𝑝adic Hodge Theory (2019)
Organizers:
Bhargav Bhatt, University of Michigan
Martin Olsson, University of California, Berkeley
This second symposium in the series focused on nonabelian aspects of 𝑝adic Hodge theory. While classical 𝑝adic Hodge theory is often phrased in terms of comparison results for cohomology, nonabelian aspects are concerned with categories of sheaves, comparisons between them and nonabelian invariants, such as algebraic completions of homotopy groups. Therefore, a major theme of the symposium was the definition and study of categories of local systems, in étale, crystalline or integral contexts, where one expects comparison results to hold.

Meeting Report
Overview of talks
The talks at the conference consisted of several organized pairs of lectures, covering important recent developments, as well as individual talks on topics related to the themes of the workshop.
Inspired by the complex theory developed by Simpson and others giving a correspondence between complex local systems and Higgs fields on algebraic varieties over C, Faltings proposed in 2005 an analogue in the 𝑝adic context using 𝑝adic Hodge theory. Gerd Faltings (A 𝑝adic Simpson Correspondence) and Ahmed Abbes (Local Structure of Almostétale φmodules) discussed developments in this area since then.
Kęstutis Česnavičius and Peter Scholze gave a twotalk series entitled “Cohomological Purity in Bad Characteristics and Some Results on the Cohomology of Finite Flat Group Schemes,” explaining their recent proof of cohomological purity for finite flat commutative group schemes. Remarkably, their proof uses Dieudonné theory over perfectoid rings to prove results over noetherian rings.
In a twotalk series titled “Coefficients in Integral 𝑝adic Hodge Theory via Generalized Ainfrepresentations and qconnections,” Matthew Morrow and Takeshi Tsuji explained their recent work developing a theory of integral 𝑝adic Hodge Theory, in the sense of earlier work of Bhatt, Morrow and Scholze, with coefficients.
Dating back to work of Coleman on 𝑝adic integration, it has long been known that 𝑝adic Hodge theory of the fundamental group can be used to study rational points on algebraic varieties. This has in recent years been studied by many people in a program initiated by Minhyong Kim. At the symposium, talks in this direction were given by Minhyong Kim (“Principal Bundles and Diophantine Geometry: Some Remarks on Effectivity”) and Amnon Besser (“Vologodsky Integration on Curves with SemiStable Reduction: Progress Report”).
A key feature of Hodge theory with coefficients and in families is the RiemannHilbert correspondence relating complex local systems to Dmodules. At the workshop, Xinwen Zhu and Ruochuan Liu explained their recent work toward a 𝑝adic version. The title of their lectures was “Padic RiemannHilbert Functors and Applications.”
Faltings’ 𝑝adic Simpson correspondence can be strengthened in the case of vector bundles corresponding to Higgs bundles with vanishing Higgs field. Christopher Deninger and Annette Werner have constructed a functorial parallel transport for vector bundles on 𝑝adic varieties satisfying a condition on their reduction. They explained this work at the conference in two talks titled “Vector Bundles and Padic Representations.”
The remaining talks were individual presentations on topics related to the workshop.
In his talk “Monodromy and Galois Actions on Deformation Rings,” Daniel Litt spoke about recent work related to results of Deligne about the fundamental question of which representations of the fundamental group of a variety over C come from geometry, in the sense that they appear as subquotients of the cohomology of a variety over the function field.
Related to this, in her talk “Arithmetic Subspaces of Moduli Spaces of Rank One Local Systems,” Hélène Esnault spoke about loci in the character variety for the fundamental group for which the associated local systems are motivic. This is related to a conjecture of Carlos Simpson.
Ofer Gabber discussed various duality results in 𝑝adic Hodge theory, including Poincare duality for mod 𝑝 cohomology of smooth proper rigid analytic spaces. The title of his talk was “Almost Duality for Nearby Cycles of \(O^+/p\).”
Jacob Lurie discussed, in his talk “Full Level Structures on Elliptic Curves,” an integral version of a result of Scholze showing that the limit of the tower of moduli spaces for elliptic curves with level structure is perfectoid.
Another approach to 𝑝adic cohomology of algebraic varieties is using the de RhamWitt complex. Arthur Ogus discussed recent developments in this area in his talk “The Saturated de RhamWitt Complex, after Bhatt, Lurie, Mathew.” There is some hope that this new de RhamWitt complex is better behaved than the classical one in the case of singular varieties, and Ogus presented new results in this direction for toric varieties.
Vadim Vologodsky discussed a new approach to defining crystalline cohomology of differential graded algebras over positive characteristic fields, using topological cyclic homology. This work is related to work of Bhatt, Morrow and Scholze and some of the central themes of the first meeting in the symposium series.
Finally, Jared Weinstein discussed a conjecture of Kottwitz making precise the Local Langlands Correspondence for representations of reductive 𝑝adic groups. He explained this recent work in his talk “The Cohomology of Local Shtuka Spaces.”
New Developments and Future Directions
The symposium was very active, with an informal and interactive atmosphere. Many of the presenters discussed both finished work as well as work in progress, and there were a lot of smaller conversations about important technical points, improvements to arguments, extensions of results, etc. — the kinds of conversations that often lead to substantial advances and require a strong group of experts in the area. For example, one series of questions during a talk led to substantial improvements in the arguments for one of the main results presented; another speaker credited the mere potential presence of an audience member as having led to the discovery of an error in a proof while the speaker was preparing the talk, eventually leading to a better proof. While several important finished results were presented at the symposium, the small size of the meeting and the informal atmosphere fostered numerous discussions about work in progress and future directions.

Agenda
MONDAY
10:00  11:00 AM Gerd Faltings  A 𝑝adic Simpson Correspondence 11:30 AM  12:30 PM Minhyong Kim  Principal Bundles and Diophantine Geometry: Some Remarks on Effectivity 5:00  6:00 PM Matthew Morrow  Coefficients in Integral 𝑝adic Hodge Theory via Generalized Ainfrepresentations and qconnections 6:15  7:15 PM Hélène Esnault  Arithmetic Subspaces of Moduli Spaces of Rank One Local Systems TUESDAY
10:00  11:00 AM Annette Werner  Vector Bundles and padic Representations I and II 11:30 AM  12:30 PM Daniel Litt  Monodromy and Galois Actions on Deformation Rings 5:00  6:00 PM Takeshi Tsuji  Coefficients in Integral Padic Hodge Theory via Generalized Ainfrepresentations and qconnections 6:15  7:15 PM Xinwen Zhu  Padic RiemannHilbert Functors and Applications I 8:30  9:30 PM Concert: Ray Chen (violin) Alice Sara Otto (piano) WEDNESDAY
9:45 AM  2:00 PM Guided Hike to Partnach Gorge 5:00  6:00 PM Christopher Deninger  Vector Bundles and padic Representations I and II 6:15  7:15 PM Amnon Besser  Vologodsky Integration on Curves with SemiStable Reduction: Progress Report THURSDAY
10:00  11:00 AM Ruochuan Liu  Padic RiemannHilbert Functors and Applications II 11:30 AM  12:30 PM Kęstutis Česnavičius  Cohomological Purity in Bad Characteristics 3:30  4:30 PM Vadim Vologodsky  On the Periodic Topological Cyclic Homology of (DG) Algebras in Characteristic p 5:00  6:00 PM Ofer Gabber  Almost duality for nearby cycles of 𝑂⁺/𝑝 6:15  7:15 PM Arthur Ogus  The Saturated de RhamWitt Complex, after Bhatt, Lurie, Mathew FRIDAY
10:00  11:00 AM Jacob Lurie  Full Level Structures on Elliptic Curves 11:30 AM  12:30 PM Jared Weinstein  The Cohomology of Local Shtuka Spaces 5:00  6:00 PM Ahmed Abbes  Local Structure of Almostétale \(\varphi\)modules 6:15  7:15 PM Peter Scholze  Some Results on the Cohomology of Finite Flat Group Schemes *Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

Abstracts
Ahmed Abbes
Centre national de la recherche scientifique & Institut des Hautes Études ScientifiquesLocal Structure of Almostétale \(\varphi\)modules
In the appendix of his 2002 Astérisque article, Faltings sketched a proof of a relative version of his main comparison theorem in 𝑝adic Hodge theory. Abbes will report on a key ingredient of this proof describing the local structure of certain almostétale \(\varphi\)modules, which is interesting in itself. This is joint work with Michel Gros.
Amnon Besser
Ben Gurion UniversityVologodsky Integration on Curves with SemiStable 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 𝑝adic 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 nonabelian 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 ParisSudCohomological 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 Rgroup 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
GoetheUniversität FrankfurtVector Bundles and 𝑝adic Representations I and II
Deninger and Werner introduce the category of vector bundles with numerically flat reduction on a 𝑝adic variety and construct a functorial étale parallel transport for such bundles. This strengthens Faltings’ higher dimensional 𝑝adic 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 BerlinArithmetic 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 𝑝adically integral and Galois invariant, are motivic. This is joint work with Moritz Kerz.
Gerd Faltings
Max Planck Institute for MathematicsA 𝑝adic Simpson Correspondence
Faltings explains his old paper in the title. The main method is a local calculation in 𝑝adic Hodge theory, which implies the result for small representations, but there are also some interesting complications for big representations.
Ofer Gabber
(CNRS & IHES)Almost duality for nearby cycles of \(O^+/p\)
We discuss several points in the approaches of Faltings and Scholze to padic Hodge theory and Grothendieck duality in this context. Let \(K\) be an algebraically closed complete rank 1 valued field with valuation ring \(O_K\) of mixed characteristic \((0, p)\), \(X\) a proper smooth connected rigid analytic space over \(K\) of dimension \(d\) with normal formal model \(\mathcal{X}\) over Spf \(O_K\). To show Poincaré duality for \(H^∗(X_{et},\mathbb{Z}/p)\) one considers the “nearby cycle” complex \(R\psi_∗(O^+/p)\) on \(\mathcal{X}/p\) ; it has bounded almost coherent cohomology and one observes that \(\mathcal{H}^dR\psi_∗(O^+/p)(d)\) has a canonical almost map to the dualizing sheaf; we show that this induces an almost autoduality by means of local uniformization by quotients of nice formal models by finite groups. We also discuss how to get a duality theorem for proper morphisms of rigid analytic spaces and complexes of \(\mathbb{Z}/p^n\) sheaves with locally bounded constructible cohomology.
Minhyong Kim
Oxford UniversityPrincipal Bundles and Diophantine Geometry: Some Remarks on Effectivity
Much of the interest surrounding the nonabelian 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 StudyMonodromy 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) ddimensional Abelian variety over K, then \(\# A(K)_{\mathrm{tors}} < N\). The FreyMazur conjecture predicts that there is a constant N′=N′(g,d) such that if A₁, A₂ are (traceless) ddimensional 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 lprimary versions of these conjectures in the function field setting.
Jacob Lurie
Harvard UniversityFull 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 JussieuParis Rive Gauche
and
Takeshi Tsuji
University of TokyoCoefficients in Integral 𝑝adic Hodge Theory via Generalized A_{inf}representations and qconnections
In these talks, Morrow and Tsuji will present an overview of their work on coefficients in integral 𝑝adic Hodge theory. To any smooth formal scheme over the ring of integers of the completed algebraic closure of a 𝑝adic field, they associate a category of ‘relative BreuilKisinFargues’ modules; these are defined as certain locally free modules over the A_{inf}period sheaf on the proétale site of the generic fiber of X. Any relative BKF module gives rise to a vector bundle with connection, an Fcrystal on the special fiber, and a crystalline lisse Z_{p}sheaf on the generic fiber; their cohomologies are then intertwined by a relative form of the A_{inf}cohomology from BhattM.Scholze. These relative BKF modules are closely related to other notions: they include Faltings’ relative FontaineLaffaille modules; on any small opening they are given by certain small generalized representations as in Faltings’ and GrosAbbesT.’s work on the 𝑝adic Simpsons correspondence, and given a choice of coordinates, they correspond to modules with a qconnection, in the sense of qde Rham cohomology, as a result of which they also admit an interpretation in terms of BhattScholze’s theory of prismatic cohomology.
Arthur Ogus
University of California, BerkeleyThe Saturated de RhamWitt Complex, after Bhatt, Lurie, Mathew
Let X/k be a smooth scheme over a perfect field of characteristic p. The de RhamWitt 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 BonnSome 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 EconomicsOn 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 F_{p} is isomorphic to the periodic (algebraic) cyclic homology of a lifting of the algebra over Z_{p}. This is joint work with Alexander Petrov.
Jared Weinstein
Boston UniversityThe Cohomology of Local Shtuka Spaces
The local Langlands correspondence predicts that representations of a reductive group G over a 𝑝adic 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 LefschetzVerdier fixed point formula. This is joint work with Tasho Kaletha.
Xinwen Zhu
California Institute of Technology𝑝adic RiemannHilbert Functors and Applications
Zhu will review his recent works in the construction of 𝑝adic RiemannHilbert functors for 𝑝adic 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.

Notes
Ahmed Abbes
Centre national de la recherche scientifique & Institut des Hautes Études Scientifiques
Local Structure of Almostétale \(\varphi\)modules (PDF)Daniel Litt
Institute for Advanced Study
Monodromy and Galois Actions on Deformation Rings (PDF)Notes taken by Kestutis Cesnavicus:
Cesnavicius: Cohomological purity in bad characteristic (PDF)
Deninger: Vector bundles and 𝑝adic parallel transport II (PDF)
Litt: Monodromy and Galois action on _pi_1^_et (PDF)
Liu: 𝑝adic RiemannHilbert correspondence and applications (PDF)
Ogus: The saturated de RhamWitt complex after Bhatt, Lurie, Mathew (PDF)
Scholze: Some results on the cohomology of finite flat group schemes (PDF)
Vologodsky: On periodic topological cyclic homology of (dg) algebras in characteristic p (PDF)
Weinstein: Cohomology of moduli spaces of local shtukas (PDF) 
Participants
Ahmed Abbes CNRS & IHÉS Amnon Besser Ben Gurion University Bhargav Bhatt University of Michigan Kęstutis Česnavičius CNRS, Université ParisSud Aise Johan de Jong Columbia University Christopher Deninger Universität Münster Hélène Esnault Freie Universität Berlin Gerd Faltings Max Planck Institute for Mathematics Ofer Gabber CNRS & IHÉS Michel Gros CNRS & Université Rennes 1 Minhyong Kim Oxford University Daniel Litt Institute for Advanced Study Ruochuan Liu BICMR/Peking University Jacob Lurie Harvard University Matthew Morrow CNRS & Institut de Mathématiques de Jussieu–Paris Rive Gauche Arthur Ogus University of California, Berkeley Martin Olsson University of California, Berkeley Peter Scholze Mathematisches Institut der Universität Bonn Takeshi Tsuji University of Tokyo Vadim Vologodsky Higher School of Economics Jared Weinstein Boston University Annette Werner Goethe University Frankfurt Xinwen Zhu California Institute of Technology