Alexander Petrov Characteristic classes of p-adic local systems
Abstract:
Continuous cohomology classes of the group GL_n(Z_p) with coefficients in Q_p give rise to a theory of characteristic classes for etale Z_p-local systems on schemes or adic spaces, valued in (absolute) etale cohomology with Q_p-coefficients. These classes can be thought of as p-adic analogs of Chern-Simons characteristic classes of complex local systems.
For local systems on a smooth variety over an algebraically closed field of characteristic zero these p-adic characteristic classes vanish in degrees >1, at least for large enough p (with the bound a priori depending on the variety, and the rank of the local systems). But for varieties over non-closed fields these classes are often non-trivial. For a variety X over a p-adic field they are related to Chern classes of the pro-etale bundle associated to the local system. For a Hodge-Tate local system on X those Chern classes are expressed in terms of Chern classes of the graded pieces of the associated Higgs bundle, which gives a partial description of the characteristic classes of the local system.
For crystalline local systems on varieties with good reduction a complete calculation of these characteristic classes would follow from the following vanishing statement: for an F-isocrystal on the special fiber the i-th Chern class of the corresponding vector bundle on the Fargues-Fontaine curve of X are zero for i>1.
This is joint work with Lue Pan.
11:00-12:00
Pol van Hoften p-divisible groups and p-adic Fourier theory
Abstract:
I will discuss joint work in progress with Andrew Graham and Sean Howe, in which we develop a p-adic Fourier transform for p-divisible rigid analytic groups. For the open unit disk, this recovers the Amice transform and for a Lubin—Tate p-divisible group of dimension 1, this recovers the p-adic Fourier theory of Schneider—Teitelbaum. We expect our formalism to work well in rigid analytic families.
2:30-3:30
Brandon Levin Weight part of Serre's conjecture and the Emerton-Gee stack
Abstract: This will be colloquium-style talk about the weight part of Serre's conjecture and its connection to moduli spaces of Galois representations. The weight part of Serre's conjectures is a modularity conjecture which predicts in what spaces of mod p modular forms a given mod p Galois representation appears. This question is expected to be purely local and controlled by the geometry of local deformation rings with p-adic Hodge theory conditions. I will describe how this conjecture can be understood through the Emerton-Gee stack of mod p Galois representations and overview progress towards the conjecture and related problems.
4:00-5:00
Jared Weinstein The Picard group of the K(n)-local category
Abstract:
Homotopy theory is centered on the symmetric monoidal category of spectra (generalized cohomology theories). The category of spectra is complicated,
so as in algebra, one profits from localizing this category at "primes", which are the Morava K-theories K(n) (relative to a rational prime p). The Picard group of the K(n)-local category was studied by Hopkins-Mahowald-Sadovsky, who calculated it completely when n = 1. In this project we calculate the Picard group of the K(n)-local category completely for all n, at least when the prime p is large relative to n. As a key input we use recent results of Colmez-Dospinescu-Niziol on the pro-etale cohomology of Drinfeld's symmetric space. This is a joint project with Barthel, Schlank, and Stapleton.
Tuesday, September 17
9:30-10:30
Tony Feng Prismatic Steenrod operations and applications to Brauer groups
Abstract:
Tate conjectured that the Brauer group of a surface over a finite field has a natural symplectic structure. This is now known in odd characteristics, but the case of characteristic 2 has remained elusive. I will describe work-in-progress with Shachar Carmeli on the construction of a syntomic Steenrod algebra, which acts on the mod p syntomic cohomology of algebraic varieties in characteristic p. For p=2, we apply our theory to affirm Tate’s conjecture. Although the applications are classical, our methods rely on recent advances in perfectoid geometry and prismatic cohomology.
11:00-12:00
Toby Gee
Cuspidal cohomology classes for GL_n(Z)
Abstract:
I will explain a construction of weight zero, level 1 automorphic representations of GL_n/Q for some values of n>1, concentrating on the aspects related to p-adic Galois representations. This is joint work with George Boxer and Frank Calegari.
2:30-3:30
Jacob Lurie Rationalized Syntomic Cohomology
Abstract:
Syntomic cohomology is an invariant p-adic formal schemes, which has a close relationship to (etale-localized) algebraic K-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed that, after inverting p, syntomic cohomology admits a concrete description in terms of more familiar invariants, such as de Rham and crystalline cohomology. In this talk, I'll explain an alternative perspective on their result, which avoids the use of K-theoretic methods.
4:00-5:00
Bogdan Zavyalov The trace morphism and Poincare Duality in p-adic non-archimedean geometry
Abstract: I will explain a construction of the trace morphism for smooth morphism
of analytic adic spaces. Then I will explain how one can use this trace to prove various
Poincare Duality type results. In particular, I will discuss a new easy proof of Poincare Duality
for F_p-cohomology groups of smooth proper p-adic rigid-analytic spaces and appropriately
generalize this result to arbitrary proper morphisms.
Wednesday, September 18
9:00-10:00
Vytas Paskunas On local Galois deformation rings: generalised reductive groups
Abstract:
I will explain recent joint work with Julian Quast on deformation rings of local Galois groups
valued in generalised reductive group schemes. The prerpint is available at
https://arxiv.org/abs/2404.14622
10:30-11:30
Naoki Imai
The syntomic realization functor for Shimura varieties
Abstract: We discuss a new characterization of the canonical integral model of
Shimura varieties of abelian type at hyperspecial level using a
realization functor in prismatic F-gauges, which we call the syntomic
realization functor. This is a joint work with Hiroki Kato and Alex
Youcis.
11:45-12:45
Ananth Shankar
Integral canonical models of Shimura varieties
Abstract:
Given an exceptional Shimura
variety S, we prove the existence of integral canonical models for S at all sufficiently large
primes. Our method passes through finite characteristic and relies on a partial generaliza-
tion of Ogus-Vologodsky. As applications, we prove analogues of Tate semisimplicity in
finite characteristic, CM lifting theorems for ordinary points, the Tate isogeny theorem
for ordinary points, and an unramified p-adic analogue of Borel's extension theorem for Shimura varieties. This is joint with Ben Bakker and Jacob Tsimerman.
Thursday, September 19
9:30-10:30
Lue Pan
Some recent developments in the Sen theory
Abstract:
Around the 1970s, Sen introduced a systematic way to study the Hodge-Tate structure of p-adic Galois representations. I want to discuss some recent development of the Sen theory and its applications in the study of p-adic geometry of Shimura varieties.
11:00-12:00
Juan Esteban Rodriguez Camargo
Fargues-Fontaine de Rham stacks
Abstract:
Joint work in progress with Johannes Anschütz, Arthur-César Le Bras and Guido Bosco.
The analytic de Rham stack is a new construction in analytic geometry (after Clausen and Scholze) whose theory of quasi-coherent
sheaves encodes a notion of p-adic D-modules, but that has the virtue that it can be defined even under lack of differentials (eg. for
perfectoid spaces or Fargues-Fontaine curves). In this talk I will mention some (expected) applications of the theory of the analytic de
Rham stack in p-adic Hodge theory in the form of Fargues-Fontaine de Rham stacks; analytic objects whose cohomology theories
refine the usual de Rham cohomology of rigid spaces in the form of the Fargues-Fontaine de Rham cohomology of Le Bras-Vezzani.
2:30-3:30
Mingjia Zhang
Intersection cohomology of Shimura varieties
Abstract:
In a joint work with Daniels, van Hoften and Kim, we have constructed "Igusa stacks" for certain Shimura varieties and applied these to study the cohomology of Shimura varieties. This turns out to be fruitful and leads to interesting results. In case of PEL type Shimura varieties, following a suggestion of Scholze, the Igusa stacks are shown to have minimal compactifications. In this talk, we investigate the possibility of studying the intersection cohomology of Shimura varieties using minimally compactified Igusa stacks. This is joint work, much in progress, with Ana Caraiani and Linus Hamann.
4:00-5:00
Gal Porat Locally analytic vectors in mixed characteristic
Abstract:
Work of Berger-Colmez describes Sen theory in terms of locally analytic vectors in Q_p-Banach representations of p-adic Lie groups. This led to applications in the study of completed cohomology (Pan, Rodriguez Camargo). I will discuss recent work (some of it in progress) which aims to achieve a generalization of the work of Berger-Colmez to a mixed characteristic setting.
Friday, September 20
9:00-10:00
Linus Hamann Finiteness Theorems for Geometric Eisenstein Series
Abstract:
A fundamental operation in the smooth representation theory of a p-adic group G is parabolic induction. This allows one to construct smooth representations of G in terms of smooth representations of a proper Levi M, in a way that is known to preserve basic finiteness properties such as finite generation and admissibility. In this talk, we will discuss how to prove the analogues of these properties in the context of the geometrization of the local Langlands correspondence. In particular, we replace parabolic induction by a geometric Eisenstein functor which takes \ell-adic sheaves on the moduli stack of M-bundles on the Fargues-Fontaine curve to \ell-adic sheaves on the moduli stack of G-bundles, and show that this functor preserves compact and ULA sheaves, the geometric analogues of finitely generated and admissible representations, respectively. This is based on joint work with David Hansen and Peter Scholze.
10:30-11:30
David Hansen Geometric Whittaker coefficients and the Fargues-Fontaine curve
Abstract:
I'll explain several versions of the Casselman-Shalika formula in the context of the Fargues-Scholze geometrization program, with an emphasis on the case of GL2. I'll also try to explain how I came to appreciate the power of doing sheaf theory on huge spaces. Joint work in progress with Linus Hamann and Lucas Mann.
11:45-12:45
Haoyang Guo Pointwise criteria of p-adic local systems
Abstract: For a family of abelian varieties over a p-adic field, it is natural to ask how does the reduction of individual fibers relate to the reduction of the entire family. More generally, one can consider the analogous question for the crystallinity and the semi-stability of a p-adic local system. In this talk, we show that for a p-adic local system over a rigid space X with a smooth (resp. semi-stable) model, it is crystalline (resp. semi-stable) if and only if its restrictions at many classical points are so. As one of the key tools, we will introduce a crystalline enhancement of the p-adic Riemann--Hilbert functor. This is a joint work with Ziquan Yang.
Local Information
There will be coffee breaks from 10:30-11:30 and 2:30-3:30 each day (except
Wednesday and Friday).
The classrooms Eckhart 131 and 133 have been reserved throughout the week for
informal discussion among the conference participants.
The closest quick option for lunch near the department is the
Reynolds
Club (immediately north), which includes a Pret a Manger as
well as several restaurants (from which one can either order in advance via
Grubhub or at a kiosk inside).