# 𝑝-adic Hodge Theory (2022)

Date & Time

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

In this final symposium, organizers Bhargav Bhatt and Martin Olsson covered recent advances in all aspects of 𝑝-adic Hodge theory, including both foundational breakthroughs internal to the subject as well as applications to other areas of mathematics. In addition to discussing important contemporary work, they highlighted directions for future research.

• Meeting Report

This third symposium in the series focused on recent advances in all aspects of p-adic Hodge theory. The symposium covered both foundational advances as well as applications of the theory. The atmosphere was informal and interactive with lively interactions during and between talks. The symposium also featured a designated “problem session” for discussion of open problems in the area, in addition to the many informal conversations.

### Overview of Talks

Several talks at the symposium explained new work related to derived categories of coefficient systems for p-adic cohomology. As in the case of étale cohomology and complex geometry, the added flexibility and constructions that one obtains from more refined coefficient systems, such as constructible sheaves and complexes or perverse sheaves, is important in many settings and applications.

In this direction, Lucas Mann explained in his talk, A p-adic 6-functor formalism in rigid-analytic geometry, that a construction of a full six-functor formalism for suitable derived categories of sheaves on rigid-analytic varieties. The approach is based on a Riemann-Hilbert type correspondence and the newly developed theory of condensed mathematics of Clausen and Scholze.

David Hansen’s talk, Constructible de Rham complexes, discussed a new p-adic analogue of Saito’s derived category of mixed Hodge modules. Hansen addressed the fact that de Rham local systems are not closed under extensions in the usual derived category, and therefore obtaining the correct derived category of de Rham complexes requires a different approach that was explained in the talk.

Jacob Lurie spoke about The Cartier-Witt stack, which is a geometric object whose quasi-coherent sheaves correspond to prismatic crystals, thus geometrizing the notion of coefficient systems in the prismatic context; the geometry of this stack enables a number of applications, including a refinement of the seminal Deligne-Illusie theorem.

In the talk Duality for p-adic pro-étale cohomology of analytic curves, Wiesława Nizioł discussed duality for pro-étale cohomology in the p-adic context. She discussed duality results both in the arithmetic and geometric context.

Johannes Anschütz and Arthur-César Le Bras jointly gave a two-part talk entitled A Fourier transform for Banach-Colmez spaces. They explained their work developing a version of the geometric Fourier transform, in the style of Deligne and Laumon, in the p-adic setting for certain perfectoid spaces. In addition to explaining the general feature, several applications were discussed.

Another recurring theme at the symposium was the connection between p-adic Hodge theory and the local Langlands correspondence, modularity and number-theoretic applications.

Matthew Emerton and Toby Gee gave two joint lectures based on their collaborative work, singularly entitled Moduli stacks of (phi, Gamma)-modules. While deformation theory of Galois representations has played a key role in number theory for several decades, Emerton and Gee introduce a new global geometric object, using the theory of (phi, Gamma)-modules, which in some sense patches together the deformation rings of such deformation rings.

These modules, and corresponding Galois representations, were also the subject of Rebecca Bellovin’s talk, Trianguline Galois representations in mixed characteristic. Bellovin explain how to study integral structure and reduction in this context.

In a different, but related, direction, Pierre Colmez discussed in his talk, Factorization of the p-adic étale cohomology of coverings of Drinfeld’s upper half-plane, the representation theoretic decomposition, in the style of the local Langlands correspondence, of the p-adic étale cohomology of coverings of Drinfeld’s upper plane given by considering level structure.

In connection with modularity, Ana Caraiani gave a talk, Modularity over CM fields, concerning local-to-global compatibility aspects of the torsion Langlands correspondence and its applications to modularity of elliptic curves generalizing classical work of Wiles and Taylor-Wiles on Fermat’s Last Theorem.

The symposium concluded with a talk by Laurent Fargues entitled Extension du domaine de la lute, in which he explained how the so-called Fargues-Fontaine curve enables one to realize geometrically, using gerbes and Tannakian categories of isocrystals, inner forms of reductive groups over p-adic fields. Fargues also discussed a “geometrization” of the local Langlands correspondence for reductive groups using this theory.

A number of important other topics were covered in the lectures at the symposium. Yves André’s lecture, A remark on the Tate conjecture, discussed extensions of work of Moonen on applications of p-adic Hodge theory to prove semi-simplicity of Galois representations arising from motives. Ofer Gabber gave a talk, Flattening for non-Noetherian formal schemes, explaining various important technical results that arise in the context of current work on p-adic geometry (which often involves working with non-Noetherian rings outside the scope of many standard algebraic tools). Kęstutis Česnavičius gave a talk entitled Adic continuity and descent for flat cohomology, wherein he explained fundamental descent results for flat cohomology — a cohomology theory which is closely related to crystalline and prismatic cohomology and which has played an important role in recent years. Kiran Kedlaya gave a talk entitled Monodromy representations of p-adic differential equations in families, discussing a new proof of an important foundational result on p-adic differential equations called the p-adic monodromy theorem; the new proof works in more general contexts, leading to applications in p-adic Hodge theory.

In sum, the talks delved into many of the most important recent developments in the subject and pointed the way for future research in the area.

• Agenda

#### MONDAY | 05.09.22

 10:00 - 11:00 AM Yves André | A remark on the Tate conjecture 11:30 - 12:30 PM Kęstutis Česnavičius | Adic continuity and descent for flat cohomology 5:00 - 6:00 PM Lucas Mann | A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry 6:15 - 7:15 PM Discussion

#### TUESDAY | 05.10.22

 10:00 - 11:00 AM Toby Gee | Moduli stacks of (phi,Gamma)-modules, Part I 11:30 - 12:30 PM Matthew Emerton | Moduli stacks of (phi,Gamma)-modules, Part II 5:00 - 6:00 PM Wieslawa Niziol | Duality for p-adic pro-etale cohomology of analytic curves 6:15 - 7:15 PM Problem Session

#### WEDNESDAY | 05.11.22

 5:00 - 6:00 PM Ofer Gabber | Flattening for non noetherian formal schemes 6:15 - 7:15 PM Jacob Lurie | The Cartier-Witt Stack

#### THURSDAY | 05.12.22

 10:00 - 11:00 AM Johannes Anschütz | A Fourier transform for Banach-Colmez spaces 11:30 - 12:30 PM Arthur-César Le Bras | A Fourier transform for Banach-Colmez spaces 5:00 - 6:00 PM Pierre Colmez | Factorisation of the p-adic étale cohomology of coverings of Drinfeld's upper half plane 6:15 - 7:15 PM Rebecca Bellovin | Trianguline Galois representations in mixed characteristic

#### FRIDAY | 05.13.22

 10:00 - 11:00 AM Ana Caraiani | Modularity over CM fields 11:30 - 12:30 PM Kiran Kedlaya | Monodromy representations of p-adic differential equations in families 5:00 - 6:00 PM David Hansen | Constructible de Rham complexes 6:15 - 7:15 PM Laurent Fargues | Extension du domaine de la lutte
• Abstracts & Slides

Arthur-César Le Bras
A Fourier transform for Banach-Colmez spaces

I will discuss the definition and some examples of an $$\ell$$-adic Fourier transform for Banach-Colmez spaces. Joint work with Johannes Anschütz.

Wieslawa Niziol
Duality for p-adic pro-etale cohomology of analytic curves

I will discuss duality theorems, both arithmetic and geometric, for p-adic pro-etale cohomology of rigid analytic curves. This is joint work with Pierre Colmez and Sally Giles.

Ofer Gabber
Flattening for non noetherian formal schemes

Let X’—–>X be an admissible blowup of formal schemes topologically of finite type over an adic* qcqs formal scheme S; then X—–>S becomes formally flat of finite presentation after an admissible blowup of S (and taking proper transform) iff the same holds for X’—–>S .We discuss foundational approaches to the rigid étale site and some expected cohomology computations.

Kiran Kedlaya
Monodromy representations of p-adic differential equations in families

The p-adic local monodromy theorem is a fundamental structure theorem for vector bundles with connection on p-adic analytic annuli. It has important foundational consequences in p-adic cohomology (finiteness of rigid cohomology) and p-adic Hodge theory (Fontaine’s conjecture C_{st}). The original version of this theorem by Andre-Kedlaya-Mebkhout depended on a “Frobenius structure” and on the base field being discretely valued. We describe a recent extension of this theorem that can be formulated without either of these hypotheses, and therefore generalizes easily to a corresponding relative statement (for a family of annuli parametrized by a rigid space). This again has consequences in p-adic cohomology (a drastically simplified proof of “cohomological semistable reduction”) and p-adic Hodge theory (a relative form of C_{st} conjectured by Liu-Zhu).

Pierre Colmez
Factorisation of the p-adic étale cohomology of coverings of Drinfeld’s upper half plane

We will report on a joint work with Gabriel Dospinescu and Wieslawa Niziol giving a factorisation à la Emerton for the p-adic étale cohomology of coverings of Drinfeld’s upper half-plane.

Yves André
A remark on the Tate conjecture

View Slides (PDF)

The Tate conjecture has two parts: i) Tate classes are generated by algebraic classes, ii) semisimplicity of Galois representations coming from pure motives. In a recent note with the same title, B. Moonen proved, with an argument from p-adic Hodge theory, that i) implies ii) in characteristic 0. I’ll elaborate and recast his result in the framework of observability theory, and discuss the case of positive characteristic.

Lucas Mann
A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry

Using the recently developed condensed mathematics by Clausen-Scholze we construct a full p-adic 6-functor formalism on rigid-analytic varieties and more generally on diamonds and small v-stacks. Instead of working directly with F_p-sheaves, this 6-functor formalism is based on a theory of “solid quasicoherent almost $\mathcal O_X^+/p$”-sheaves on $X$. By proving a version of a p-torsion Riemann-Hilbert correspondence we relate this category to actual F_p-sheaves, which in particular provides a purely local proof of p-adic Poincaré duality in the rigid setting. We also expect many applications of our 6-functor formalism to the p-adic Langlands program.

Kęstutis Česnavičius
Adic continuity and descent for flat cohomology

I will discuss properties of flat cohomology of adically complete rings with coefficients in commutative, finite, locally free group schemes. The talk is based on joint work with Peter Scholze.

Johannes Anschuetz
A Fourier transform for Banach-Colmez spaces

We will explain the definition of an \ell-adic Fourier transform for Banach-Colmez spaces and discuss some examples.

Jacob Lurie
The Cartier-Witt Stack

View Slides (PDF)

The formalism of prismatic cohomology has an associated theory of coefficient objects, called prismatic crystals. Prismatic crystals on the formal scheme Spf(Z_p) can be viewed quasi-coherent sheaves on an algebro-geometric object called the Cartier-Witt stack. In this talk, I’ll describe the geometry of this stack and note some consequences for the theory of absolute prismatic cohomology. Joint work with Bhargav Bhatt.

Toby Gee (+Matthew Emerton)
Moduli stacks of (phi,Gamma)-modules

Two talks on moduli stacks of (phi,Gamma)-modules, the first by TG and the second by ME. We’ll explain what we know about these stacks and their relationship to the p-adic Langlands program.

David Hansen
Constructible de Rham complexes

Let X be a variety over a p-adic field. By work of Scholze, Liu-Zhu, Diao-Lan-Liu-Zhu, and others, we have a good working theory of de Rham Q_p-local systems on X. But what should it mean for a general object in D^b_c(X,Q_p) to be de Rham? I’ll propose an answer to this question, and sketch some applications of the resulting theory. I’ll also try to emphasize the analogy with Saito’s theory of mixed Hodge modules.

Rebecca Bellovin
Trianguline Galois representations in mixed characteristic

$$p$$-adic Galois representations have associated $$(\varphi,\Gamma)$$-modules, and it is possible for an irreducible Galois representation to have a reducible $$(\varphi,\Gamma)$$-module; trianguline $$(\varphi,\Gamma)$$-modules are successive extensions of rank-$$1$$ objects. Although the trianguline structure on the $$(\varphi,\Gamma)$$-module is, in general, not compatible with an integral structure on the Galois representation, I will discuss an extension of this theory to Galois representations with coefficients in mixed characteristic and positive characteristic. I will give applications to extended eigenvarieties.

Ana Caraiani
Modularity over CM fields

I will discuss joint work in progress with James Newton, where we prove a local-global compatibility result in the crystalline case for Galois representations attached to torsion classes occurring in the cohomology of locally symmetric spaces. I will then explain an application to the modularity of elliptic curves over imaginary quadratic fields, which also builds on recent work of Allen-Khare-Thorne.

Laurent Fargues
Extension du domaine de la lutte

I will explain a new object that extends the Kottwitz set and allows us to reach all inner forms of a given reductive group over a p-adic field. This involves a Tannakian category of extended isocrystals that contains the usual category of isocrystals. This is linked to the so called Kaletha gerb and we give a geometric interpretation of this gerb using the curve. At the end I will explain how to formulate a Geometrization conjecture of the local Langlands correspondence for any reductive group.

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