đť‘ťadic Hodge Theory (2022)
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 đť‘ťadic 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 padic 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 â€śproblem sessionâ€ť 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 padic cohomology. As in the case of Ă©tale 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 padic 6functor formalism in rigidanalytic geometry, that a construction of a full sixfunctor formalism for suitable derived categories of sheaves on rigidanalytic varieties. The approach is based on a RiemannHilbert type correspondence and the newly developed theory of condensed mathematics of Clausen and Scholze.
David Hansenâ€™s talk, Constructible de Rham complexes, discussed a new padic analogue of Saitoâ€™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 CartierWitt stack, which is a geometric object whose quasicoherent 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 DeligneIllusie theorem.
In the talk Duality for padic proĂ©tale cohomology of analytic curves, WiesĹ‚awa NizioĹ‚ discussed duality for proĂ©tale cohomology in the padic context. She discussed duality results both in the arithmetic and geometric context.
Johannes AnschĂĽtz and ArthurCĂ©sar Le Bras jointly gave a twopart talk entitled A Fourier transform for BanachColmez spaces. They explained their work developing a version of the geometric Fourier transform, in the style of Deligne and Laumon, in the padic 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 padic Hodge theory and the local Langlands correspondence, modularity and numbertheoretic 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 Bellovinâ€™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 padic Ă©tale cohomology of coverings of Drinfeldâ€™s upper halfplane, the representation theoretic decomposition, in the style of the local Langlands correspondence, of the padic Ă©tale cohomology of coverings of Drinfeldâ€™s upper plane given by considering level structure.
In connection with modularity, Ana Caraiani gave a talk, Modularity over CM fields, concerning localtoglobal compatibility aspects of the torsion Langlands correspondence and its applications to modularity of elliptic curves generalizing classical work of Wiles and TaylorWiles on Fermatâ€™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 socalled FarguesFontaine curve enables one to realize geometrically, using gerbes and Tannakian categories of isocrystals, inner forms of reductive groups over padic fields. Fargues also discussed a â€śgeometrizationâ€ť 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 AndrĂ©â€™s lecture, A remark on the Tate conjecture, discussed extensions of work of Moonen on applications of padic Hodge theory to prove semisimplicity of Galois representations arising from motives. Ofer Gabber gave a talk, Flattening for nonNoetherian formal schemes, explaining various important technical results that arise in the context of current work on padic geometry (which often involves working with nonNoetherian rings outside the scope of many standard algebraic tools). KÄ™stutis ÄŚesnaviÄŤius gave a talk entitled Adic continuity and descent for flat cohomology, wherein he explained fundamental descent results for flat cohomology â€” 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 padic differential equations in families, discussing a new proof of an important foundational result on padic differential equations called the padic monodromy theorem; the new proof works in more general contexts, leading to applications in padic 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 AndrĂ©  A remark on the Tate conjecture 11:30  12:30 PM KÄ™stutis ÄŚesnaviÄŤius  Adic continuity and descent for flat cohomology 5:00  6:00 PM Lucas Mann  A pAdic 6Functor Formalism in RigidAnalytic 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 padic proetale 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 CartierWitt Stack THURSDAY  05.12.22
10:00  11:00 AM Johannes AnschĂĽtz  A Fourier transform for BanachColmez spaces 11:30  12:30 PM ArthurCĂ©sar Le Bras  A Fourier transform for BanachColmez spaces 5:00  6:00 PM Pierre Colmez  Factorisation of the padic Ă©tale 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 padic 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
ArthurCĂ©sar Le Bras
A Fourier transform for BanachColmez spacesI will discuss the definition and some examples of an \(\ell\)adic Fourier transform for BanachColmez spaces. Joint work with Johannes AnschĂĽtz.
Wieslawa Niziol
Duality for padic proetale cohomology of analytic curvesI will discuss duality theorems, both arithmetic and geometric, for padic proetale cohomology of rigid analytic curves. This is joint work with Pierre Colmez and Sally Giles.
Ofer Gabber
Flattening for non noetherian formal schemesLet 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 Ă©tale site and some expected cohomology computations.
Kiran Kedlaya
Monodromy representations of padic differential equations in familiesThe padic local monodromy theorem is a fundamental structure theorem for vector bundles with connection on padic analytic annuli. It has important foundational consequences in padic cohomology (finiteness of rigid cohomology) and padic Hodge theory (Fontaine’s conjecture C_{st}). The original version of this theorem by AndreKedlayaMebkhout 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 padic cohomology (a drastically simplified proof of “cohomological semistable reduction”) and padic Hodge theory (a relative form of C_{st} conjectured by LiuZhu).
Pierre Colmez
Factorisation of the padic Ă©tale cohomology of coverings of Drinfeld’s upper half planeWe will report on a joint work with Gabriel Dospinescu and Wieslawa Niziol giving a factorisation Ă la Emerton for the padic Ă©tale cohomology of coverings of Drinfeld’s upper halfplane.
Yves AndrĂ©
A remark on the Tate conjectureThe 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 padic Hodge theory, that i) implies ii) in characteristic 0. Iâ€™ll elaborate and recast his result in the framework of observability theory, and discuss the case of positive characteristic.
Lucas Mann
A pAdic 6Functor Formalism in RigidAnalytic GeometryUsing the recently developed condensed mathematics by ClausenScholze we construct a full padic 6functor formalism on rigidanalytic varieties and more generally on diamonds and small vstacks. Instead of working directly with F_psheaves, this 6functor formalism is based on a theory of “solid quasicoherent almost $\mathcal O_X^+/p$”sheaves on $X$. By proving a version of a ptorsion RiemannHilbert correspondence we relate this category to actual F_psheaves, which in particular provides a purely local proof of padic PoincarĂ© duality in the rigid setting. We also expect many applications of our 6functor formalism to the padic Langlands program.
KÄ™stutis ÄŚesnaviÄŤius
Adic continuity and descent for flat cohomologyI 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 BanachColmez spacesWe will explain the definition of an \elladic Fourier transform for BanachColmez spaces and discuss some examples.
Jacob Lurie
The CartierWitt StackThe 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 quasicoherent sheaves on an algebrogeometric object called the CartierWitt 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)modulesTwo 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 padic Langlands program.
David Hansen
Constructible de Rham complexesLet X be a variety over a padic field. By work of Scholze, LiuZhu, DiaoLanLiuZhu, and others, we have a good working theory of de Rham Q_plocal 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 fieldsI will discuss joint work in progress with James Newton, where we prove a localglobal 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 AllenKhareThorne.
Laurent Fargues
Extension du domaine de la lutteI 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 padic 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.