Let $X$ be a smooth connected algebraic curve over an algebraically
closed field, let $S$ be a finite closed subset in $X$, and let
$\mathcal F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the
deformation of $\mathcal F_0$. 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 $\mathcal F$ is irreducible and physically rigid, then it is
cohomologically rigid, under the extra condition that $\mathcal F\mod \ell$ is absolutely
irreducible
or that $\mathcal F$ has finite monodromy.