On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

Drinfeld, Vladimir

doi:10.1016/j.aim.2017.06.029

Cited by 13 publications

(27 citation statements)

References 43 publications

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“…Using the result on Frobenius tori, we do not use Hartl-Pál's theorem. It is also worth mentioning that Drinfeld proved the independence of the entire arithmetic monodromy groups (not only the neutral component) over Q , [19]. He uses a stronger representation-theoretic reconstruction theorem (see Remark 4.3.10).…”

Section: Relations With Previous Workmentioning

confidence: 99%

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

D’Addezio

2020

Sel. Math. New Ser.

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It has been proven by Serre, Larsen-Pink and Chin, that over a smooth curve over a finite field, the monodromy groups of compatible semi-simple pure lisse sheaves have "the same" π 0 and neutral component. We generalize their results to compatible systems of semi-simple lisse sheaves and overconvergent F-isocrystals over arbitrary smooth varieties. For this purpose, we extend the theorem of Serre and Chin on Frobenius tori to overconvergent F-isocrystals. To put our results into perspective, we briefly survey recent developments of the theory of lisse sheaves and overconvergent F-isocrystals. We use the Tannakian formalism to make explicit the similarities between the two types of coefficient objects.

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Section: Relations With Previous Workmentioning

confidence: 99%

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

D’Addezio

2020

Sel. Math. New Ser.

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“…From this we shall deduce Theorem 6.11 in Subsection 6.3, which says that any absolutely irreducible compatible system has absolutely irreducible reduction for almost all λ. The proof of this was first published by Drinfeld in [Dri15], and we basically give a more detailed version of his proof. The final Subsection 6.4 contains the proof of Theorem 6.14, which is part (c) of Theorem 1.1 in the absolutely irreducible case.…”

Section: Introductionmentioning

confidence: 93%

“…(a) The definition of the motivic group M should also include conditions at the places of E above p; this is possible using the work [Abe] of Abe that attaches certain isocrystals at places above p and thus extends the work of L. Lafforgue; cf. also [Dri15]. For the present work these places are irrelevant, and so we omit them.…”

Section: The Motivic Groupmentioning

confidence: 99%

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

Böckle

Gajda

Petersen

2019

Trans. Amer. Math. Soc.

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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.

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“…For a general reductive group G the notion of compatible family is subtle (because the obvious condition on the conjugacy classes of the Frobenius elements is not sufficient). In [Dri15] Drinfeld gave the right conditions to define compatible families and proved that any continuous semisimple morphism Gal(F /F ) → G(Q ℓ ) factorizing through π 1 (U, η) for some open dense U ⊂ X (and such that the Zariski closure of its image is semisimple) belongs to a unique compatible family.…”

Section: Independence On ℓmentioning

confidence: 99%

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

Lafforgue

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text).The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [Gel84,Bum97,BeGe03,Tay04,Fre07,Art14]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck's vision of motives, special values of L-functions, Ramanujan-Petersson conjecture, generalized Riemann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been established. Moreover the Langlands correspondence over function fields admits a geometrization, the "geometric Langlands program", which is related to conformal field theory in Theoretical Physics.Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split.The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later,

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On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field

Cited by 13 publications

References 43 publications

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

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

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

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