Titles, Abstracts, and Slides

June 7–11, 2021

Ahmed Abbes The relative Hodge-Tate spectral sequence

I will report on a joint work with Michel Gros providing a generalization of the Hodge-Tate spectral sequence to morphisms. The latter takes place in Faltings topos. Its construction requires the introduction of a relative variant of this topos which is the main novelty of our work.

Alexander Beilinson Height pairing and vanishing cycles

(No abstract available)

Vladimir Berkovich Hodge theory for non-Archimedean analytic spaces

In a work in progress, I defined integral “etale” cohomology and de Rham cohomology for so called bounded non-Archimedean analytic spaces over the field of formal Laurent power series with complex coefficients. The former are local systems of finitely generated abelian groups on a certain log formal complex analytic space, and the latter are finite free modules over the ring of formal power series provided with a Gauss-Manin connection. Both give rise to vector bundles with connections on that log formal complex analytic space, and the bundles are shown to be isomorphic. Furthermore, if a bounded non-Archimedean analytic space has no boundary, its integral and de Rham cohomology form a so called mixed Hodge structure over the log formal complex analytic space. This structure depends functorially on the non-Archimedean space, and in the case when it comes from a complex algebraic variety over a punctured disc, the associated mixed Hodge structure is an extension of the corresponding classical object.

Bhargav Bhatt The absolute prismatic site

The absolute prismatic site of a p-adic formal scheme carries and organizes interesting arithmetic and geometric information attached to the formal scheme. In this talk, after recalling the definition of this site, I will discuss a stacky approach to absolute prismatic cohomology and its concomitant structures (joint with Lurie, and due independently to Drinfeld). As a geometric application, I'll explain Drinfeld's refinement of the Deligne-Illusie theorem on Hodge-to-de Rham degeneration. On the arithmetic side, I'll describe a new classification of crystalline Galois representations of the Galois group of a local field in terms of F-crystals on the site (joint with Scholze).

Kęstutis Česnavičius Grothendieck–Serre in the quasi-split unramified case

The Grothendieck–Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We settle it in the case when G is quasi-split and R is unramified. To overcome obstacles that have so far kept the mixed characteristic case out of reach, we adapt Artin's construction of "good neighborhoods" to the setting where the base is a discrete valuation ring, build equivariant compactifications of tori over higher dimensional bases, and study the geometry of the affine Grassmannian in bad characteristics.

Miaofen Chen Newton stratification and weakly admissible locus in p-adic Hodge theory

Rapoport and Zink introduce the p-adic period domain (also called the admissible locus) inside the rigid analytic p-adic flag varieties. The weakly admissible locus is an approximation of the admissible locus in the sense that these two spaces have the same classical points. In this talk, we will try to describe the relation between the weakly admissible locus and the Newton stratification in the flag variety. This is a joint work in progress with Jilong Tong.

Hélène Esnault Motivic connections over a finite field (work in progress with Michael Groechenig)

If $X$ is a smooth variety over a perfect field $k$ of characteristic $p>0$ which lifts to $W_2(k)$, its de Rham complex up to a certain level splits. It is the spectacular theorem of Deligne-Illusie from 1987, which led to manifold developments. It enabled Ogus-Vologodsky to develop a version of the Simpson correspondence, and later to Lan-Sheng-Zuo to define when $X$ is projective Higgs-de Rham flows, which over finite fields for flat connections with vanishing Chern classes, are preperiodic.

On the other hand, over the field of complex numbers, Brunebarbe-Klingler-Totaro proved, in answer to a question I had posed, that if $X$ smooth projective has no degree $\ge 1$ symmetric differential forms, then all flat connections are motivic, in fact they have finite monodromy. Their proof is transcendental.

We investigate a version of this theorem over a finite field. Under Deligne-Illusie $W_2(\mathbb{F}_q)$ assumption, we suspect that precisely the same vanishing condition on global symmetric differential forms forces flat connections with vanishing Chern classes to have finite monodromy. As of today, we can prove it in rank $\le 3$.

Ofer Gabber Comparison of oriented products and rigid toposes

(No abstract available)

Yongquan Hu On a generalization of Colmez’s functor

In 2005, Colmez defined an exact functor from the category of finite length admissible smooth representations of GL_2(Q_p) over a field of characteristic p to the category of finite length continuous representations of the absolute Galois group of Q_p. This functor has played a crucial role in the p-adic Langlands program for GL_2(Q_p). In this talk, I will review the construction of Colmez’s functor and a generalization due to Breuil. I will discuss the exactness and finiteness of this (generalized) functor. This is a joint work with Breuil, Herzig, Morra and Schraen.

Kazuya Kato Logarithmic abelian varieties

This is a joint work with T. Kajiwara and C. Nakayama. Logarithmic abelian varieties are degenerate abelian varieties which live in the world of log geometry of Fontaine-Illusie. They have group structures which do not exist in the usual algebraic geometry. By using the group structures, we give a new formulation of the work of K-W Lan on the toroidal compactification of the moduli space of abelian varieties with PEL structures.

Nicholas Katz Exponential sums and finite groups

This is joint work with Antonio Rojas Leon and Pham Huu Tiep, where we look for “interesting” finite groups arising as monodromy groups of “simple to remember” families of exponential sums”.

Gérard Laumon On the derived Lusztig correspondence

We will study the relationships between the $\ell$-adic derived category of the stack $[\mathfrak{t}/N_{G}(T)]$ and the $\ell$-adic derived category of the stack $[\mathfrak{g}/G]$.

Akhil Mathew Remarks on p-adic logarithmic cohomology theories

Many p-adic cohomology theories (e.g., de Rham, crystalline, prismatic) are known to have logarithmic analogs. I will explain how the theory of the “infinite root stack” (introduced by Talpo-Vistoli) gives an alternate approach to building the logarithmic theory (from the non-logarithmic one). As a consequence, one obtains an integral version of (log-)syntomic cohomology with comparisons to p-adic nearby cycles. Joint with Bhargav Bhatt and Dustin Clausen.

Sophie Morel Intersection cohomology of Shimura varieties and pizza

Given a disc in the plane select any point in the disc and cut the disc by four lines through this point that are equally spaced. We obtain eight slices of the disc, each having angle π/4 at the point. The pizza theorem says that the alternating sum of the areas of these slices is equal to zero. I will talk about higher-dimensional versions of this theorem, where the lines are replaced by a Coxeter arrangement and the pizza by a ball (or a more general shape), and explain how this problem sheds light on the combinatorics that appear in the spectral description of the intersection cohomology of Shimura varieties. This is joint work with Richard Ehrenborg and Margaret Readdy.

Ngô Bảo Châu Jet bundles and differential calculus

(No abstract available)

Arthur Ogus Prisms, prismatic neighborhoods, and p-de Rham cohomology

Prismatic cohomology, as proposed by B. Bhatt and P. Scholze, provides a uniform framework for many of the cohomoogy theories involved in p-adic Hodge theory. I will focus on the crystalline incarnation of prismatic cohomology and its relation to p-de Rham cohomology, as suggested by B. Bhatt. I will also attempt to relate crystalline prisms to previoius work on the p-adic Cartier transform due to several authors. The main new ingredient is a detailed study of the geometry of Frobenius lifts and prismatic neighborhoods.

Martin Olsson Representability results for flat cohomology

Let $f:X\rightarrow S$ be a proper morphism of schemes over a field $k$ of positive characteristic, and let $G$ be a finite flat abelian group scheme over $X$. In this talk I will discuss recent representability results for the derived pushforwards $R^if_*G$. Key ingredients in proving our results is the development of a theory of compactly supported flat cohomology and description in terms of the cotangent complex in some cases. This is joint work with Daniel Bragg.

Takeshi Saito Micro support of a constructible sheaf in mixed characteristic

One of the obstacles in the definition of the singular support of a constructible sheaf in mixed characteristic is the absence of the cotangent bundle. We define a ‘Frobenius pull-back’ of the cotangent bundle restricted on the closed fiber of a flat regular scheme over a discrete valuation ring of mixed characteristic by using the sheaf of Frobenius-Witt differentials. We give some examples of constructible sheaves whose singular supports exist.

Yunqing Tang Basic reductions of abelian varieties

Elkies proved that an elliptic curve over Q has infinitely many supersingular reductions. The generalization of the 0-dimensional supersingular locus of the modular curve is the so called basic locus of a Shimura curve at a good prime. In this talk, we generalize Elkies’s theorem to some abelian varieties over totally real fields parametrized by certain unitary Shimura curves arising from the moduli spaces of cyclic covers of the projective line ramified at 4 points. This is joint work in progress with Wanlin Li, Elena Mantovan, and Rachel Pries.

Michael Temkin Logarithmic geometry and resolution of singularities

I will tell about recent developments in resolution of singularities achieved in a series of works with Abramovich and Wlodarczyk – resolution of log varieties, resolution of morphisms and a no-history (or dream) algorithm for resolution of varieties. I will try to especially emphasize the role of logarithmic geometry in these algorithms and in the quest after them.

Yichao Tian Cohomology of prismatic crystals

Prismatic crystals are natural analogues of classical crystalline crystals on prismatic sites, which were introduced by Bhatt and Scholze. In this talk, I will explain some general properties such objects on the prismatic site for a smooth variety over the base. The main result is a finiteness theorem for the cohomology of prismatic crystals on proper and smooth varieties.

Cong Xue Cohomology of stacks of shtukas

Let X be a geometrically connected smooth projective curve over Fq and G a reductive group over the function field of X. For any finite set I we have the stacks of shtukas over X^I, and the Satake sheaves over the stacks of shtukas.

Let N be a level structure. We prove that the relative cohomology sheaf of the stack of shtukas is ind-smooth over (X - N)^I. Moreover, we hope to prove that the cohomology of the special fiber of the stack of shtukas at a point in the level with coefficients in the nearby cycles is isomorphic to the cohomology of the generic fiber of the stack of shtukas.