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

Abstract

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. Let E E be a sufficiently large finite extension of Q p \mathbf {Q}_p and ρ p \rho _p be a p p -adic semi-stable representation G a l ( Q p ¯ / Q p ) → G L 3 ( E ) \mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E) such that the associated Weil–Deligne representation W D ( ρ p ) \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 of ρ p \rho _p depends on three invariants in E E . We construct a family of locally analytic representations Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) of G L 3 ( Q p ) \mathrm {GL}_3(\mathbf {Q}_p) depending on three invariants L 1 , L 2 , L 3 ∈ E \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E , such that each representation in the family contains the locally algebraic representation A l g ⊗ S t e i n b e r g \mathrm {Alg}\otimes \mathrm {Steinberg} determined by W D ( ρ p ) \mathrm {WD}(\rho _p) (via classical local Langlands correspondence for G L 3 ( Q p ) \mathrm {GL}_3(\mathbf {Q}_p) ) and the Hodge–Tate weights of ρ p \rho _p . When ρ p \rho _p comes from an automorphic representation π \pi of a unitary group over Q \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 with π \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 representation Π \Pi in Breuil’s family embeds into the Hecke eigenspace above, the embedding of Π \Pi extends uniquely to an embedding of a Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) into the Hecke eigenspace, for certain L 1 , L 2 , L 3 ∈ E \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in E uniquely determined by Π \Pi . This gives a purely representation theoretical necessary condition for Π \Pi to embed into completed cohomology. Moreover, certain natural subquotients of Σ m i n ( λ , L 1 , L 2 , L 3 ) \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 object Σ ( 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 representation Σ m i n ( λ , L 1 , L 2 , L 3 ) \Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3) gives a relation between the higher L \mathscr {L} -invariants studied in [Amer. J. Math. 141 (2019), pp. 611–703] as well as the work of Breuil and Ding and the p p -adic dilogarithm function which appears in the construction of Σ ( λ , L _ ) \Sigma (\lambda , \underline {\mathscr {L}}) in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145].