2003/12/29 by Gebhard Boeckle, Boeckle, Gebhard, Chandrashekhar Khare +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT
paper · pdf · doi:10.48550/arxiv.math/0312490
This revised version is cleaner, although not substantially different. We check that our arguments work for \ell=2
openalex publication_date 2003/12/29 · arxiv created 2004/04/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
As a sequel to our proof of the analog of Serre's conjecture for function fields in Part I of this work, we study in this paper the deformation rings of n-dimensional mod ℓ representations ρ of the arithmetic fundamental group π1(X) where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p (≠ ℓ). We are able to show in many cases that the resulting rings are finite flat over \BZ_ℓ. The proof principally uses a lifting result of the authors in Part I of this two-part work, Taylor-Wiles systems and the result of Lafforgue. This implies a conjecture of A.J. ~de Jong for representations with coefficients in power series rings over finite fields of characteristic ℓ, that have this mod ℓ representation as their reduction.