On the semisimplicity of reductions and adelic openness for EEE-rational compatible systems over global function fields (original) (raw)
Transactions of the American Mathematical Society
Let X be a normal geometrically connected variety over a finite field κ of characteristic p. Let (ρ λ : π1(X) → GLn(E λ)) λ be any semisimple E-rational compatible system where E is a number field and λ ranges over the finite places of E not above p. We derive new properties on the monodromy groups of such systems for almost all λ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. A key input to our results are automorphic methods and the Langlands correspondence over global function fields proved in [Laf02] by L. Lafforgue. To say more, let (ρ λ : π1(X) → GLn(k λ)) λ be the corresponding mod-λ system, where for every λ by O λ and k λ we denote the valuation ring and the residue field of E λ , and where the reduction is done with respect to some π1(X)-stable O λ-lattice Λ λ of E n λ. Let also G geo λ be the Zariski closure of ρ λ (π1(Xκ)) in GLn,E and let G geo λ be its schematic closure in AutO λ (Λ λ). Assume in the following that the algebraic groups G geo λ are connected. We prove that for almost all λ the group scheme G geo λ is semisimple over O λ and its special fiber agrees with the Nori envelope of ρ λ (π1(Xκ)). A comparable result under different hypotheses was proved in [CHT17] by Cadoret, Hui and Tamagawa using other methods. As an intermediate result, we show for X a curve that any potentially tame compatible system of mod-λ representations can be lifted to a compatible system over a number field, cf. [Dri15]; this implies for almost all λ the semisimplicity of the restriction ρ λ | π 1 (X κ). Finally we establish adelic openness for (ρ λ | π 1 (X κ)) λ in the sense of Hui-Larsen [HL15], for E = Q in general, and for E Q under additional hypotheses. Contents 1 Introduction 2 2 Notation 7 3 Basic results on compatible systems over function fields 8 3.
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