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# The Main Theorem

Lemma 7.1. Let G/k be a form of Gm, and N in Parith ι-pure of weight zero and arithmetically semisimple. The quotient group Garith,N /Ggeom,N is a group of multiplicative type, in which a Zariski dense subgroup is generated by the image of any single Frobenius conjugacy class Frobk,χ. If the quotient is finite, say of order n, then it is canonically Z/nZ, and the image in this quotient of any Frobenius conjugacy class FrobE,χ is deg(E/k) mod n.

Proof. Representations of the quotient Garith,N /Ggeom,N are objects in <N>arith which are geometrically trivial, i.e., those objects V ⊗ δ1, for V some completely reducible representation of Gal(k/k), which lie in <N>arith. Such an object is a finite direct sum of one-dimensional ...

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