Pith. sign in

REVIEW 6 minor 43 references

An introduction to p-adic period rings

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The notes construct the p-adic period rings $B_{\mathrm{crys}}$ and $B_{\mathrm{dR}}$ and state the comparison theorems that identify p-adic étale cohomology with de Rham and crystalline cohomology.

desk verdict Solid, honest course notes on p-adic period rings; no new results, but a careful exposition that deserves referee time as a pedagogical reference. read the letter →

arxiv 1908.08424 v1 pith:REU4A6AR submitted 2019-08-22 math.NT math.RT

Xavier Caruso (IMB) This is my paper · ORCID
classification math.NTmath.RT MSC 11S2011F8014F3014F40
keywords p-adicHodgetheoryperiodringscrystallinecohomologydeRhamétaleGaloisrepresentationsHodge-TatefilteredFrobeniusmodules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to establish, in a lecture format, that the p-adic period rings $B_{\mathrm{crys}}$ and $B_{\mathrm{dR}}$ exist with the structures the B-admissibility strategy requires, and that they support the comparison theorems central to p-adic Hodge theory. For a proper smooth variety over a finite extension of $\mathbb{Q}_p$, it states that $B_{\mathrm{dR}}$ tensored with algebraic de Rham cohomology is canonically isomorphic to $B_{\mathrm{dR}}$ tensored with étale cohomology, and that under good reduction $B_{\mathrm{crys}}$ similarly identifies crystalline cohomology with étale cohomology. A sympathetic reader should care because these rings turn the Galois action on étale cohomology into linear-algebra data: de Rham representations become filtered vector spaces, and crystalline representations become admissible filtered Frobenius modules. The paper's contribution is architectural, showing how the two rings are built and what the main theorems say.

What carries the argument

The machinery is the construction of the two period rings. Starting from $\mathbb{C}_p$, one forms the tilt $R$, a perfect characteristic-$p$ ring obtained by taking the inverse limit of $\mathcal{O}_{\mathbb{C}_p}/p$ under Frobenius; the Witt vectors of $R$ give $A_{\mathrm{inf}}=W(R)$ and $B^+_{\mathrm{inf}}=A_{\mathrm{inf}}[1/p]$. A distinguished surjective map $\theta:B^+_{\mathrm{inf}}\to \mathbb{C}_p$ evaluates series, and $B^+_{\mathrm{dR}}$ is the completion of $B^+_{\mathrm{inf}}$ with respect to $\ker\theta$, whose fraction field is $B_{\mathrm{dR}}$. The ring $B_{\mathrm{crys}}$ is built from the divided-power envelope of the same ideal, completed and then localized away from the element $t=\log[\varepsilon]$, a period for the cyclotomic character. The load-bearing properties are that powers of $t$ generate the graded pieces of the de Rham filtration and that the Galois invariants compute to $K$ and $K_0$; these facts make the comparison isomorphisms possible.

What would settle it

Compute, for some explicit proper smooth variety over a finite extension of $\mathbb{Q}_p$, the dimension of $(B_{\mathrm{dR}} \otimes_{\mathbb{Q}_p} H^r_{\mathrm{et}}(X_{\bar K},\mathbb{Q}_p))^{G_K}$; if it differed from $\dim_K H^r_{\mathrm{dR}}(X)$, the $C_{\mathrm{dR}}$ isomorphism of Equation (36) would fail. Alternatively, exhibit a filtered Frobenius module satisfying the numerical conditions of Theorem 4.3.11 that is not in the essential image of $D_{\mathrm{crys}}$.

Watch

Extended reading notes

Core claim

The central claim is that the period ring $B_{\mathrm{dR}}$ is a complete discrete valuation field with residue field $\mathbb{C}_p$, carrying a Galois action, a distinguished element $t$ on which Galois acts through the cyclotomic character, a de Rham filtration whose graded ring is $B_{\mathrm{HT}} = \mathbb{C}_p[t,t^{-1}]$, and fixed points $(B_{\mathrm{dR}})^{G_K}=K$. The subring $B_{\mathrm{crys}}$ carries a Frobenius endomorphism $\phi$, has fixed points $K_0$, contains $t$, and satisfies $(B_{\mathrm{crys}} \cap \mathrm{Fil}^0 B_{\mathrm{dR}})^{\phi=1}=\mathbb{Q}_p$. Given these structures, the paper states the comparison theorems: for $X$ proper smooth over $K$, $B_{\mathrm{dR}} \otimes_K H^r_{\mathrm{dR}}(X) \simeq B_{\mathrm{dR}} \otimes_{\mathbb{Q}_p} H^r_{\mathrm{et}}(X_{\bar K},\mathbb{Q}_p)$ compatibly with filtration and Galois action, and for good reduction $B_{\mathrm{crys}} \otimes_{W(k)} H^r_{\mathrm{crys}}(\bar X) \simeq B_{\mathrm{crys}} \otimes_{\mathbb{Q}_p} H^r_{\mathrm{et}}(X_{\bar K},\mathbb{Q}_p)$ compatibly with Frobenius and Galois action.

Load-bearing premise

The survey stands on the correctness of the cited comparison theorems and the cited admissibility criterion, since it states them without giving independent proofs.

Editorial extensions

If this is right

  • Under the $C_{\mathrm{dR}}$ theorem, every étale cohomology group of a proper smooth p-adic variety is de Rham, and the de Rham cohomology is recovered as the Galois invariants of $B_{\mathrm{dR}} \otimes H^r_{\mathrm{et}}$.
  • Under the $C_{\mathrm{crys}}$ theorem, good reduction forces the étale cohomology to be crystalline, and the crystalline cohomology is recovered compatibly with Frobenius and Galois action.
  • Every de Rham representation is Hodge–Tate, with Hodge–Tate weights read off from the filtration on $D_{\mathrm{dR}}(V)$, so the period-ring dictionary transfers cohomological information into filtered vector spaces.
  • Crystalline representations are classified by admissible filtered Frobenius modules, with the numerical criterion of Theorem 4.3.11 deciding admissibility.
  • A representation that is not de Rham cannot come from the étale cohomology of a proper smooth variety, giving a concrete obstruction to geometric origin.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the comparison theorems are accepted, the period-ring construction suggests that p-adic Galois representations are best studied through their associated linear-algebra objects, so classification questions reduce to filtered vector spaces and Frobenius modules.
  • The tilt-and-Witt-vector route to $B_{\mathrm{dR}}$ indicates that perfectoid methods are not decoration: the same period rings should control étale cohomology beyond proper smooth varieties, for instance for rigid-analytic varieties.
  • A testable extension is to ask whether the semi-stable ring $B_{\mathrm{st}}$ yields an equally complete dictionary once a monodromy operator is added to the filtered Frobenius modules.
  • Because the paper's comparison maps are stated rather than proved, a reader could test the stated functoriality on an explicit family, such as an elliptic curve with split multiplicative reduction or a product of such curves.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. These lecture notes introduce the p-adic period rings B_crys and B_dR and explain their role in p-adic Hodge theory. Section 1 sets up local fields, Galois representations, and Fontaine's B-admissibility formalism; Section 2 proves the Ax–Sen–Tate theorem and the characterization of C_p-admissible representations, and develops Sen's theory; Section 3 constructs B_inf^+, B_crys, the rings B_mu, and B_dR via the tilt and Witt vectors, and computes their invariants; Section 4 states the C_dR and C_crys comparison theorems, defines crystalline and de Rham representations, and states the Colmez–Fontaine admissibility criterion. The deep comparison theorems are quoted from the literature rather than proved; the paper's internal claims are the foundational local results in Sections 1 through 3.

Significance. If the results it surveys are correct—and they are standard—this is a useful, generally careful exposition of a central subject. It proves the main local theorems (ramification in Z_p-extensions, Ax–Sen–Tate, C_p-admissibility, Sen's operator) in a self-contained way, constructs the period rings in enough detail to support their listed properties, and gives precise statements of the comparison theorems and the Colmez–Fontaine criterion. The dependence on external results is transparent and attributed, so the survey's value is not diminished by not re-proving Faltings–Tsuji–Scholze or Colmez–Fontaine. I found no circularity and no internally inconsistent new claim.

minor comments (6)
  1. [§3.3.2] The identification of B_dR^+ with C_p((t)) as a ring is incorrect as stated: B_dR^+ is a complete discrete valuation ring with residue field C_p and uniformizer t, so the ring-isomorphic object is C_p[[t]]; it is the fraction field B_dR that is isomorphic to C_p((t)). Please correct this.
  2. [§3.2.2] The displayed formula 'v_p(n!) = n/(p-1)' is not an equality; the exact formula is v_p(n!) = (n-s_p(n))/(p-1), and only the asymptotic estimate is used. Please add '≈' or make the estimate explicit.
  3. [§3.3.7] The passage from principality of ker theta to the DVR property of B_dR^+ deserves a sentence or a reference: principality alone is not sufficient for a completion to be a DVR; one also uses that the completion is a domain and m-adically separated.
  4. [§3.1.7] In the proof of Proposition 3.1.7, the symbol x is reused for both the element being tested and the product omega * phi^{-1}(omega) * ... * phi^{-n}(omega); please rename one of them.
  5. [§1.2] In the display after Eq. (5), 'were ĀK' should read 'where ĀK'.
  6. [§4.1] In Conjecture 4.1.2, the same letter p denotes both the prime number and a prime ideal of O_F; using a different symbol for prime ideals would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper is an expository survey whose comparison theorems and admissibility criterion are explicitly attributed to external proofs; its internal constructions are self-contained and no prediction is fitted to its inputs.

full rationale

The paper's substantive claims—existence of B_crys and B_dR with the listed properties and the comparison isomorphisms (36) and (39)—are presented as known theorems, not derived from the survey's own assertions. Section 4.1 states 'Our ambition is only to state the relevant theorems in this direction and definitely not to prove them,' and attributes C_dR to Faltings [15] and Tsuji [41], C_crys to Fontaine–Messing/Tsuji, the perfectoid proof to Scholze [37], and the admissibility criterion to Colmez–Fontaine [11]. The internal chain (construction of B_inf^+, B_crys, B_dR, filtration, invariant computations) is carried out within the notes, with proofs for the computed invariants (Theorems 3.4.1 and 3.4.2) and for the fixed-point statement of Proposition 3.4.4; none of these internal steps is equivalent by construction to the comparison theorems being quoted. There is no fitted parameter later renamed as a prediction, no self-citation chain used to forbid alternatives (the only self-referential citations are to Berger–Colmez [4] and to other lectures in the same volume, and they are not load-bearing), and the Colmez–Fontaine criterion is cited as an external theorem rather than imported from the author's own prior work. The only apparent slip, 'as a ring, B_dR^+ is isomorphic to C_p((t))' in Section 3.3.2 (should be C_p[[t]]), is a harmless copyedit and does not bear on circularity. Hence the derivation chain is self-contained up to explicitly attributed external theorems, and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No data are fitted and no free parameters appear. The paper is an exposition: its central statements are imported from established literature, and the ledger reflects that reliance rather than any new postulate.

assumptions (4)
  • standard math Standard background on local fields, ramification, Witt vectors, and local class field theory as in Serre [39].
    Invoked throughout Sections 1 and 2; the paper states reminders but does not prove these foundations.
  • domain assumption Tate's Hodge-like decomposition for p-divisible groups and Fontaine's Honda-system classification.
    Section 1.2 uses these cited results as motivational seeds for the period-ring formalism.
  • domain assumption The comparison theorems C_dR and C_crys (Theorems 4.1.1 and 4.1.3), attributed to Faltings, Tsuji, and Scholze.
    Section 4.1 states them without proof; the survey's geometric consequences inherit their correctness.
  • domain assumption The Colmez-Fontaine admissibility criterion for filtered phi-modules (Theorem 4.3.11).
    Stated at the end of Section 4.3 with citation to [11]; the classification discussion depends on it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An introduction to p-adic period rings." pith.science (2026). https://pith.science/paper/REU4A6AR

@misc{pith2026190808424,
  author       = {Pith},
  title        = {Pith review of: An introduction to p-adic period rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REU4A6AR}},
  note         = {Machine review of arXiv:1908.08424}
}
read the original abstract

This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{\'e}tale and crystalline or de Rham cohomologies for p-adic varieties.

Figures

Figures reproduced from arXiv: 1908.08424 by the authors.

Figure 1
Figure 1. The graph of the function ψr (r ≥ 4) whenever u > eF p−1 . Letting s go to infinity, we end up with ρ(u + eF ) = ρ(u) + 1 for u > eF p−1 . This relation, combined with the facts that ρ is nondecreasing, left-continuous and takes integral values, implies that there exists a real constant a such that ρ(u) = ⌈ u−a eF ⌉ for u > eF p−1 . The fact that a is indeed an integer is a consequence of the Hasse–Arf theorem. Rema… view at source ↗
Figure 2
Figure 2. Diagram of period rings; all arrows are injective [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. The Newton polygon of [ε] − 1 Proposition 3.1.5. The element ω = [ε] − 1 [ε 1/p] − 1 = [ε 1/p] + [ε 1/p] 2 + · · · + [ε 1/p] p−1 satisfies the condition of Proposition 3.1.3.(i). Proof. We want to check that θ(ω) = 0 and v♭ (ω mod p) = 1. The first equality follows from the fact that θ([ε]) = 1 and the fact that θ([ε 1/p]) is a primitive p-th root of unity. Let us now prove that v♭ (ωmodp) = 1. Reducing modulo p, we… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Convergence conditions for elements in Acrys Topology and additional structures. Since Acrys is defined as a p-adic completion, it is quite natural to endow Acrys (and B+ crys) with the p-adic topology. Noticing that we can obviously write [p ♭ ] n = n! · [p ♭ ] n n! ,…
Figure 5
Figure 5. Figure 5: The Newton polygon of t 3.2.3 The element t An essential property of Acrys is that it contains a period for the cyclotomic character, that is a special element on which Galois acts by multiplication by χcycl. This distinguished element is: t = log [ε] = X∞ i=1 (−1)i−1 …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 42 canonical work pages

  1. [1]

    Andreatta, R

    F. Andreatta, R. Brasca, O. Brinon, B. Chiarellotto, N. M azzari, S. Panozzo, M. Seveso, An introduc- tion to perfectoid spaces , this volume

    work page

  2. [2]

    Ax, Zeros of polynomials over local fields , J

    J. Ax, Zeros of polynomials over local fields , J. of Algebra 15 (1970), 417–428

    work page 1970

  3. [3]

    Berger , ´Equations diff´ erentiellesp-adiques et (ϕ,N )-modules filtr´ es, Ast´ erisque319 (2008), 13–38

    L. Berger , ´Equations diff´ erentiellesp-adiques et (ϕ,N )-modules filtr´ es, Ast´ erisque319 (2008), 13–38

    work page 2008

  4. [4]

    Berger , P

    L. Berger , P. Colmez, Th´ eorie de Sen et vecteurs localement analytiques , Ann. Sci. ´Ecole Norm. Sup. 49 (2016), 947–970

    work page 2016

  5. [5]

    Berthelot, Cohomologie cristalline des sch´ emas de caract´ eristiquep, Lecture notes in math

    P. Berthelot, Cohomologie cristalline des sch´ emas de caract´ eristiquep, Lecture notes in math. 407 (1974)

    work page 1974

  6. [6]

    Berthelot, Alt´ erations des vari´ et´ es alg´ ebriques (d’apr` es A.J

    P. Berthelot, Alt´ erations des vari´ et´ es alg´ ebriques (d’apr` es A.J. de Jong) , Ast´ erisque241 (1997), 273– 311

    work page 1997

  7. [7]

    Berthelot, L

    P. Berthelot, L. Breen, W. Messing, Th´ eorie de Dieudonn´ e cristalline II, Lecture notes in math. 930 (1982)

    work page 1982

  8. [8]

    Berthelot, A

    P. Berthelot, A. Ogus, Notes on crystalline cohomology , Princeton University Press, Princeton (1978)

    work page 1978

Show all 43 references
  1. [9]

    Brinon, Filtered (ϕ,N )-modules and semi-stable representations , this volume

    O. Brinon, Filtered (ϕ,N )-modules and semi-stable representations , this volume

  2. [10]

    Colmez, Une construction de BdR, Rend

    P. Colmez, Une construction de BdR, Rend. Sem. Mat. Univ . Padova 128 (2012), 109–130 57

    2012

  3. [11]

    Colmez and J.-M

    P. Colmez and J.-M. Fontaine, Construction des repr´ esentations p-adiques semi-stables, Invent. Math. 140 (2000), 1–43

    2000

  4. [12]

    Conrad, Finite group schemes over bases with low ramification , Compositio Math

    B. Conrad, Finite group schemes over bases with low ramification , Compositio Math. 119 (1999), 239–320

    1999

  5. [13]

    Demazure, Lectures on p-divisible groups , LNM 302 (1972)

    M. Demazure, Lectures on p-divisible groups , LNM 302 (1972)

    1972

  6. [14]

    Emerton, Local-global compatibility in the p-adic Langlands programme for GL 2/Q, preprint (2011)

    M. Emerton, Local-global compatibility in the p-adic Langlands programme for GL 2/Q, preprint (2011)

    2011

  7. [15]

    Faltings, p-adic Hodge theory , J

    G. Faltings, p-adic Hodge theory , J. Amer . Math. Soc., 1 (1988), 255–299

    1988

  8. [16]

    Faltings, Crystalline cohomology andp-adic Galois-representations, in Algebraic analysis, geometry and number theory (Baltimore, MD, 1988), 25–80

    G. Faltings, Crystalline cohomology andp-adic Galois-representations, in Algebraic analysis, geometry and number theory (Baltimore, MD, 1988), 25–80

    1988

  9. [17]

    Fargues, J.-M

    L. Fargues, J.-M. Fontaine, Courbes et fibr´ es vectoriels en th´ eorie de Hodge p-adique , to appear in Ast´ erisque

  10. [18]

    Fontaine, Groupesp-divisibles sur les corps locaux , Ast´ erisque47–48 (1977)

    J.-M. Fontaine, Groupesp-divisibles sur les corps locaux , Ast´ erisque47–48 (1977)

    1977

  11. [19]

    Fontaine, Modules galoisiens, modules filtr´ es et anneaux de Barsotti -T ate, in Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes, Ast´ erisque65 (1979) , 3–80

    J.-M. Fontaine, Modules galoisiens, modules filtr´ es et anneaux de Barsotti -T ate, in Journ´ ees de G´ eom´ etrie Alg´ ebrique de Rennes, Ast´ erisque65 (1979) , 3–80

    1979

  12. [20]

    Fontaine, Formes diff´ erentielles et modules de T ate des vari´ et´ es ab´ eliennes sur les corps locaux, Invent

    J.-M. Fontaine, Formes diff´ erentielles et modules de T ate des vari´ et´ es ab´ eliennes sur les corps locaux, Invent. Math. 65 (1981), 379–409

    1981

  13. [21]

    Fontaine, P´ eriodesp-adiques, Ast´ erisque223 (1994)

    J.-M. Fontaine, P´ eriodesp-adiques, Ast´ erisque223 (1994)

    1994

  14. [22]

    Fontaine, Arithm´ etique des repr´ esentations galoisiennesp-adiques, Ast´ erisque297 (2004), 1– 115

    J.-M. Fontaine, Arithm´ etique des repr´ esentations galoisiennesp-adiques, Ast´ erisque297 (2004), 1– 115

    2004

  15. [23]

    Fontaine, W

    J.-M. Fontaine, W. Messing, p-adic periods and p-adic ´ etale cohomology, in Current trends in arith- metical algebraic geometry (Arcata, Calif., 1985) , Contemp. Math. 67 (1987), 179–207,

    1987

  16. [24]

    Fontaine, B

    J.-M. Fontaine, B. Mazur , Geometric Galois representations , in Elliptic curves, modular forms & Fer- mat’s last theorem (Hong Kong, 1993) , Ser . Number Theory , I, Int. Press, Cambridge, MA (1995), 41–78,

    1995

  17. [25]

    Freixas i Montplet, Hodge–T ate decomposition, this volume

    G. Freixas i Montplet, Hodge–T ate decomposition, this volume

  18. [26]

    Grothendieck, Groupes de Barsotti-Tate et cristaux , Actes du Congr` es International des Math´ ematiciens de Nice (1970), p

    A. Grothendieck, Groupes de Barsotti-Tate et cristaux , Actes du Congr` es International des Math´ ematiciens de Nice (1970), p. 431–436

    1970

  19. [27]

    Hyodo, K

    O. Hyodo, K. Kato, “Semi-stable reduction and crystalline cohomology with lo garithmic poles , Ast´ erisque223 (1994), 221–268

    1994

  20. [28]

    de Jong Smoothness, semi-stability and alterations , Inst

    A.J. de Jong Smoothness, semi-stability and alterations , Inst. Hautes ´Etudes Sci. Publ. Math. 83 (1996), 51–93

    1996

  21. [29]

    de Jong Families of curves and alterations , Ann

    A.J. de Jong Families of curves and alterations , Ann. Inst. Fourier 47 (1997), 599–621

    1997

  22. [30]

    Kato Logarithmic structure of Fontaine-Illusie , Algebraic, Analysis, Geometry and Number Theory , John Hopkins University Press (1989), 191–224

    K. Kato Logarithmic structure of Fontaine-Illusie , Algebraic, Analysis, Geometry and Number Theory , John Hopkins University Press (1989), 191–224

    1989

  23. [31]

    Kisin, Crystalline representations andF -crystals, Algebraic geometry and number theory , Progress in Mathematics, 253 (2006), 459–496

    M. Kisin, Crystalline representations andF -crystals, Algebraic geometry and number theory , Progress in Mathematics, 253 (2006), 459–496

    2006

  24. [32]

    Kisin, The Fontaine-Mazur conjecture for GL 2, J

    M. Kisin, The Fontaine-Mazur conjecture for GL 2, J. Amer . Math. Soc. 22 (2009), 641–690

    2009

  25. [33]

    Lazard, Les z´ eros des fonctions analytiques d’une variable sur un corps valu´ e complet, Inst

    M. Lazard, Les z´ eros des fonctions analytiques d’une variable sur un corps valu´ e complet, Inst. Hautes Etudes Sci. Publ. Math. 14 (1962), 47–75

    1962

  26. [34]

    Le Borgne, Optimisation du th´ eor` eme d’Ax–Sen–T ate et application ` a un calcul de cohomologie galoisiennep-adique, Ann

    J. Le Borgne, Optimisation du th´ eor` eme d’Ax–Sen–T ate et application ` a un calcul de cohomologie galoisiennep-adique, Ann. Inst. Fourier , Grenoble 60 (2010), 1105–1123

    2010

  27. [35]

    Sen, Continuous cohomology and p-adic Galois representations , Invent

    S. Sen, Continuous cohomology and p-adic Galois representations , Invent. Math., 62 (1980), 89–116

    1980

  28. [36]

    Scholze, Perfectoid spaces, Publ

    P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes ´Etudes Sci. 116 (2012), 245–313

    2012

  29. [37]

    Scholze, p-adic Hodge theory for rigid-analytic varieties , Forum Math

    P. Scholze, p-adic Hodge theory for rigid-analytic varieties , Forum Math. Pi 1 (2013), 77 pp

    2013

  30. [38]

    Scholze, J

    P. Scholze, J. Weinstein, Berkeley lectures on p-adic geometry , available at http://www.math.uni-bonn.de/people/scholze/Berkeley .pdf 58

  31. [39]

    Serre, Corps locaux, Hermann Paris (1962)

    J.-P. Serre, Corps locaux, Hermann Paris (1962)

    1962

  32. [40]

    Tate, p-divisible groups , Proc

    J. Tate, p-divisible groups , Proc. Conf. Local Fields (Driebergen, 1966), Springer , Be rlin (1967), 158–183

    1967

  33. [41]

    Tsuji, p-adic ´ etale cohomology and crystalline cohomology in the semi-stable reduction case , Invent

    T. Tsuji, p-adic ´ etale cohomology and crystalline cohomology in the semi-stable reduction case , Invent. Math. 137 (1999), 233–411

    1999

  34. [42]

    Tsuji, Semi-stable conjecture of Fontaine-Jannsen: a survey , in Cohomologies p-adiques et applica- tions arithm´ etiques, II,no

    T. Tsuji, Semi-stable conjecture of Fontaine-Jannsen: a survey , in Cohomologies p-adiques et applica- tions arithm´ etiques, II,no. 279, Soc. Math. France (2002), 323–370

    2002

  35. [43]

    Yamashita, An introduction to p-adic Hodge theory for open varieties via syntomic cohomolo gy, this volume 59

    G. Yamashita, An introduction to p-adic Hodge theory for open varieties via syntomic cohomolo gy, this volume 59

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.