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# Symplectic Examples

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. Wedenoteby G := j?F its middle extension to G. Then the object N := G(1/2)[1] Parith is pure of weight zero and geometrically irreducible.

Theorem 18.1. Suppose that N is not geometrically isomorphic to any nontrivial multiplicative translate of itself, and that N is symplectically self-dual. Suppose further that for either of the two possible geometric isomorphisms G/kGm/k, both F(0)unip and F()unip are a single Jordan block Unip(e) of the same size e ≥ 1. For ...

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