Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens

Étesse, Jean-Yves

doi:10.1016/s0012-9593(02)01099-6

Cited by 14 publications

(15 citation statements)

References 17 publications

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“…In this case, the result is proven in [19, Proposition 3.6.1] for lisse sheaves. The proof is the same for overconvergent Fisocrystals, as they satisfy étale descent by [23].…”

Section: Lemma 346mentioning

confidence: 89%

“…In the latter case, Crew have already studied the problem when X 0 is a smooth curve, [12]. Later in [23], Étesse proved that overconvergent isocrystals (with and without Frobenius structure) over smooth varieties of arbitrary dimension satisfy étale descent. 5 This allows a generalization of Crew's work.…”

Section: Comparison With the éTale Fundamental Groupmentioning

confidence: 99%

“…By [23], overconvergent isocrystals with and without F-structure satisfy étale descent. Therefore, there exist fully faithful embeddings E → f * E H and…”

Section: Comparison With the éTale Fundamental Groupmentioning

confidence: 99%

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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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“…In this case, the result is proven in [19, Proposition 3.6.1] for lisse sheaves. The proof is the same for overconvergent Fisocrystals, as they satisfy étale descent by [23].…”

Section: Lemma 346mentioning

confidence: 89%

Section: Comparison With the éTale Fundamental Groupmentioning

confidence: 99%

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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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“…Also, by [5, Théorème 3], (Spec A I , Spec A I /pA I ) is a Henselian couple in the sense of[7, 18.5.5]. Then we have the assertion (3) by [4, Corollaire 1 du Théorème 3] and [4, Lemme in p.573] (see also the proof of[6, Proposition 3]).…”

mentioning

confidence: 93%

Purity for overconvergence

Shiho

2011

Sel. Math. New Ser.

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Let X → X be an open immersion of smooth varieties over a field of characteristic p > 0 such that the complement is a simple normal crossing divisor and Z ⊆ Z ⊆ X closed subschemes of codimension at least 2. In this paper, we prove that the canonical restriction functor between the categories of overconvergent F-iso-is an equivalence of categories. We also give an application of our result to the equivalence of certain categories. Mathematics Subject Classification (2010)12H25 · 14F35 IntroductionLet X be a regular scheme and Z ⊆ X a closed subscheme of codimension at least 2. Then, by the famous Zariski-Nagata purity, any locally constant constructible sheaf on (X \Z ) et extends uniquely to a locally constant constructible sheaf on X et . As a p-adic analog of this fact, Kedlaya proved in [11, 5.3.3] the following result on the purity for overconvergent isocrystals. Let k be a field of characteristic p > 0, X → X an open immersion of k-varieties with X smooth and Z ⊆ X a closed subscheme of X of codimension at least 2 such that X \Z is dense in X . Then, the restriction functor between the categories of overconvergent isocrystals Isoc † (X, X ) −→ Isoc † (X \Z , X ) is an equivalence of categories.

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“…The latter is the trivial character, so any horizontal section of f * E ⊗n descends to E ⊗n , forcing the latter to be constant on U. Moreover, by a theorem of Étesse [Et,Théorème 4], any horizontal section of E ⊗n over an open dense subset of X extends to X. This yields the desired result.…”

Section: Weights and Determinantal Weightsmentioning

confidence: 91%

Fourier transforms and $p$-adic ‘Weil II’

Kedlaya¹

2006

Compositio Math.

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We give a purity theorem in the manner of Deligne's 'Weil II' theorem for rigid cohomology with coefficients in an overconvergent F -isocrystal; the proof mostly follows Laumon's Fourier-theoretic approach, transposed into the setting of arithmetic D-modules. This yields in particular a complete, purely p-adic proof of the Weil conjectures when combined with recent results on p-adic differential equations by André, Christol, Crew, Kedlaya, Matsuda, Mebkhout and Tsuzuki.

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Descente étale des F-isocristaux surconvergents et rationalité des fonctions L de schémas abéliens

Cited by 14 publications

References 17 publications

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

The monodromy groups of lisse sheaves and overconvergent F-isocrystals

Purity for overconvergence

Fourier transforms and $p$-adic ‘Weil II’

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