Perfectoid spaces

Scholze, Peter; Weinstein, Jared

doi:10.23943/princeton/9780691202082.003.0007

Cited by 90 publications

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“…• 𝑋 = Spa( 𝐴, 𝐴 + ): a 1-dimensional smooth affinoid adic space over Spa(𝐶, O 𝐶 ); • G: a finite-dimensional compact p-adic Lie group; • 𝑋 = Spa(𝐵, 𝐵 + ): an affinoid perfectoid algebra over Spa(𝐶, O 𝐶 ), which is a 'log G-Galois pro-étale perfectoid covering' of X. More precisely, this means that there is a finite set S of classical points in X and 𝑋 ∼ lim ← − −𝑖∈𝐼 𝑋 𝑖 in the sense of Definition 7.14 of [Sch12] for some index set I, where each 𝑋 𝑖 = Spa(𝐵 𝑖 , 𝐵 + 𝑖 ) is a finite Galois covering of X unramified outside of S, and 𝐵 + is the p-adic completion of lim − − →𝑖 𝐵 + 𝑖 . Moreover, the inverse limit of the Galois group of 𝑋 𝑖 over X is identified with G. When S is nonempty, we further assume for each point s in S, the ramification-index 𝑒 𝑖 of 𝑋 𝑖 → 𝑋 at s is a p-power for any i and {𝑒 𝑖 } 𝑖 ∈𝐼 is unbounded; • assume X is small in the sense that S contains at most one element and there is an étale map…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…Since the higher cohomology of O + X 𝐾 𝑝 almost vanishes on any affinoid perfectoid open subset (Theorem 1.8. (iv) of [Sch12]), we can compute 𝐻 𝑖 (X 𝐾 𝑝 , O + X 𝐾 𝑝 ) and 𝐻 𝑖 (X 𝐾 𝑝 , O + X 𝐾 𝑝 /𝑝 𝑛 ) by Čech cohomology. Take a finite affinoid perfectoid cover of X 𝐾 𝑝 and let 𝑀 • be the Čech complex for O + X 𝐾 𝑝 with respect to this cover.…”

Section: Corollary 443mentioning

confidence: 99%

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On locally analytic vectors of the completed cohomology of modular curves

Pan¹

2022

Forum of Mathematics, Pi

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We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak {gl}_2(\mathbb {Q}_p)$ . As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight k, we show its corresponding Galois representation has Hodge–Tate–Sen weights $0,k-1$ and prove a converse result.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Section: Corollary 443mentioning

confidence: 99%

On locally analytic vectors of the completed cohomology of modular curves

Pan¹

2022

Forum of Mathematics, Pi

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“…We record this as an open question (see Question 6.10). This is the motivation behind this section and therefore we start by defining the Fontaine rings using the language of perfectoid spaces of Scholze [60]. At the moment, there are certain obstructions mentioned in the introduction in generalizing signed Selmer groups to any p-adic Lie extension.…”

Section: Signed Selmer Groups and P-adic Hodge Theorymentioning

confidence: 99%

“…The Fontaine's ring E is the tilt of the perfectoid field Q p (µ p ∞ ) in the terminology of Scholze. In Section 5, we start by recalling the notion of perfectoid spaces of Scholze [60] which gives a geometric understanding of Fontaine's p-adic Hodge theory and Fontaine-Winterberger theorem [24]. Then, we use the language of perfectoids to give the definition of signed Selmer groups.…”

Section: Introductionmentioning

confidence: 99%

Conjecture A andμ-invariant for Selmer groups of supersingular elliptic curves

Hamidi¹,

Ray²

2022

Journal De Théorie Des Nombres De Bordeaux

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L'accès aux articles de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jtnb. centre-mersenne.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedramArticle mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.centre-mersenne.org/

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“…Theorem 3.11 [21,Theorem 1.3]. The tilting functor induces an equivalence between the smallétale site over K and the smallétale site over K .…”

Section: The Motivic Tilting Equivalence With Z[1/ P]-coefficientsmentioning

confidence: 99%

Rigidity for Rigid Analytic Motives

Bambozzi¹,

Vezzani²

2019

J. Inst. Math. Jussieu

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fields, introduced in [5] by Ayoub, both in their version with and without transfers. More precisely, given any normal rigid analytic variety S over K, we denote by RigDAé t pS, Λq (resp. RigDMé t pS, Λq) the category ofétale motives without transfers (resp. with transfers) over S with coefficients in the ring Λ. The precise definition of these categories is recalled in the first section of the paper. Our main result is the following theorem.Theorem (2.1). Let S be a normal rigid analytic variety over a non-Archimedean field K with ℓ-finite cohomological dimension, for all primes ℓ invertible in the residue field k of K, and let Λ be a N-torsion ring, where N is a positive integer invertible in k. The functors:As in the algebraic situation, DpSé t , Λq denotes the derived category of unbounded complexes ofétale sheaves of Λ-modules over the smallétale site and the functors Lι˚arise naturally from the inclusion of the smallétale topos into the big one.We remark that the theorem above is a generalization of the usual Rigidity Theorem, corresponding to the case in which K is trivially valued. Nonetheless, to our knowledge the original algebraic proofs can not be adapted easily to the non-Archimedean context. Our strategy is rather to use algebraic Rigidity to deduce the rigid one, by means of the analytification functors and the relation between rigid varieties and formal schemes. We also remark that, even for proving our statement over a field S " Spa K for motives without transfers, the full relative Rigidity Theorem for schemes is used. Indeed, the six functors formalism plays a crucial role in our proof (see Section 2.2). This is no longer true for motives with transfers, as we show in the appendix, for which a more direct and geometric proof is possible in the absolute case.Just like its algebraic versions, the theorem above has some interesting immediate consequences, discussed in the last section of the paper. They constitute our main motivation for proving the Rigidity Theorem in the non-Archimedean setting.

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Perfectoid spaces

Cited by 90 publications

References 0 publications

On locally analytic vectors of the completed cohomology of modular curves

On locally analytic vectors of the completed cohomology of modular curves

Conjecture A andμ-invariant for Selmer groups of supersingular elliptic curves

Rigidity for Rigid Analytic Motives

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