P-adic Hodge Theory

Date & Time


Location

Schloss Elmau
Krün, Germany

May 7-13, 2017

Note: Attendance is by invitation only.

Organizers:
Bhargav Bhatt, University of Michigan
Martin Olsson, UC Berkeley

Organized by Bhargav Bhatt (University of Michigan) and Martin Olsson (UC Berkeley), the symposium will bring together experts to explore recent developments in p-adic Hodge theory and understand the emerging relationship of p-adic Hodge theory with other subjects in mathematics.

The topics to be discussed include:
  • The relationship between p-adic Hodge theory, algebraic K-theory, and topological Hochschild homology.
  • Recent developments in integral p-adic Hodge theory.
  • The connection between p-adic Hodge theory and derived algebraic geometry.

Click here for notes for the talks.

  • Agendaplus--large

    Download agenda PDF here.

    SUNDAY
    7:30 -9:30 PM Dinner at La Salle
    MONDAY
    8:30 – 10:30 AM Breakfast
    10:30 – 11:30 AM Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series
    #1: Matthew Morrow

    An overview of “Integral p-adic Hodge Theory”
    11:30 -12:00 PM Break
    12:00 – 1:00 PM Laurent Fargues | Simple connectedness of the fibers of an Abel-Jacobi morphism and local class field theory
    1:00 – 2:00 PM Lunch
    2:00 – 4:30 PM Discussion & Recreation*
    5:00 – 6:00 PM Bryden Cais | Breuil-Kisin modules and crystalline cohomology
    6:00 – 6:30 PM Break
    6:30 – 7:30 PM Minhyong Kim | Reciprocity laws and principal bundles
    8:00 – 9:30 PM Dinner at La Salle
    TUESDAY
    8:30 – 10:30 AM Breakfast
    10:30 – 11:30 AM Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series
    2: Peter Scholze

    Hochschild homology, cyclic homology and relations to de Rahm cohomology
    11:30 -12:00 PM Break
    12:00 – 1:00 PM Michel Gros | Simpson correspondance in characteristic p>0 and splittings of the algebra of PD-differential operators
    1:00 – 2:00 PM Lunch
    2:00 – 4:30 PM Discussion & Recreation*
    5:00 – 6:00 PM Ahmed Abbes | Lifting the Cartier transform of Ogus-Vologodsky modulo p^n
    6:00 – 6:30 PM Break
    6:30 – 7:30 PM Dmitry Kaledin | Co-periodic cyclic homology
    8:00 – 9:30 PM Dinner at Wintergarden
    WEDNESDAY
    7:30 – 9:30 AM Breakfast
    9:45 – 2:00 PM Guided Hike to Partnach Gorge
    2:00 – 3:00 PM Lunch at Wintergarden
    3:00 – 5:00 PM Recreation & Discussion
    5:00 – 5:30 PM Tea
    5:30 – 6:30 PM Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series
    3: Jacob Lurie

    Topological Hochschild Homology
    6:30 – 7:30 PM Wieslawa Niziol | Cohomology of p-adic Stein spaces
    8:00 – 9:30 PM Dinner at La Salle
    THURSDAY
    8:30 – 10:30 AM Breakfast
    10:30 – 11:30 AM Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series
    4: Lars Hesselholt

    THH and cyclotomic spectra
    11:30 -12:00 PM Break
    12:00 – 1:00 PM Pierre Colmez | Cohomology of p-adic analytic curves
    1:00 – 2:00 PM Lunch
    2:00 – 4:30 PM Discussion & Recreation*
    5:00 – 6:00 PM Ana Caraiani | Galois representations and torsion classes
    6:00 – 6:30 PM Break
    6:30 – 7:30 PM Kiran Kedlaya | Tautological local systems and (phi, gamma)-modules
    8:00 – 9:30 PM Dinner at La Salle
    FRIDAY
    8:30 – 10:30 AM Breakfast
    10:30 – 11:30 AM Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series
    5: Peter Scholze

    A “weight” filtration on THH and its relation with crystalline cohomology and \(A\Omega\)
    11:30 -12:00 PM Break
    12:00 – 1:00 PM Ruochuan Liu | Logarithmic OBdR
    1:00 – 2:00 PM Lunch
    2:00 – 4:30 PM Discussion & Recreation*
    5:00 – 6:00 PM Takeshi Tsuji | The relative Fontaine-Laffaille theory and Ainf representations with Frobenius
    8:00 – 9:30 PM Dinner at Kaminstüberl
    LOCATIONS
    SESSIONS Pavillion located at the Schloss Elmau Retreat
    MEALS La Salle unless otherwise noted
    TEA & DISCUSSION Pavillion located at the Schloss Elmau Retreat
    EXCURSION Meet in Schloss Elmau Lobby
    SATURDAY DEPARTURE Meet in Schloss Elmau Lobby

    *Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

    Start Time

    The meeting will begin on Monday morning at 10:30 AM in the Pavillon meeting room located at the Schloss Elmau Retreat. Breakfast will start at 8:30 AM in La Salle restaurant.

    Meeting Location

    All symposium activities will take place in the Pavillon meeting room located at the Schloss Elmau Retreat.

    AV

    Participants will have access to a projector and screen for computer based talks as well as blackboards for those who prefer to give board-based talks. High-speed Internet access is available as well.

    Wednesday Excursion

    On Wednesday symposia activities will be shortened so that anyone interested may attend a hiking excursion. Participants will embark on a fully guided hiking tour from Schloss-Elmau through the Partnach Gorge. The hike difficulty is moderate and will take approximately 3 hours.

    Dress Code

    Business casual clothing should be worn during the symposia. The weather can change very quickly so we also advise bringing warm-weather clothing appropriate for spring in the mountains.If you plan on taking part in the hike to Partnach Gorge we advise you to wear hiking boots, or even better, light mountain boots, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and take waterproofs (e.g. raincoat or umbrella) with you. Bringing along a small backpack or satchel in which to carry your water, camera and other items may also be useful to you.

  • Participantsplus--large

    Download participant list PDF here.

    Ahmed Abbes Institut des Hautes Études Scientifiques
    Bhargav Bhatt University of Michigan
    Bryden Cais University of Arizona
    Ana Caraiani University of Bonn
    Pierre Colmez Institut de Mathématiques de Jussieu
    Brian Conrad Stanford University
    Aise Johan de Jong Columbia University
    Gerd Faltings Max-Planck-Institute for Mathematics
    Laurent Fargues Institut de Mathématiques de Jussieu
    Jean-Marc Fontaine Université Paris-Sud
    Ofer Gabber Institut des Hautes Études Scientifiques
    Michel Gros Université Rennes 1
    Lars Hesselholt University of Copenhagen
    Dmitry Kaledin Steklov Math Institute
    Kiran Kedlaya UC San Diego
    Minhyong Kim University of Oxford
    Ruochuan Liu Beijing International Center for Mathematical Research
    Jacob Lurie Harvard University
    Matthew Morrow CNRS, Jussieu
    Wiesława Nizioł École Normale Supérieure de Lyon
    Martin Olsson UC Berkeley
    Peter Scholze Universität Bonn
    Takeshi Tsuji University of Tokyo
  • Abstractsplus--large

    \(\)

    Ahmed Abbes: Lifting the Cartier transform of Ogus-Vologodsky modulo p^n [following H. Oyama, A. Shiho and D. Xu]

    In their seminal work, A. Ogus and V. Vologodsky constructed an analogue of Simpson correspondence in characteristic \(p>0\). More precisely, given a smooth scheme \(X\) over a perfect field \(k\) of characteristic \(p>0\), together with a lifting modulo \(p^2\), they constructed a functor (the Cartier transform) from the category of modules with integrable connection on \(X\) to the category of Higgs modules on \(X’\) (the base change of \(X\) by the Frobenius of \(k\)), each subject to a suitable nilpotence condition.Almost simultaneously, G. Faltings proposed a \(p\)-adic analogue of the Simpson correspondence. The relationship between these two correspondences remains mysterious, and the first challenge is to lift the Cartier transform modulo \(p^n\). A. Shiho did the first step by lifting the “local” correspondence modulo \(p^n\), given a lifting of the relative Frobinus modulo \(p^{n+1}\). Independently, in his thesis under the supervision of T. Tsuji, H. Oyama proposed a very beautiful interpretation of the Cartier transform (modulo \(p\)) as the pull-back by a morphism of ringed topoi. In his PhD thesis, my student D. Xu uses Oyama topos to “glue” Shiho’s local constructions and hence lift the Cartier transform modulo \(p^n\), under the (only) assumption that \(X\) lifts to a smooth formal scheme over the Witt vectors of k.

    Abbes will report on the works of Shiho, Oyama and Xu.

     


     

    Bryden Cais: Breuil—Kisin modules and crystalline cohomology

    Let \(k\) be a perfect field of characteristic \(p\), and \(K\) a totally ramified extension of \(W(k)[1/p]\). The theory of Breuil—Kisin modules provides a classification of stable lattices in crystalline \(p\)-adic representations of the absolute Galois group, \(G_K\), of \(K\) via finite-height Frobenius modules over the power series ring \(W(k)[[u]]\). On the other hand, the \(i\)-th integral \(p\)-adic etale cohomology of a smooth and proper formal scheme \(X\) over the ring of integers \(O_K\) in \(K\) provides a stable lattice in a crystalline \(p\)-adic \(G_K\)-representation, and so has a Breuil—Kisin module attached to it. In this case, it is natural to ask if the associated Breuil—Kisin module can be described in terms of the crystalline cohomology of \(X\). Recent work of Bhatt, Morrow, and Scholze provides such a description after extending scalars to the period ring A_inf.In this talk, Cais will explain how to descend this result to obtain the Breuil—Kisin module over \(W(k)[[u]]\) when \(i < p-1\) and the crystalline cohomology of the special fiber of \(X\) is \(p\)-torsion-free in degrees \(i\) and \(i+1\).

    This is joint work with Tong Liu.

     


     

    Ana Caraiani: Galois representations and torsion classes

    Caraiani will describe joint work in progress with Allen, Calegari, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne on potential modularity for elliptic curves over imaginary quadratic fields. The key ingredients are a version of the Taylor-Wiles patching argument due to Calegari and Geraghty, and a result on torsion in the cohomology of Shimura varieties that is joint with Scholze. She will focus on explaining how to prove cases of local-global compatibility for torsion classes.

     


     

    Pierre Colmez: Cohomology of \(p\)-adic analytic curves

     


     

    Laurent Fargues: Simple connectedness of the fibers of an Abel-Jacobi morphism and local class field theory

     


     

    Michel Gros: Simpson correspondance in characteristic \(p>0\) and splittings ​of the algebra of PD-differential operators

    We will present the Simpson correspondance in positive characteristic set out by Ogus and Vologodsky from the perspective of splittings of a completion of the algebra of PD-differential operators viewed as an Azumaya algebra over its center.

     


     

    Dmitry Kaledin: Co-periodic cyclic homology

     


     

    Kiran Kedlaya: Tautological local systems and (phi, Gamma)-modules

    In his 2010 ICM lecture, Kedlaya proposed a construction of “tautological” relative (phi, Gamma)-modules on a suitable period domain, which would on the admissible locus correspond to the tautological crystalline local system. This was based on an at-the-time rudimentary and unnamed theory, now recognizable as the theory of perfectoid spaces. In light of the vast improvements in the theory since then, we finally revisit this proposal.

     


     

    Minhyong Kim: Reciprocity laws and principal bundles

    The study of principal bundles and their moduli is ubiquitous in number theory. The problem of determining the locus of global moduli spaces inside local ones gives rise to precise questions on reciprocity laws with definite applications to Diophantine problems. This talk will give a rather speculative summary of these ideas and results.

     


     

    Ruochuan Liu: Logarithmic \(OB_{dR}\)

    We will discuss a construction of the logarithmic version of \(OB_{dR}\) and some related applications.

     


     

    Wiesia Niziol: Cohomology of \(p\)-adic Stein spaces

    Niziol will describe how the geometric \(p\)-adic pro-etale cohomology of \(p\)-adic Stein spaces with semistable reduction can be recovered from their de Rham and Hyodo-Kato cohomologies. She will also discuss some examples. This is based on a joint work with Pierre Colmez and Gabriel Dospinescu.

     


     

    Takeshi Tsuji: The relative Fontaine-Laffaille theory and Ainf representations with Frobenius.

    Motivated by the work by Bhatt-Morrow-Scholze on integral \(p\)-adic Hodge theory, we study an analogue of Breuil-Kisin-Fargues theory for free \(\mathbb{Z}_p\)-representations appearing in the relative Fontaine-Laffaille theory by Faltings. We show that Galois cohomology décalée in the sense of BMS is related to de Rham complex with coefficients similarly to their work on the constant coefficients. We also obtain a new proof of the fully faithfulness of the functor from the category of “filtered Frobenius crystals” to that of free \(\mathbb{Z}_p\) representations.

     


     

    Integral \(p\)-adic Hodge theory and topological cyclic homology (series)

    1) Matthew Morrow: An overview of “Integral \(p\)-adic Hodge theory”

    Morrow will provide an overview of some of the main constructions of the paper “Integral \(p\)-adic Hodge theory” joint with Bhatt and Scholze. In particular, he will explain the construction — via décalage functors, almost purity, and period sheaves on the pro-étale site — of \(A\Omega\), relating \(p\)-adic etale cohomology and Langer—Zink’s relative de Rham—Witt sheaves. Its alternative construction, via topological cyclic homology, will be given in Scholze’s second talk.

     


     

    2) Peter Scholze: Hochschild homology, cyclic homology, and relations to de Rham cohomology

    This talk will recall some classical facts: First, we give the definitions of Hochschild homology and (negative, and periodic) cyclic homology. Then, for smooth commutative algebras over a characteristic 0 field, we will compute these theories in terms of de Rham cohomology.

     


     

    3) Jacob Lurie: Topological Hochschild Homology

    Hochschild homology is an algebraic invariant of pairs \((R,A)\), where \(R\) is a commutative ring and \(A\) is an associative algebra over \(R\). In particular, one can consider the “absolute” Hochschild homology of an associative ring \(A\), taking \(R\) to be the ring of integers. However, it turns out that there is an even more “absolute” version of the same construction, called topological Hochschild homology, which is obtained by taking \(R\) to be the sphere spectrum.In this talk, Lurie will give a brief introduction to the language of spectra, outline the construction of topological Hochschild homology, and give some sense of why it might be preferable to its algebraic cousin.

     


     

    4) Lars Hesselholt: \(THH\) and cyclotomic spectra

    Topological Hochschild homology was defined by Bökstedt in the mid-eighties to be Hochschild homology relative to the initial ring \(\mathbb{S}\) of higher algebra. As noticed by Bökstedt-Hsiang-Madsen, topological Hochschild homology, surprisingly, comes equipped with a Frobenius operator \(\varphi_p\) for every prime number \(p\). This structure is encoded in the notion of a cyclotomic spectrum, the proper definition of which was given only recently by Nikolaus-Scholze. In addition to its cyclotomic spectrum structure, one expects topological Hochschild homology to admit a natural “weight” filtration such that, for schemes over \(\mathbb{F}_p\), the geometric Frobenius acts as \(p^w\varphi_p\) on the \(w\)th graded piece for the filtration. Bhatt-Morrow-Scholze have now constructed this filtration, at least for schemes over \(\mathbb{Z}_p\).

     


     

    5) Peter Scholze: A “weight” filtration on \(THH\), and its relation with (crystalline cohomology and) \(A\Omega\)

    (joint work with Bhargav Bhatt and Matthew Morrow)For \(p\)-complete commutative rings, we define a “weight” filtration on THH and related objects like \(TR^n\), \(TF\), \(THH^{hS^1}\) and \(TC\) (the latter two under mild hypothesis on the ring). For smooth \(F_p\)-algebras, the graded pieces recover objects known from crystalline cohomology, such as (truncations of) \(\Omega\) (for \(THH\)), \(W_n\Omega\) (for \(TR^n\)), \(W\Omega\) (truncated for \(TF\), untruncated for \(THH^{hS^1})\), along with the Nygaard filtration on it, which can be used to define Milne’s sheaves \(Z_p(r)=\nu_r[-r]\) (which are \(r\)-th graded piece of \(TC\)). Similarly, for smooth \(O_{C_p}\)-algebras, the graded pieces recover objects known from our paper on integral \(p\)-adic Hodge theory, such as (truncations of) \(\tilde{\Omega}\) (for \(THH\)), \(\tilde{W_n\Omega}\) (for \(TR^n\)) and \(A\Omega\) (truncated for \(TF\), untruncated for \(THH^{hS^1}\)), along with a Nygaard filtration on it, which can be used to construct certain syntomic complexes \(Z_p(r)\) (which are the \(r\)-th graded piece of \(TC\)).

    Scholze will explain the construction of the filtration, and outline the proof of these comparison results. (Already the crystalline case is new.) An application is an extended definition of \(A\Omega\), which gives the descent of Breuil—Kisin—Fargues modules to Breuil—Kisin modules for proper smooth (formal) schemes defined over a discretely valued field.

  • Travelplus--large

     

    Air

    The foundation will coordinate and purchase your air travel to the symposium. Please provide your travel specifications by clicking the registration link above.Participants arriving via plane will arrive via Munich International Airport (MUC).

    Train

    The closet train station to Schloss Elmau is Klais. Those traveling by train are advised to book their tickets directly and request reimbursement from the foundation at the conclusion of the meeting.All participants are required to arrive on Sunday prior to the meeting’s start and attend the entire week. No accommodations will be made for partial participation.

    Additional Night at Elmau

    The official departure date for all participants is Saturday (meeting activities conclude Friday evening). As a special bonus participants may depart on Sunday at no additional cost. Please note your desired departure date when registering.

    Passports

    Passports are required.

    Local Ground Transportation

    Arrival at Munich International Airport

    The foundation will arrange for your transfer from MUC airport to Schloss Elmau via a private car. After claiming your luggage at baggage claim you will proceed to the exit where a uniformed driver holding a Schloss Elmau sign will greet you. Upon arrival at MUC, if you are unable to locate your driver please call Taxi Mathe directly.Taxi Mathe, Mr. Mathe
    taxi.mathe@googlemail.com
    phone 08821 9663691
    mobile 0170 8399 933

    Travel time from Munich to Schloss Elmau is 90 minutes.

    Arrival at Klais Train Station

    The foundation will arrange for your transfer from Klais train station to Schloss Elmau via a hotel shuttle.Travel time from Kalis to Elmau is approximately 15 minutes.

    Departure from Schloss Elmau

    The day prior to your departure you will receive a departure letter advising on your departure time to the airport or train station via private car. Please meet your driver in the hotel lobby ten minutes prior to your noted departure time. If for any reason your flight departure time changes, please visit the front desk to update hotel staff.
  • Hotelplus--large

    Schloss Elmau Luxury Spa & Cultural Hideaway
    In Elmau 2, 82493 Krün
    Germany
    Tel: +49 (0) 8823 18-0
    www.schloss-elmau.de

    Check-in time is at 4:30 PM
    Check-out time is 11:30 AM

  • Mealsplus--large

     

    While Traveling

    On days when participants are traveling to or from Germany the foundation will reimburse up to US $50 per day for meals.

    Day of Arrival (Sunday)

    Participants may dine on their own at any one of Schloss Elmau’s onsite restaurants, the cost of which will be covered by the foundation.

    Monday through Friday

    All meals occurring during the meeting week will be covered directly by the Simons Foundation. Participants are expected to be present for all foundation-hosted meals.Breakfast will be served daily starting at 7:30 AM at La Salle restaurant. Food at La Salle is served buffet-style and is open to all hotel guests and a table has been set aside so that symposia participants may dine together.

    Lunch will be served daily at La Salle.

    Dinner will be hosted around the hotel property at one of the restaurants. Locations are noted on the agenda.

    Day of Departure (Saturday)

    If departing on Saturday, your breakfast will be covered by the foundation. If you choose to depart on Sunday, you are responsible for all your meals from Saturday lunch onward.
  • Contactsplus--large

     

    Registration and Travel Assistance

    Christophe Vergnol
    christophe.vergnol@protravelinc.com
    (646) 747-9767

    Registration, Hotel and General Meeting Assistance

    Meghan Fazzi
    Senior Executive Assistant, Simons Foundation
    mfazzi@simonsfoundation.org
    (212) 524-6080
  • Reimbursement and Travel Policyplus--large

    Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the travel policy. An email will be sent within a week following the conclusion of the symposium with further instructions on submitting your expenses via the foundation’s web-based expense reimbursement platform.

    Receipts are required for any expenses over $50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. Should you have any questions, please contact Meghan Fazzi.

    Guest Policy

    It is the foundation’s policy to discourage spouses, families or others from joining participants during the meeting. Participants who choose to invite guests to come before or stay on after the meeting do so at their own expense. The Simons Foundation will not cover expenses associated with such extensions.

    Additional Nights

    Should you decide to extend your stay, Schloss Elmau will honor the conference rate of €337.50/night for up to two nights. Any nights beyond those associated with the conference will not be reimbursed by the foundation. Please note on your registration form if you wish to extend your stay.
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