Deformations and Rigidity of $ell$-adic Sheaves

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 $mathrm{dim}, H^1(X,j_astmathcal End(mathcal F))=2g$, where $j:X-Sto X$ is the open immersion, and $g$ is the genus of $X$.