Séminaire de Géométrie Arithmétique Paris-Pékin-Tokyo
# Deformation and rigidity of $\ell$-adic sheaves

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Centre de conférences Marilyn et James Simons (IHES)
### Centre de conférences Marilyn et James Simons

#### IHES

Le Bois Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

Let X be a smooth connected algebraic curve over an algebraically closed field, let S be a finite closed subset in X, and let *F*_0 be a lisse *l*-torsion sheaf on X-S. We study the deformation of *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 $\overline{Q}_\ell$-sheaf *F* is irreducible and physically rigid, then it is cohomologically rigid in the sense that \chi(X,j_*End(*F*))=2, where j:X-S--> X is the open immersion.

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