Mod $\mathrm{\ell }$ representations of arithmetic fundamental groups II (A conjecture of A.J. de Jong)

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 $\mathrm{\ell }$ representations $\rho$ of the arithmetic fundamental group ${\pi }_1(X)$ where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($\mathrm{\ne }\mathrm{\ell }$). We are able to show in many cases that the resulting rings are finite flat over ${\text{BZ}}_{\mathrm{\ell }}$. 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 $\mathrm{\ell }$, that have this mod $\mathrm{\ell }$ representation as their reduction.