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We consider the rigid monoidal category of character sheaves on a smoothcommutative group scheme $G$ over a finite field $k$ and expand the scope ofthe function-sheaf dictionary from connected commutative algebraic groups tothis 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 acohomology group determined by the component group scheme of $G$. We alsoclassify all morphisms in the category character sheaves on $G$. As anapplication, we study character sheaves on Greenberg transforms of locallyfinite type N\'eron models of algebraic tori over local fields. This provides ageometrization of quasicharacters of $p$-adic tori.

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