On the semisimplicity of reductions and adelic openness for 𝐸-rational compatible systems over global function fields

Abstract

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.

Cyclic base change of cuspidal automorphic representations over function fields

“…We can associate to each v ∈ S a partition Jord(π v ) of n, namely the one given by the Jordan decomposition of the nilpotent part of the Weil-Deligne representation rec Kv (π v ). By [BGP,Corollary 6.12], we can find an integer N 0 > max(2, q, n) such that for each prime number ℓ > N and for each place λ of Q of residue characteristic ℓ, ρ λ | G KFq is irreducible. Since S is finite, it suffices to fix a single w ∈ S and show that the number of primes ℓ > N 0 such that there is a place λ|ℓ of Q such that H 2 (K w , ad ρ λ ) = 0 is finite.…”

Independence of algebraic monodromy groups in compatible systems

“…Acknowledgements The author would like to thank his PhD supervisor Andrew Wiles for introducing him to the study of these problems, and for his constant guidance and encouragement. The author would also like to thank Wojciech Gajda for bringing [2] to his attention, and Laura Capuano, Toby Gee, Minhyong Kim, Giacomo Micheli, Damian Rössler, and Jack Thorne for interesting conversations and useful suggestions. Finally, the author would like to thank the anonymous referee for the very helpful comments and suggestions.…”