On the p-adic cohomology of the Lubin-Tate tower

TL;DR

This paper establishes a finiteness result for the p-adic cohomology of the Lubin-Tate tower and constructs a functor linking representations of GL_n(F) to Galois and division algebra representations, confirming local-global compatibility.

Contribution

It introduces a canonical functor from admissible p-adic representations of GL_n(F) to those of Galois groups and division algebras, and verifies compatibility with existing patching methods.

Findings

Proves a finiteness result for p-adic cohomology of Lubin-Tate tower.

Constructs a functor linking admissible p-adic representations to Galois and division algebra representations.

Verifies local-global compatibility and compatibility with patching constructions.

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.