Padic Hodge Theory (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 padic Hodge theory and understand the emerging relationship of padic Hodge theory with other subjects in mathematics.
 The relationship between padic Hodge theory, algebraic Ktheory, and topological Hochschild homology.
 Recent developments in integral padic Hodge theory.
 The connection between padic Hodge theory and derived algebraic geometry.
Click here for notes for the talks.

Agenda
SUNDAY 7:30 9:30 PM Dinner at La Salle MONDAY 8:30 – 10:30 AM Breakfast 10:30 – 11:30 AM Integral padic Hodge Theory & Topological Cyclic Homology: A Fivetalk Series
#1: Matthew Morrow
An overview of “Integral padic Hodge Theory”11:30 12:00 PM Break 12:00 – 1:00 PM Laurent Fargues  Simple connectedness of the fibers of an AbelJacobi 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  BreuilKisin 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 padic Hodge Theory & Topological Cyclic Homology: A Fivetalk Series
2: Peter Scholze
Hochschild homology, cyclic homology and relations to de Rahm cohomology11:30 12:00 PM Break 12:00 – 1:00 PM Michel Gros  Simpson correspondance in characteristic p>0 and splittings of the algebra of PDdifferential 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 OgusVologodsky modulo p^n 6:00 – 6:30 PM Break 6:30 – 7:30 PM Dmitry Kaledin  Coperiodic 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 padic Hodge Theory & Topological Cyclic Homology: A Fivetalk Series
3: Jacob Lurie
Topological Hochschild Homology6:30 – 7:30 PM Wieslawa Niziol  Cohomology of padic Stein spaces 8:00 – 9:30 PM Dinner at La Salle THURSDAY 8:30 – 10:30 AM Breakfast 10:30 – 11:30 AM Integral padic Hodge Theory & Topological Cyclic Homology: A Fivetalk Series
4: Lars Hesselholt
THH and cyclotomic spectra11:30 12:00 PM Break 12:00 – 1:00 PM Pierre Colmez  Cohomology of padic 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 padic Hodge Theory & Topological Cyclic Homology: A Fivetalk 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 OB_{dR} 1:00 – 2:00 PM Lunch 2:00 – 4:30 PM Discussion & Recreation* 5:00 – 6:00 PM Takeshi Tsuji  The relative FontaineLaffaille 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 boardbased talks. Highspeed 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 SchlossElmau 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 warmweather 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. 
Participants
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 MaxPlanckInstitute for Mathematics Laurent Fargues Institut de Mathématiques de Jussieu JeanMarc Fontaine Université ParisSud 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 
Abstracts
\(\)
Ahmed Abbes: Lifting the Cartier transform of OgusVologodsky 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 pullback 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 finiteheight 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 \(i < p1\) and the crystalline cohomology of the special fiber of \(X\) is \(p\)torsionfree 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 TaylorWiles 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 localglobal compatibility for torsion classes.
Pierre Colmez: Cohomology of \(p\)adic analytic curves
Laurent Fargues: Simple connectedness of the fibers of an AbelJacobi morphism and local class field theory
Michel Gros: Simpson correspondance in characteristic \(p>0\) and splittings of the algebra of PDdifferential 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 PDdifferential operators viewed as an Azumaya algebra over its center.
Dmitry Kaledin: Coperiodic 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 atthetime 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 proetale cohomology of \(p\)adic Stein spaces with semistable reduction can be recovered from their de Rham and HyodoKato cohomologies. She will also discuss some examples. This is based on a joint work with Pierre Colmez and Gabriel Dospinescu.
Takeshi Tsuji: The relative FontaineLaffaille theory and Ainf representations with Frobenius.
Motivated by the work by BhattMorrowScholze on integral \(p\)adic Hodge theory, we study an analogue of BreuilKisinFargues theory for free \(\mathbb{Z}_p\)representations appearing in the relative FontaineLaffaille 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 proétale 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 mideighties to be Hochschild homology relative to the initial ring \(\mathbb{S}\) of higher algebra. As noticed by BökstedtHsiangMadsen, 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 NikolausScholze. 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. BhattMorrowScholze 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.