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

Abstract

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].

On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$

“…We will study this second filtration as well as its relative position with respect to the aforementioned lattice of canonical subspaces in a forthcoming work, which should lead to a proof of Conjecture 5.18 which generalizes Théorème 6.23 and Remarque 6.24 of [Schr11]. These computations also shed light on the construction (in the GL n case) of locally analytic representations in Breuil's Ext 1 conjecture as well as those in Theorem 1.1 of [Qian21].…”

“…Then there clearly exists some 1 ≤ t ≤ t ′ such that y α t ′ = x ∞ α t ′ , which implies that y α t ′ ∈ W Iα t ′ ,∅ and thus Remark 5.12. Conjecture 5.11 is known for n = 2 with K = Q p by Breuil in [Bre04] and [Bre10]), by Schraen and Ding for n = 2 with general K in [Schr10] and [Ding16]), and for n = 3 with K = Q p by [Schr11], [Bre19], [BD20] and [Qian21]. We refer further details to Remark 1.4.…”

“…Remark 5.13. As in [Bre10], [Ding16], [Bre19], [BD20] and [Qian21], the representation W x (λ) is expected to satisfy certain p-adic local-global compatibility. Let F be a number field, v|p a finite place of p, G /F a reductive group satisfying G(F v ) ∼ = GL n (F v ) and U v ⊆ G(A ∞,v ) a compact open subgroup.…”

“…for the illustration of this subrepresentation with each factor defined in paragraphs before it). Then we can combine Proposition 6.8, Theorem 7.1 of [Qian21] with Theorem 1.1 of [BD20] and obtain (1.14) E ∆,∅ = Ext 2 PGL 3 (Qp) (1 3 , St an 3 ) ։ Ext 2 PGL 3 (Qp) (1 3 , Π − (D)) ∼ = Ext 2 PGL 3 (Qp) (1 3 , Π(D))…”

“…In another direction, Schraen has done extensive computations in his thesis [Schr11] and find higher L -invariants inside certain (higher) Ext-groups between locally analytic generalized Steinberg representations. Afterwards, part of the author's thesis has connected Breuil-Ding and Schraen's approaches in an explicit way, and in particular leading to a new family of locally analytic representations Π Q (ρ) (see (1.1) of [Qian21]) which contains Π BD (ρ). Breuil-Ding's approach has the advantage of constructing explicit true representations (instead of objects in certain derived category), but its generalization to GL n (K) is à priori difficult due to complexity of Π BD (ρ) and Π Q (ρ).…”