Date & Time

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

In this final symposium, organizers Bhargav Bhatt and Martin Olsson plan to cover 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 aim to highlight directions for future research.

• 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

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

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

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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