[Paper Review] On the p-adic cohomology of the Lubin-Tate tower

Peter Scholze|arXiv (Cornell University)|Jun 12, 2015

Algebraic Geometry and Number Theory14 references5 citations

TL;DR

This paper establishes a canonical functor from admissible $p$-adic representations of $\mathrm{GL}_n(F)$ to admissible $p$-adic representations of $\mathrm{Gal}_F \times D^\times$, where $D$ is the central division algebra of invariant $1/n$, via the $p$-adic cohomology of the Lubin-Tate tower. The key result is the finiteness and Galois-equivariance of the cohomology of the associated sheaf on $\mathbb{P}^{n-1}_{\breve{F}}$, which realizes a local $p$-adic Langlands correspondence and verifies local-global compatibility with the Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin patching construction.

ABSTRACT

We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.

Motivation & Objective

• To construct a canonical functor from admissible $p$-adic $\mathrm{GL}_n(F)$-representations to admissible $p$-adic $\mathrm{Gal}_F \times D^\times$-representations via $p$-adic cohomology of the Lubin-Tate tower.
• To establish local-global compatibility between this correspondence and the global patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.
• To verify that the cohomology groups are admissible and carry continuous Galois actions, extending the Weil group action.
• To provide evidence for a $p$-adic local Langlands correspondence beyond $\mathrm{GL}_2(\mathbb{Q}_p)$, including a $p$-adic Jacquet-Langlands correspondence.
• To clarify the geometric and representation-theoretic structure of the Lubin-Tate tower and its period map in the context of Rapoport-Zink spaces and period domains.

Proposed method

• Constructs a Weil-equivariant sheaf $\mathcal{F}_\pi$ on $\mathbb{P}^{n-1}_{\breve{F}}$ from a smooth $\mathrm{GL}_n(F)$-representation $\pi$ on an $\mathbb{F}_p$-vector space.
• Uses the Gross-Hopkins period map $\pi_{\mathrm{GH}}: \mathcal{M}_{\mathrm{LT},\infty} \to \mathbb{P}^{n-1}_{\breve{F}}$ to pull back $\mathcal{F}_\pi$ to the inverse limit of the Lubin-Tate tower.
• Applies finiteness results for $\acute{e}$tale cohomology of proper rigid-analytic spaces to show that $H^i_{\acute{e}t}(\mathbb{P}^{n-1}_C, \mathcal{F}_\pi)$ is finite-dimensional and vanishes for $i > 2(n-1)$.
• Establishes that the cohomology groups carry admissible $D^\times$-actions and extend continuously to $\mathrm{Gal}_F$-actions via the Weil descent datum.
• Relies on the properness of $\mathbb{P}^{n-1}$ and the surjectivity of the period map to ensure cohomological finiteness.
• Uses the theory of accessible and weakly accessible period domains (Appendix by Rapoport) to analyze the structure of the period domain and its image under the crystalline period map.

Experimental results

Research questions

• RQ1 Does the $p$-adic cohomology of the Lubin-Tate tower yield a canonical functor from $\mathrm{GL}_n(F)$-representations to $\mathrm{Gal}_F \times D^\times$-representations?
• RQ2 Is the resulting cohomology admissible and compatible with the Galois action, extending the Weil group action?
• RQ3 Does this construction satisfy local-global compatibility with the global patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin?
• RQ4 Can this construction be interpreted as a $p$-adic Jacquet-Langlands correspondence between $\mathrm{GL}_n(F)$ and $D^\times$-representations?
• RQ5 Under what conditions is the image of the crystalline period map equal to the admissible locus in the period domain?

Key findings

• The cohomology groups $H^i_{\acute{e}t}(\mathbb{P}^{n-1}_C, \mathcal{F}_\pi)$ are independent of the choice of algebraically closed complete extension $C$ of $\breve{F}$, and vanish for $i > 2(n-1)$.
• For all $i \geq 0$, the cohomology $H^i_{\acute{e}t}(\mathbb{P}^{n-1}_{\mathbb{C}_p}, \mathcal{F}_\pi)$ is an admissible $D^\times$-representation.
• The action of the Weil group $W_F$ on the cohomology extends continuously to an action of the absolute Galois group $\mathrm{Gal}_F$.
• The construction gives a canonical functor from admissible $\mathrm{GL}_n(F)$-representations to admissible $\mathrm{Gal}_F \times D^\times$-representations, realizing a $p$-adic local Langlands correspondence.
• The correspondence is compatible with the patching construction: composing the patching construction with this functor recovers the original Galois representation for $n=2$, as shown in Corollary 9.3.
• The image of the crystalline period morphism coincides with the admissible locus $\mathcal{F}(G,b,\{\mu\})^{\rm a}$ when the local Shimura variety arises from an RZ-space of type EL or PEL.