From the function-sheaf dictionary to quasicharacters of $p$-adic tori

Clifton Cunningham, David Roe

Published 2013-10-10, updated 2014-10-01Version 3

We introduce the category of quasicharacter sheaves on smooth commutative group schemes $G$ over finite fields $k$ and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We show that the group of isomorphism classes of quasicharacter sheaves is an extension of the group of characters of $G(k)$ by a cohomology group determined by the component group scheme of $G$. As an application, we find a category of sheaves for which isomorphism classes correspond to quasicharacters in the case of algebraic tori, abelian varieties and unipotent wound groups over local fields. In this geometrization of quasicharacters, our sheaves are local systems on Greenberg transforms of locally finite type N\'eron models.