Change of coefficients in the $p\ne \ell$ local Langlands correspondence for $G{L}_n$

Abstract

Let $\ell$ and $p$ be distinct primes, $n$ a positive integer, $F_{\ell }$ an $\ell$-adic local field of characteristic $0,$ and let $W(k)$ denote the ring of Witt vectors over an algebraically closed field of characteristic $p$. Work of Emerton-Helm, Helm and Helm-Moss defines and constructs a smooth $A[{GL}_n(F_{\ell })]$-module $\tilde{\pi }({\rho }_A)$ for a continuous Galois representation ${\rho }_A:G_{F_{\ell }}\to {GL}_n(A)$ over a $p$-torsionfree reduced complete local $W(k)$-algebra $A$ interpolating the local Langlands correspondence. However, since $\tilde{\pi }$ is not a functor, there is no clear way to speak about the local Langlands correspondence over non-reduced or finite characteristic $W(k)$-algebras. We describe two natural and reasonable variants of the local Langlands correspondence with arbitrary complete local $W(k)$-algebras as coefficients. They are isomorphic when evaluated on the universal framed deformation of a Galois representation $\overline{\rho }$ over $k$, and more generally we find a surjection in one direction. In many cases, including $n=2$ or $3,$ they both recover $\tilde{\pi }(\rho )$ when $\rho$ has coefficients in a finite extension of $W(k)[p^{-1}].$ On the Galois side, this requires finding minimal lifts between Galois deformations.

23 pages; comments welcome

References in corpus (1)

• Converse theorems and the local Langlands correspondence in families