Deformations and Rigidity of $\ell$-adic Sheaves

Abstract

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 $X$.