Slopes of indecomposable $F$-isocrystals

Drinfeld, Vladimir; Kedlaya, Kiran S.

doi:10.4310/pamq.2017.v13.n1.a5

Cited by 16 publications

(35 citation statements)

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“…By [DrKe16] the conjectures above would also imply p-adic estimates on Hecke eigenvalues which would sharper than those in [Laf11].…”

Section: Conjectures On Arthur Parametersmentioning

confidence: 93%

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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“…By [DrKe16] the conjectures above would also imply p-adic estimates on Hecke eigenvalues which would sharper than those in [Laf11].…”

Section: Conjectures On Arthur Parametersmentioning

confidence: 93%

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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“…In the limit, one gets (7.2). For each open subgroup U ⊂ Π one has the group π [DK,Prop. B.7.6(ii)], one has a canonical isomorphism…”

Section: Definition Of Parmentioning

confidence: 99%

“…In the first part of Appendix D we reformulate these data in more "elementary" terms (such as Dynkin diagrams and extensions of Π by finite abelian groups). In §D.2-D.3 we translate some results of [Laf,Laf2,DK] into this language.…”

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confidence: 99%

“…26 It is similar to the "Newton polygon" in the sense of [Ka]. Laf2,DK]. As usual, we identify the set Hom(P C , Q)/W C with the dominant cone Hom + (P C , Q) ⊂ Hom(P C , Q), and we equip it with the following partial order: given dominant rational coweightsμ 1 andμ 2 we say thatμ 1 ≤μ 2 ifμ 2 −μ 1 is a linear combination of simple coroots with non-negative rational coefficients.…”

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confidence: 99%

“…The proposition is proved in [DK,] using F -isocrystals and the main theorem of T. Abe's work [Ab2] (existence of "crystalline companions" in the case dim X = 1).…”

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confidence: 99%

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

Drinfeld

2018

Advances in Mathematics

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Abstract. Let Π be the fundamental group of a smooth variety X over Fp. Let Q be an algebraic closure of Q. Given a non-Archimedean place λ of Q prime to p, consider the λ-adic pro-semisimple completion of Π as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on λ. If dim X = 1 we also prove a crystalline generalization of this fact.We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-LarsenVarshavsky.We also formulate two related Conjectures E.8.1 and E.9.1. Each of them is a kind of "reciprocity law" involving a sum over all ℓ-adic cohomology theories (including the crystalline theory for ℓ = p).

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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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Slopes of indecomposable $F$-isocrystals

Cited by 16 publications

References 19 publications

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

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

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

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