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FROM THE FUNCTION-SHEAF DICTIONARY TO QUASICHARACTERS OF
$p$-ADIC TORI
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 13 October 2015, pp. 1-37
- Print publication:
- February 2018
-
- Article
- Export citation
-
We consider the rigid monoidal category of character sheaves on a smooth commutative group scheme
$G$ over a finite field 
$k$, and expand the scope of the function-sheaf dictionary from connected commutative algebraic groups to this setting. We find the group of isomorphism classes of character sheaves on 
$G$, and show that it is an extension of the group of characters of 
$G(k)$ by a cohomology group determined by the component group scheme of 
$G$. We also classify all morphisms in the category character sheaves on 
$G$. As an application, we study character sheaves on Greenberg transforms of locally finite type Néron models of algebraic tori over local fields. This provides a geometrization of quasicharacters of 
$p$-adic tori.
