Mod $\ell$ representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong

Abstract

We study deformation rings of an n-dimensional representation $\overline{\rho}$, defined over a finite field of characteristic $\ell$, of the arithmetic fundamental group $\pi_1(X)$, where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p ($ \neq \ell$). When $\overline{\rho}$ has large image, we are able to show that the resulting rings are finite flat over $\mathbf{Z}_\ell$. The proof principally uses a Galois-theoretic lifting result of the authors in Part I of this two-part work, a lifting result for cuspidal mod $\ell$ forms of Ogilvie, Taylor–Wiles systems and the result of Lafforgue. This implies a conjecture of de Jong for representations of $\pi_1(X)$ with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation $\overline{\rho}$ as their reduction. A proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory. The methods are different.