Perfectoid spaces: A survey

Scholze, Peter

doi:10.4310/cdm.2012.v2012.n1.a4

Cited by 48 publications

(65 citation statements)

References 31 publications

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“…As an application of the construction of the B + dR -cohomology theory, we can prove degeneration of the Hodge-Tate spectral sequence, [59], in general. Theorem 13.3.…”

Section: Rational P-adic Hodge Theory Revisitedmentioning

confidence: 99%

Integral $p$-adic Hodge theory — announcement

Bhatt

Morrow

Scholze

2015

Mathematical Research Letters

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We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of Cp. It takes values in a mixed-characteristic analogue of Dieudonné modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crystalline, de Rham and étale cohomology, which allows us to prove strong integral comparison theorems.The construction of the cohomology theory relies on Faltings' almost purity theorem, along with a certain functor Lη on the derived category, defined previously by Berthelot-Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham-Witt complexes of Langer-Zink, and can be computed as a q-deformation of de Rham cohomology. Contents 1. Introduction 1 2. Some examples 13 3. Algebraic preliminaries on perfectoid rings 19 4. Breuil-Kisin-Fargues modules 31 5. Rational p-adic Hodge theory 45 6. The Lη-operator 49 7. Koszul complexes 56 8. The complex Ω X 61 9. The complex AΩ X 69 10. The relative de Rham-Witt complex 80 11. The comparison with de Rham-Witt complexes 86 12. The comparison with crystalline cohomology over A crys 96 13. Rational p-adic Hodge theory, revisited 104 14. Proof of main theorems 118 References 123

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“…As an application of the construction of the B + dR -cohomology theory, we can prove degeneration of the Hodge-Tate spectral sequence, [59], in general. Theorem 13.3.…”

Section: Rational P-adic Hodge Theory Revisitedmentioning

confidence: 99%

Integral $p$-adic Hodge theory — announcement

Bhatt

Morrow

Scholze

2015

Mathematical Research Letters

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“…We refer to [18] for this implication in the semistable case and to [11] in the general regular case. Thus the local invariant cycle theorem also holds in the mixed characteristic case if dim(X η ) ≤ 2 by the results of Rapoport and Zink [23] and in many more cases by the recent work of Scholze [24]. Further unconditional, and probably well known, results were recorded by Flach and Morin in [11]: If X is regular and l = p then sp induces an isomorphism on W 1 and is an isomorphism for i = 0, 1.…”

Section: Introductionmentioning

confidence: 77%

On the p-adic local invariant cycle theorem

2016

Math. Z.

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The aim of this paper is to consider the p-adic local invariant cycle theorem in the mixed characteristic case.In the first part of the paper, via case-by-case discussion, we construct the p-adic specialization map, and then write out the complete conjecture in p-adic case. We proved the theorem in good reduction and semistable reduction cases.In the second part of the paper, by using Berthelot, Esnault and Rülling's trace morphisms in [BER], we first prove the case of coherent cohomology, then we extend it to the Witt vector cohomology, and we then get a result on the Frobenius-stable part of the Witt vector cohomology, which corresponds the slope 0 part of the rigid cohomology, we then get the general p-adic local invariant cycle theorem.We also give another approach in the H 0 and H 1 cases in the general case.In the last part of the paper, based on Flach and Morin's work on the weight filtration in the l-adic case, we consider the p-adic analogous result (which, together with the l-adic's result, serves as a part to prove the compatibility of the Weil-etale cohomology with the Tamagawa number conjecture). This is a direct corollary of the local invariant cycle theorem by taking the weight filtration. And we also consider some typical examples that the weight filtration statement could be verified by direct computations.v

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“…Algebraists have actually managed to capture the essence of there resemblances in the notion of discrete valuation rings/fields and completion of those [74]. Recently, Scholze defined the notion of perfectoid which allows us (under several additional assumptions) to build an actual bridge between the characteristic 0 and the characteristic p. It is not the purpose of this lecture to go further in this direction; we nevertheless refer interested people to [71,72,9] for the exposition of the theory.…”

Section: Similarities With Formal and Laurent Seriesmentioning

confidence: 99%

Computations with p-adic numbers

Caruso¹

2017

Cours du CIRM

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This document contains the notes of a lecture I gave at the "Journées Nationales du Calcul Formel 1 " (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts: first, we present various implementations of p-adic numbers and compare them and second, we introduce a general framework for studying precision issues and apply it in several concrete situations. (French) National Computer Algebra DaysTheory and Arithmetic Geometry. After Diophantine equations, other typical examples come from the study of number fields: we hope deriving interesting information about a number field K by studying carefully all its p-adic incarnations K ⊗ Q Q p . The ramification of K, its Galois properties, etc. can be -and are very often -studied in this manner [69,65]. The class field theory, which provides a precise description of all Abelian extensions 2 of a given number field, is also formulated in this language [66]. The importance of p-adic numbers is so prominent today that there is still nowadays very active research on theories which are dedicated to purely p-adic objects: one can mention for instance the study of p-adic geometry and p-adic cohomologies [6,58], the theory of p-adic differential equations [50], Coleman's theory of padic integration [24], the p-adic Hodge theory [14], the p-adic Langlands correspondence [5], the study of p-adic modular forms [34], p-adic ζ-functions [52] and L-functions [22], etc. The proof of Fermat's last Theorem by Wiles and Taylor [81, 78] is stamped with many of these ideas and developments. Over the last decades, p-adic methods have taken some importance in Symbolic Computation as well. For a long time, p-adic methods have been used for factoring polynomials over Q [56]. More recently, there has been a wide diversification of the use of p-adic numbers for effective computations: Bostan et al. [13] used Newton sums for polynomials over Z p to compute composed products for polynomials over F p ; Gaudry et al. [32] used p-adic lifting methods to generate genus 2 CM hyperelliptic curves; Kedlaya [49], Lauder [54] and many followers used p-adic cohomology to count points on hyperelliptic curves over finite fields; Lercier and Sir vent [57] computed isogenies between elliptic curves over finite fields using p-adic differential equations.The need to build solid foundations to the algorithmics of p-adic numbers has then emerged. This is however not straightforward because a single p-adic number encompasses an infinite amount of information (the infinite sequence of its digits) and then necessarily needs to be truncated in order to fit in the memory of a computer. From this point of view, p-adic numbers behave very similarly to real numbers and the questions that emerge when we are trying to implement p-adic numbers are often the same as the questions arising when dealing with rounding errors in the real setting [62,26,63]. The algorithmic study of p-adic numbers is then located at the frontier between Symbolic Computa...

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Perfectoid spaces: A survey

Cited by 48 publications

References 31 publications

Integral $p$-adic Hodge theory — announcement

Integral $p$-adic Hodge theory — announcement

On the p-adic local invariant cycle theorem

Computations with p-adic numbers

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