which were inspired by the constructions in Section 2.2 of [11]. Results of Dat (cf.…”

Section: Introductionmentioning

confidence: 99%

Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

Gehrmann

2022

Res Math Sci

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Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld’s upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of $$\mathcal {L}$$ L -invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups.

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Simple ℒ-invariants for GL_{𝓃}

Cited by 14 publications

References 31 publications

Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})

Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})

p-adic étale cohomology of period domains

Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

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