P-adic Hodge Theory (2019)

Date & Time


Location

Schloss Elmau
In Elmau 2, 82493 Krün Germany

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Registration Closed
Invitation Only

Participation is by invitation only. All participants must register.

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

Previous (2017) symposium »

Organized by Bhargav Bhatt (U. Michigan) and Martin Olsson (UC Berkeley), this second symposium in the series on p-adic Hodge theory will focus on non-abelian aspects of the theory. While traditional formulations of the main results in p-adic Hodge theory are cohomological, or abelian, in nature, there has been significant progress over the last decade on categorial or non-abelian formulations. This symposium will bring together experts in this and related areas to explore what is known and directions for the future.

Specific topics to be discussed include:

  • The p-adic Simpson’s correspondence.
  • Nonabelian reciprocity laws and applications to questions about rational points.

This symposium will help bring broader arithmetic questions to the attention of researchers in p-adic Hodge theory, as well as enable researchers working on rational points to benefit from interactions with the p-adic Hodge theory community.

 

  • Agendaplus--large

    SUNDAY

    8:30 - 9:30 PMDinner at La Salle

    MONDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMGerd Faltings | A p-adic Simpson Correspondence
    11:00-11:30 AMBreak
    11:30 - 12:30 PMMinhyong Kim | Principal Bundles and Diophantine Geometry: Some Remarks on Effectivity
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMMatthew Morrow | Coefficients in Integral P-adic Hodge Theory via Generalized Ainf-representations and q-connections
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMHélène Esnault | Arithmetic Subspaces of Moduli Spaces of Rank One Local Systems
    8:00 - 9:30 PMDinner: Ganesha Restaurant in the Retreat

    TUESDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMAnnette Werner | Vector Bundles and p-adic Representations I and II
    11:00-11:30 AMBreak
    11:30 - 12:30 PMDaniel Litt | Monodromy and Galois Actions on Deformation Rings
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMTakeshi Tsuji | Coefficients in Integral P-adic Hodge Theory via Generalized Ainf-representations and q-connections
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMVadim Vologodsky | On the Periodic Topological Cyclic Homology of (DG) Algebras in Characteristic p
    7:30 - 8:15 PMDinner
    8:30 - 9:30 PMConcert: Ray Chen (violin) Alice Sara Otto (piano)

    WEDNESDAY

    7:30 - 9:30 AMBreakfast
    9:45 - 2:00 PMGuided Hike to Partnach Gorge
    2:00 - 3:00 PMLunch at Ganesha Restaurant in the Retreat
    3:00 - 4:30 PMRecreation & Discussion
    4:30 - 5:00 PMTea
    5:00 - 6:00 PMChristopher Deninger | Vector Bundles and p-adic Representations I and II
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMAmnon Besser | Vologodsky Integration on Curves with Semi-Stable Reduction: Progress Report
    8:00 - 9:30 PMDinner

    THURSDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMXinwen Zhu | P-adic Riemann-Hilbert Functors and Applications
    11:00-11:30 AMBreak
    11:30 - 12:30 PMKęstutis Česnavičius | Cohomological Purity in Bad Characteristics
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMOfer Gabber | TBA
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMArthur Ogus | The Saturated de Rham-Witt Complex, after Bhatt, Lurie, Mathew
    8:00 - 9:30 PMDinner

    FRIDAY

    7:30 - 9:45 AMBreakfast at La Salle
    10:00 - 11:00 AMJacon Lurie | Full Level Structures on Elliptic Curves
    11:00-11:30 AMBreak
    11:30 - 12:30 PMJared Weinstein | The Cohomology of Local Shtuka Spaces
    12:30 - 1:30 PMLunch at La Salle
    1:30 - 4:30 PMDiscussion & Recreation*
    4:30- 5:00 PMTea
    5:00 - 6:00 PMAhmed Abbes | Local Structure of Almost-étale \(\varphi\)-modules
    6:00 - 6:15 PMBreak
    6:15 - 7:15 PMPeter Scholze | Some Results on the Cohomology of Finite Flat Group Schemes
    8:00 - 9:30 PMDinner: Kaminstüberl Fondue Restaurant

    LOCATIONS

    SESSIONSPavilion located at the Schloss Elmau Retreat
    MEALSLa Salle unless otherwise noted
    TEA & DISCUSSIONPavilion located at the Schloss Elmau Retreat
    EXCURSION Meet in Schloss Elmau Lobby
    SATURDAY DEPARTUREMeet in Schloss Elmau Lobby

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

  • Abstractsplus--large

    Ahmed Abbes
    Centre national de la recherche scientifique & Institut des Hautes Études Scientifiques

    Local Structure of Almost-étale \(\varphi\)-modules

    In the appendix of his 2002 Astérisque article, Faltings sketched a proof of a relative version of his main comparison theorem in p-adic Hodge theory. Abbes will report on a key ingredient of this proof describing the local structure of certain almost-étale \(\varphi\)-modules, which is interesting in itself. This is joint work with Michel Gros.

     

    Amnon Besser
    Ben Gurion University

    Vologodsky Integration on Curves with Semi-Stable Reduction: Progress Report

    Vologodsky extended Coleman’s integration theory to give canonical paths on the unipotent de Rham fundamental groupoid of any smooth variety over a p-adic field. Besser expects this to have a strong connection with the unipotent Albanese map for such varieties and, therefore, for the study of rational points using non-abelian techniques at primes of bad reduction. Besser will provide some evidence for that. A major problem with Vologodsky integration is that it is less explicit than Coleman integration. Besser will report on progress, due in part to work with several researchers, on the relation between Vologodsky integration and Coleman integration.

     

    Kęstutis Česnavičius
    Centre national de la recherche scientifique, Université Paris-Sud

    Cohomological Purity in Bad Characteristics

    An fppf cohomological purity conjecture predicts that for a regular (or, more generally, complete intersection) Noetherian local ring (R, m) and a commutative finite flat R-group scheme G, one should have \(H^i_m(R, G) = 0\) for \(i < dim(R)\). Česnavičius will discuss several cases of this conjecture. The talk is based on joint work with Peter Scholze.

     

    Christopher Deninger
    Universität Münster
    -and-
    Annette Werner
    Goethe-Universität Frankfurt

    Vector Bundles and p-adic Representations I and II

    Deninger and Werner introduce the category of vector bundles with numerically flat reduction on a p-adic variety and construct a functorial étale parallel transport for such bundles. This strengthens Faltings’ higher dimensional p-adic Simpson correspondence in the case of vanishing Higgs fields. The first talk by Annette Werner is expository and also explains some stronger results and examples in the case of curves, while the second, by Christopher Deninger, outlines the proof.

     

    Hélène Esnault
    Freie Universität Berlin

    Arithmetic Subspaces of Moduli Spaces of Rank One Local Systems

    Esnault shows that closed subsets of the character variety of a complex variety with negatively weighted homology, which are p-adically integral and Galois invariant, are motivic. This is joint work with Moritz Kerz.

     

    Gerd Faltings
    Max Planck Institute for Mathematics

    A p-adic Simpson Correspondence

    Faltings explains his old paper in the title. The main method is a local calculation in p-adic Hodge theory, which implies the result for small representations, but there are also some interesting complications for big representations.

     

    Minhyong Kim
    Oxford University

    Principal Bundles and Diophantine Geometry: Some Remarks on Effectivity

    Much of the interest surrounding the non-abelian method of Chabauty comes from the hope of using it to compute completely the set of rational points on a curve of higher genus. Kim will make some observations on this.

     

    Daniel Litt
    Institute for Advanced Study

    Monodromy and Galois Actions on Deformation Rings

    Let K be a number field (resp. function field of a curve over a fixed finite field k) of degree (resp. gonality) g. The torsion conjecture predicts that there is a constant N=N(g,d) such that if A is a (traceless) d-dimensional Abelian variety over K, then \(\# |A(K)|_{\mathrm{tors}} < N\). The Frey-Mazur conjecture predicts that there is a constant N′=N′(g,d) such that if A₁, A₂ are (traceless) d-dimensional Abelian varieties over K, then A₁ is isogenous to A₂ if and only if A₁[lʳ] is isomorphic to A₂[lʳ], for lʳ > N′ and prime to the characteristic of K. Litt discusses new results toward the l-primary versions of these conjectures in the function field setting.

     

    Jacob Lurie
    Harvard University

    Full Level Structures on Elliptic Curves

    Let p be a prime number, and let M denote the moduli stack of elliptic curves equipped with an infinite level structure at the prime p. Scholze has shown that the generic fiber of M is a perfectoid space. In this talk, Lurie will discuss an integral refinement of this result, as well as some ‘approximate’ results that hold at finite level.

     

    Matthew Morrow
    Centre national de la recherche scientifique & Institut de Mathématiques de Jussieu-Paris Rive Gauche
    -and-
    Takeshi Tsuji
    University of Tokyo

    Coefficients in Integral P-adic Hodge Theory via Generalized Ainf-representations and q-connections

    In these talks, Morrow and Tsuji will present an overview of their work on coefficients in integral p-adic Hodge theory. To any smooth formal scheme over the ring of integers of the completed algebraic closure of a p-adic field, they associate a category of ‘relative Breuil-Kisin-Fargues’ modules; these are defined as certain locally free modules over the Ainf-period sheaf on the pro-étale site of the generic fiber of X. Any relative BKF module gives rise to a vector bundle with connection, an F-crystal on the special fiber, and a crystalline lisse Zp-sheaf on the generic fiber; their cohomologies are then intertwined by a relative form of the Ainf-cohomology from Bhatt-M.-Scholze. These relative BKF modules are closely related to other notions: they include Faltings’ relative Fontaine-Laffaille modules; on any small opening they are given by certain small generalized representations as in Faltings’ and Gros-Abbes-T.’s work on the p-adic Simpsons correspondence, and given a choice of coordinates, they correspond to modules with a q-connection, in the sense of q-de Rham cohomology, as a result of which they also admit an interpretation in terms of Bhatt-Scholze’s theory of prismatic cohomology.

     

    Arthur Ogus
    University of California, Berkeley

    The Saturated de Rham-Witt Complex, after Bhatt, Lurie, Mathew

    Let X/k be a smooth scheme over a perfect field of characteristic p. The de Rham-Witt complex, constructed by Illusie (with roots in work by Bloch, Lubkin and Deligne) is a canonical sheaf of differential graded algebras on the Zariski site, which computes the crystalline cohomology of X/W and which reveals a great deal of information about the action of the Frobenius endomorphism of X. Bhatt, Lurie and Mathew have recently given a simple, new construction of this complex which, unlike the original, seems to give reasonable answers for (at least some) singular schemes. Ogus will explain the main points of the new construction and how it works for schemes with toric singularities.

     

    Peter Scholze
    Mathematisches Institut der Universität Bonn

    Some Results on the Cohomology of Finite Flat Group Schemes

    Scholze proves some results on the cohomology of finite flat group schemes, such as fpqc descent, formal gluing and some purity results. This gives new proofs of Grothendieck’s purity conjectures in etale cohomology and for the Brauer group, and some new results (e.g., that, on complete intersections, torsion line bundles extend over codimension 3, and Brauer classes extend over codimension 4, as conjectured by Gabber). The proofs proceed by reduction to perfectoid rings. This is joint with Kęstutis Česnavičius.

     

    Vadim Vologodsky
    Higher School of Economics

    On the Periodic Topological Cyclic Homology of (DG) Algebras in Characteristic p

    Vologodsky proves that the periodic topological cyclic homology of a smooth proper DG algebra over Fp is isomorphic to the periodic (algebraic) cyclic homology of a lifting of the algebra over Zp. This is joint work with Alexander Petrov.

     

    Jared Weinstein
    Boston University

    The Cohomology of Local Shtuka Spaces

    The local Langlands correspondence predicts that representations of a reductive group G over a p-adic field are related to Galois representations into the Langlands dual of G. A suitably generalized conjecture of Kottwitz asserts that this relationship appears in a precise way in the cohomology of Scholze’s local shtuka spaces. We don’t know how Galois acts on this cohomology yet, but we can verify much of the rest of the conjecture, using a Lefschetz-Verdier fixed point formula. This is joint work with Tasho Kaletha.

     

    Xinwen Zhu
    California Institute of Technology

    P-adic Riemann-Hilbert Functors and Applications

    Zhu will review his recent works in the construction of p-adic Riemann-Hilbert functors for p-adic local systems on rigid analytic varieties. He will also discuss a few applications of the general theory. The talk is based on joint work with R. Liu, H. Diao, K.W. Lan and R. Liu.

  • Participantsplus--large
    Ahmed AbbesCNRS & IHÉS
    Amnon BesserBen Gurion University
    Bhargav BhattUniversity of Michigan
    Kęstutis ČesnavičiusCNRS, Université Paris-Sud
    Aise Johan de JongColumbia University
    Christopher DeningerUniversität Münster
    Hélène EsnaultFreie Universität Berlin
    Gerd FaltingsMax Planck Institute for Mathematics
    Ofer GabberCNRS & IHÉS
    Michel GrosCNRS & Université Rennes 1
    Minhyong KimOxford University
    Daniel LittInstitute for Advanced Study
    Ruochuan LiuBICMR/Peking University
    Jacob LurieHarvard University
    Matthew MorrowCNRS & Institut de Mathématiques de Jussieu–Paris Rive Gauche
    Arthur OgusUniversity of California, Berkeley
    Martin OlssonUniversity of California, Berkeley
    Peter ScholzeMathematisches Institut der Universität Bonn
    Takeshi TsujiUniversity of Tokyo
    Vadim VologodskyHigher School of Economics
    Jared WeinsteinBoston University
    Annette WernerGoethe University Frankfurt
    Xinwen ZhuCalifornia Institute of Technology
  • AVplus--large

    Participants will have access to a projector and screen for computer-based talks and blackboards for those who prefer to give board-based talks.

    High-speed Internet access is available.

  • Wednesday Excursionplus--large

    On Wednesday symposium activities are shortened for a guided hiking tour from Schloss-Elmau through the Partnach Gorge. The hike difficulty is moderate and will take approximately 3 hours.

    If you will participate in the hike, hiking boots or shoes appropriate for traversing rocky/wet terrain, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and waterproofs (e.g. raincoat or umbrella) are strongly recommended.

    A small satchel containing a light snack and water is provided, but feel free to bring your own backpack in which to carry a camera and other items that may be useful to you.

  • Travelplus--large

    Air

    The foundation will coordinate and purchase air and train travel to the symposium. Travel specifications can be provided by clicking the registration link above.

    Participants arriving via plane will arrive via Munich International Airport (MUC) Airport.

    Train

    The closet train station to Schloss Elmau is in Klais. Those traveling by train should book their tickets directly and request reimbursement at the conclusion of the meeting. Should you require booking assistance, please contact Meghan Fazzi.

    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.

    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 through customs and to the exit where a uniformed driver holding a Simons Symposia sign will greet you.

    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 Klais to Elmau is approximately 10 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

    Additional Night at Elmau

    The meeting concludes on Friday evening. Participants have the option to depart on Saturday following the conclusion of the meeting or depart on Sunday at no additional cost. Please note your desired departure date when registering.

    Dress Code

    Business casual clothing should be worn during the symposium.

    The weather can change very quickly; please bring cold-weather clothing appropriate for spring in the mountains.

  • Mealsplus--large

    Sunday and Saturday: Meals on arrival and departure days (including while traveling) are reimbursable by the foundation. Your meals on these days are not to exceed $115 USD.

    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 is served starting at 7:30 AM at La Salle restaurant. Food at La Salle is served buffet-style and is open to all hotel guests; tables are reserved for all symposium participants so that the group may dine together.

    Lunch is served daily at La Salle.

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

  • Contactsplus--large

    Registration and Travel Assistance

    Christophe Vergnol, Protravel
    chrisvgroup@protravelinc.com
    (973) 334-9300 ext. 10

    Meeting Questions and Assistance

    Meghan Fazzi
    Senior Executive Assistant, Simons Foundation
    mfazzi@simonsfoundation.org
    (212) 524-6080

  • Reimbursement and Travel Policyplus--large

    Reimbursement

    Any expenses not directly paid for by the Simons Foundation are subject to reimbursement based on the travel policy. Please review the policy and complete when submitting your expenses. Receipts are required for any expenses over $50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. All expenses are reconciled through the foundation’s online reimbursement platform. Additional information regarding reimbursement is sent via email on the final day 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 are not reimbursable by the foundation.

    Please note on your registration form if you wish to extend your stay.

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