Here we work on either the split or the nonsplit form. We begin with a lisse sheaf F on a dense open set j : U ⊂ G which is geometrically irreducible, pure of weight zero, and not geometrically isomorphic to (the restriction to U of) any Kummer sheaf Lχ. We denote by G := j? F its middle extension to G. Then the object N := G(1/2) ∈ Parith is pure of weight zero and geometrically irreducible.
Theorem 17.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself. Suppose further that for one of the two possible geometric isomorphisms G/k ≅ Gm/k, F(0)unip is a single Jordan block Unip(e) for some e ≥ 1, and F(∞)unip = 0. For n := dim(!(N)) we have
Ggeom,N = Garith,N = GL