Mod ℓ representations of arithmetic fundamental groups II (A conjecture of A.J. de Jong)

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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 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.