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 *L _{χ}*. We denote by

**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* ≅ G* _{m}/k, F*(0)

*G _{geom,N}* =

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