**Lemma 7.1.** *Let G/k be a form of* G* _{m}, and N in P_{arith} ι-pure of weight zero and arithmetically semisimple. The quotient group G_{arith,N} /G_{geom},N is a group of multiplicative type, in which a Zariski dense subgroup is generated by the image of any single Frobenius conjugacy class Frob_{k,χ}. If the quotient is finite, say of order n, then it is canonically* Z/

*Proof.* Representations of the quotient *G _{arith,N} /G_{geom},N* are objects in <

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