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

[摘要] We study deformation rings of an n-dimensional representation $overline{ho}$, 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 ($ eq ell$). When $overline{ho}$ 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 $overline{ho}$ 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.

[学科分类] 数学（综合）