2018/12/11 by Backhausz, Tibor
#11F70 #11F80 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1812.04208
Let ℓ and p be distinct primes, n a positive integer, F_ℓ an ℓ-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[GLn(F_ℓ)]-module π(ρA) for a continuous Galois representation ρA : GF_ℓ → GLn(A) over a p-torsionfree reduced complete local W(k)-algebra A interpolating the local Langlands correspondence. However, since π 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 ρ over k, and more generally we find a surjection in one direction. In many cases, including n=2 or 3, they both recover π(ρ) when ρ has coefficients in a finite extension of W(k)[p-1]. On the Galois side, this requires finding minimal lifts between Galois deformations.