[Paper Review] Towards the finite slope part for $\mathrm{GL}_n$

Christophe Breuil, Florian Herzig|arXiv (Cornell University)|Jun 7, 2018

Advanced Algebra and Geometry23 references3 citations

TL;DR

This paper constructs a finite-length locally $ϵ_p$-analytic representation $Π(\rho)^{\mathrm{fs}}$ of $\mathrm{GL}_n(L)$ for a crystalline $n$-dimensional Galois representation $\rho$ over a $p$-adic field $L$, using the locally analytic socle of principal series. It proves that this representation embeds into spaces of $p$-adic automorphic forms under mild genericity conditions, providing strong evidence for the $p$-adic local Langlands conjecture in the finite slope case.

ABSTRACT

Let $L$ be a finite extension of $\mathbb{Q}_p$ and $n\geq 2$. We associate to a crystabelline $n$-dimensional representation of $\mathrm{Gal}(\overline L/L)$ satisfying mild genericity assumptions a finite length locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$. In the crystalline case and in a global context, using the recent results on the locally analytic socle from [BHS17a] we prove that this representation indeed occurs in spaces of $p$-adic automorphic forms. We then use this latter result in the ordinary case to show that certain "ordinary" $p$-adic Banach space representations constructed in our previous work appear in spaces of $p$-adic automorphic forms. This gives strong new evidence to our previous conjecture in the $p$-adic case.

Motivation & Objective

• To construct a finite-length locally $\mathbb{Q}_p$-analytic representation $\Pi(\rho)^{\mathrm{fs}}$ of $\mathrm{GL}_n(L)$ associated to a crystalline Galois representation $\rho$ with mild genericity assumptions.
• To show that $\Pi(\rho)^{\mathrm{fs}}$ occurs as a subrepresentation in spaces of $p$-adic automorphic forms when $\rho$ is crystalline.
• To extend this result to the ordinary case, proving that certain $p$-adic Banach space representations from prior work appear in $p$-adic automorphic forms.
• To provide strong evidence for the $p$-adic local Langlands conjecture by realizing the finite slope part as a subrepresentation in automorphic $p$-adic Banach spaces.

Proposed method

• Constructs $\Pi(\rho)^{\mathrm{fs}}$ as a quotient of a direct sum of maximal subrepresentations of principal series, filtered by irreducible constituents in the locally analytic socle of $\rho$.
• Uses the locally analytic socle $\mathcal{C}^{\operatorname{soc}}(\rho)$, which consists of irreducible representations $C(w^{\mathrm{alg}}, \mathcal{F})$ parameterized by permutations of Hodge–Tate weights and refinements of eigenvalues of Frobenius.
• Applies results from [BHS17a] on the locally analytic socle to identify the largest known subrepresentation of $\widehat{S}(U^p,E)_{\mathbb{Q}_p\text{-an}}[\mathfrak{m}_r]$.
• Employs a partial adjunction between universal unitary completions and locally analytic representations to relate the construction to automorphic forms.
• Uses the isomorphism $\operatorname{Hom}_{\underline{G}(F_p^+)}(\widehat{\otimes}\Pi(r_{\widetilde{v}})_{C_{r_{\widetilde{v}}},w_{\widetilde{v}}^{-1}}(\varepsilon^{n-1}), \widehat{S}(U^p,E)[\mathfrak{m}_r]) \cong \varprojlim \operatorname{Hom}(\widehat{\otimes}\mathrm{PS}(w_{\widetilde{v}}, \sigma_{\widetilde{v}}w_{\widetilde{v}})^{\wedge}(\varepsilon^{n-1}), \cdots)$ to extend embeddings from socle to full representations.
• Relies on Conjecture 5.10 and Theorem 5.11 to establish the existence of the required $\mathrm{GL}_n(L)$-representations in the automorphic space.

Experimental results

Research questions

• RQ1 Can a finite-length locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$ be explicitly constructed for a crystalline Galois representation $\rho$ satisfying mild genericity conditions?
• RQ2 Does this constructed representation $\Pi(\rho)^{\mathrm{fs}}$ appear as a subrepresentation in spaces of $p$-adic automorphic forms?
• RQ3 In the ordinary case, do the $p$-adic Banach space representations previously constructed by the authors occur in $p$-adic automorphic forms?
• RQ4 Does the finite slope part of the automorphic $p$-adic Banach space representation have finite length and embed into the automorphic space?
• RQ5 To what extent does this construction provide evidence for the $p$-adic local Langlands correspondence for $\mathrm{GL}_n$?

Key findings

• The representation $\Pi(\rho)^{\mathrm{fs}}$ is constructed as a finite-length, admissible, locally $\mathbb{Q}_p$-analytic representation of $\mathrm{GL}_n(L)$, using the locally analytic socle of principal series associated to $\rho$.
• For crystalline $\rho$, $\Pi(\rho)^{\mathrm{fs}}$ embeds into the locally analytic vectors of $\widehat{S}(U^p,E)[\mathfrak{m}_r]$, providing the first explicit realization of the finite slope part in automorphic forms for $n \geq 2$.
• In the ordinary case, the theorem proves that $\widehat{\otimes}_{v|p} \Pi(r_{\widetilde{v}})^{\mathrm{ord}}(\varepsilon^{n-1})$ injects into $\widehat{S}(U^p,E)[\mathfrak{m}_r]$, showing that ordinary $p$-adic Banach space representations appear in automorphic forms.
• The injection is given by $\bigoplus_w \left(\widehat{\otimes}_{v|p} \Pi(r_{\widetilde{v}})_{C_{r_{\widetilde{v}}},w_{\widetilde{v}}^{-1}}(\varepsilon^{n-1})\right)^{\oplus n_w} \hookrightarrow \widehat{S}(U^p,E)[\mathfrak{m}_r]$, where $n_w$ counts the multiplicity of $\mathrm{GL}_n(L)$-invariant vectors in the socle.
• The construction confirms the conjecture in [BH15, Conj. 4.2.2] in the crystabelline case, under Conjecture 5.10, and fixes an error in the original proof of [BH15, Thm. 4.4.8].
• The finite slope part $\Pi(\rho)^{\mathrm{fs}}$ is the largest known subrepresentation of $\widehat{S}(U^p,E)_{\mathbb{Q}_p\text{-an}}[\mathfrak{m}_r]$ for $n \geq 2$, and further progress likely requires new ideas.