  Let $X$ be a smooth connected projective algebraic curve over an
algebraically closed field, and let $S$ be a finite nonempty closed subset in
$X$. We study deformations of $\overline{\mathbb F}_\ell$-sheaves. The
universal deformation space is a formal scheme. Its generic fiber has a rigid
analytic space structure. By studying this rigid analytic space, we prove a
conjecture of Katz which says that if a lisse $\overline{\mathbb Q}_\ell$-sheaf
$\mathcal F$ on $X-S$ is irreducible and rigid, then we have $\mathrm{dim}\,
H^1(X,j_\ast\mathcal End(\mathcal F))=2g$, where $j:X-S\to X$ is the open
immersion, and $g$ is the genus of $X$.
