Représentations p-adiques cristallines et de de Rham dans le cas relatif

Brinon, Olivier

doi:10.24033/msmf.424

Cited by 46 publications

(186 citation statements)

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“…Theorem 1.3) it is enough to make use of D 0 dR , we need to study all D i dR simultaneously in order to prove the above statements (for D 0 dR ). (ii) The local version of D 0 dR essentially appeared in the work of Brinon [8].…”

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confidence: 99%

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

2016

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We construct a functor from the category of p-adic étale local systems on a smooth rigid analytic variety X over a p-adic field to the category of vector bundles with an integrable connection on its "base change to B dR ", which can be regarded as a first step towards the sought-after p-adic RiemannHilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. R.

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confidence: 99%

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

2016

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“…by tensoring with R + ̟ and its siblings). This kind of rings [63,14] originate in linearizations of differential operators.2.6.1. Structure theorem.…”

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confidence: 99%

Syntomic complexes and p-adic nearby cycles

Colmez

Nizioł

2016

Invent. math.

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Abstract. We compute syntomic cohomology of semistable affinoids in terms of cohomology of (ϕ, Γ)-modules which, thanks to work of Fontaine-Herr, Andreatta-Iovita, and Kedlaya-Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness of etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction. Let O K be a complete discrete valuation ring with fraction field K of characteristic 0 and with perfect residue field k of characteristic p. Let O F = W (k) and F = O F [ 1 p ] so that K is a totally ramified extension of F ; let e = [K : F ] be the absolute ramification index of K. Let O K denote the integral closure of O K in K. Set G K = Gal(K/K), and let ϕ = ϕ W (k) be the absolute Frobenius on W (k). For a log-scheme X over O K , X n will denote its reduction mod p n , X 0 will denote its special fiber.1.1. Statement of the main results.1.1.1. The Fontaine-Messing map. Let X be a fine and saturated log-scheme log-smooth over O K equipped with the log-structure coming from the closed point. Denote by X tr the locus where the log-structure is trivial. This is an open dense subset of the generic fiber of X. For r ≥ 0, let S n (r) X be the (log) syntomic sheaf modulo p n on X 0,ét . It can be thought of as a derived Frobenius and filtration eigenspace of crystalline cohomology or as a relative Fontaine functor. , Kato [36] have constructed period morphisms (i : X 0 ֒→ X, j : X tr ֒→ X) α FM r,n : S n (r) X → i * Rj * Z/p n (r) ′ Xtr , r ≥ 0. from logarithmic syntomic cohomology to logarithmic p-adic nearby cycles. Here we set Z p (r) ′ := 1 p a(r) Z p (r), for r = (p − 1)a(r) + b(r), 0 ≤ b(r) ≤ p − 1. Assume now that X has semistable reduction over O K or is a base change of a scheme with semistable reduction over the ring of integers of a subfield of K. That is, locally, X can be written as Spec(A) for a ring Aétale overIf we put D := {X a+b+1 · · · X d = 0} ⊂ Spec(A) then the log-structure on Spec A is associated to the special fiber and to the divisor D. We have Spec(A) tr = Spec(A K ) \ D K . We prove in this paper that the Fontaine-Messing period map α FM r,n , after a suitable truncation, is essentially a quasi-isomorphism. More precisely, we prove the following theorem. for a universal constant N (not depending on p, X, K, n or r) and a constant c p depending only on p (and d if p = 2).(ii) In general, the kernel and cokernel of this map are annihilated by p N for an int...

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“…See also [5,7,10,12,13] and [25]. Throughout this section, fiber products of fine log schemes are always considered in the category of fine log schemes (cf.…”

Section: Hodge-tate Representationsmentioning

confidence: 99%

“…5 We prove (1) and (2) simultaneously. In the case (1), choose n ∈ N d+1 ∩ I B and r ∈ N such that m ≥ n, n i = m i , n d+1 ≥ n i +r and x ∈ B n+r 1 i ,K .…”

Section: Lemma 61 There Exists An Integer Cmentioning

confidence: 99%

Purity for Hodge-Tate representations

Tsuji

2010

Math. Ann.

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We prove that a kind of purity holds for Hodge-Tate representations of the fundamental group of the generic fiber of a semi-stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field. As an application, we see that the relative p-adic étale cohomology with proper support of a scheme separated of finite type over the generic fiber is Hodge-Tate if it is locally constant.

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Représentations p-adiques cristallines et de de Rham dans le cas relatif

Cited by 46 publications

References 3 publications

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

Rigidity and a Riemann–Hilbert correspondence for p-adic local systems

Syntomic complexes and p-adic nearby cycles

Purity for Hodge-Tate representations

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