Dilogarithm and higher ℒ-invariants for 𝒢ℒ₃(𝐐_{𝐩})
Authors: Qian Zicheng 
Published: May 2021

The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. LetEEbe a sufficiently large finite extension ofQp\mathbf {Q}_pandρp\rho _pbe app-adic semi-stable representation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper G normal a normal l left-parenthesis bold upper Q Subscript p Baseline overbar slash bold upper Q Subscript p Baseline right-parenthesis right-arrow normal upper G normal upper L Subscript 3 Baseline left-parenthesis upper E right-parenthesis">Gal(Qp¯/Qp)→GL3(E)\mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E)such that the associated Weil–Deligne representationWD(ρp)\mathrm {WD}(\rho _p)has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one(φ,Γ)(\varphi , \Gamma )-modules shows that the Hodge filtration ofρp\rho _pdepends on three invariants inEE. We construct a family of locally analytic representationsΣmin(λ,L1,L2,L3)\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)ofGL3(Qp)\mathrm {GL}_3(\mathbf {Q}_p)depending on three invariantsL1,L2,L3∈E\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E, such that each representation in the family contains the locally algebraic representationAlg⊗Steinberg\mathrm {Alg}\otimes \mathrm {Steinberg}determined byWD(ρp)\mathrm {WD}(\rho _p)(via classical local Langlands correspondence forGL3(Qp)\mathrm {GL}_3(\mathbf {Q}_p)) and the Hodge–Tate weights ofρp\rho _p. Whenρp\rho _pcomes from an automorphic representationπ\piof a unitary group overQ\mathbf {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 withπ\pi) 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 representationΠ\Piin Breuil’s family embeds into the Hecke eigenspace above, the embedding ofΠ\Piextends uniquely to an embedding of aΣmin(λ,L1,L2,L3)\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)into the Hecke eigenspace, for certainL1,L2,L3∈E\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in Euniquely determined byΠ\Pi. This gives a purely representation theoretical necessary condition forΠ\Pito embed into completed cohomology. Moreover, certain natural subquotients ofΣmin(λ,L1,L2,L3)\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)give an explicit complex of locally analytic representations that realizes the derived objectΣ(λ,L_)\Sigma (\lambda , \underline {\mathscr {L}})in (1.14) of [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145]. Consequently, the locally analytic representationΣmin(λ,L1,L2,L3)\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)gives a relation between the higherL\mathscr {L}-invariants studied in [Amer. J. Math. 141 (2019), pp. 611–703] as well as the work of Breuil and Ding and thepp-adic dilogarithm function which appears in the construction ofΣ(λ,L_)\Sigma (\lambda , \underline {\mathscr {L}})in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145].