Backhausz, Tibor András; (2022) The local Langlands correspondence in singular families of representations. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

Let ℓ and p be distinct primes, n a positive integer, Fℓ 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(Fℓ)]-module π(ρA) for a continuous Galois representation ρA : GFℓ → 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 straightforward way to speak about the local Langlands correspondence over non-reduced or finite characteristic W(k)- algebras (beyond what is known by considering the Bernstein centre). We describe two natural ways to attach smooth admissible representations to Galois representations with arbitrary complete local W(k)-algebras as coefficients, one defined through the universal deformation ring, and another defined by families of representations defined over discrete valuation rings. We show that these are isomorphic when evaluated on the universal framed deformation of a Galois representation ¯ρ over k. On the Galois side, this requires finding minimal lifts of Galois deformations on irreducible components of the deformation space.

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