Titles and abstracts
The slides of some talks are available: see links below.
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.
Tomoyuki Abe :
Characteristic cycles and pushforward formula
Characteristic cycles for an l-adic sheaf was defined by T. Saito using the existence result of singular support by Beilinson. It is natural to expect that the characteristic cycle is compatible with pushforward along a proper morphism, but it turned out that this question is not as straightforward as in the complex situation. A main reason for this is that, contrary to the complex case, the characteristic cycle is not necessarily be Lagrangian. In this talk, I wish to discuss this compatibility using homotopical methods. In the core of the discussion, I use the method of Kedlaya on the proof of the semistable reduction theorem.
Piotr Achinger :
The Riemann-Hilbert correspondence for rigid spaces
Let $X$ be a qcqs smooth rigid space over the complex formal Laurent series field $\mathbb{C}((t))$.
In joint work with Talpo we used log geometry to associate
to $X$ a "Betti homotopy type" $\Psi(X)$.
In the talk, we will discuss representations of its
fundamental group $\pi_1^{Betti}(X) := \pi_1(\Psi(X))$.
The main result is an equivalence between the category of
finite dimensional $\mathbb{C}$-linear representations of $\pi_1^{Betti}(X)$ and
the category of vector bundles with a $\mathbb{C}$-linear integrable connection on $X$
which are "regular at infinity" in a certain not-so-straightforward sense.
As a byproduct, we can define admissible variations of mixed Hodge structure on $X$.
Fabrizio Andreatta :
Syntomic formalism with coefficients
Motivated by the study of p-adic Abel-Jacobi maps in ongoing work with Bertolini,
Seveso and Venerucci I will provide a definition of syntomic complexes and provide a comparison
with absolute étale cohomology. I will also explain the relation with other definitions.
Yves André :
Mathematics between sacred and profane
This title was given by Bruno Chiarellotto to the speaker as a theme for a fugue. The fugue will be played during the conference.
Gilles Christol :
The relative theory for p-adic differential equations in the Robba case
(slides)
The first step toward a theory of p-adic differential equations in several variables is to look at the one variable case but with relative coefficients, namely depending on parameters. Using the Dwork-Robba terminology, the case of « admissible » coefficients is rather easy but the case of « super admissible » coefficients is much more involved. It is a joint work (in progress) with Mebkhout.
Richard Crew :
Duality for differential equations on Berkovich curves
(slides)
I will first review the work of Andrea Pulita and Jerome Poineau on the convergence properties of differential equations on a rig-smooth Berkovich curve, and its applications to finiteness questions. As an application I will prove a duality theorem for non-solvable equations on such curves. This is joint work with Andrea Pulita.
Veronika Ertl :
Poincaré duality in log rigid cohomology
Poincaré duality is an important feature of a "good" cohomology that plays a role in many applications.
In joint work with Kazuki Yamada (Keio University) we investigate it in the context of log rigid cohomology.
I will explain our construction of log rigid cohomology with compact support for several types of coefficients.
In particular, I will explain its compatibility with the structures of Hyodo--Kato theory.
This approach has the advantage, that the constructions are explicit yet versatile, and hence suitable for computations.
Luc Illusie :
New advances on de Rham cohomology in positive or mixed characteristic, after Bhatt-Lurie, Drinfeld, and Petrov
(slides)
The de Rham complex of a smooth scheme over a perfect field of characteristic $p >0$ holds many mysteries. I'll give a survey of recent discoveries in this domain.
Chris Lazda :
Good reduction of Kummer surfaces modulo 2
In residue characteristic different from 2, the good reduction of a given Kummer surface is (essentially) equivalent to the good reduction of the abelian surface it is constructed from, and is therefore completely understood via the classical Neron-Ogg-Shafarevich criterion. In residue characteristic 2, however, things are a little bit more interesting, and good reduction of a given abelian surface is not enough to ensure good reduction of its associated Kummer surface. In this talk I will explain a more refined criterion for the good reduction of K3 surfaces in general, and use it to solve the problem of good reduction for Kummer surfaces associated to abelian surfaces with good, non-supersingular reduction in residue characteristic 2. This is joint work with Alexei Skorobogatov.
Bernard Le Stum :
Descent in rigid cohomology
(slides, see also arXiv:1707.02797, arXiv:2209.07875)
Descent is a sophisticated glueing technic that applies to the coefficients of a cohomological theory (effective descent) as well as to the cohomology of these coefficients (cohomological descent). In rigid cohomology, cohomological descent is due to Chiarellotto-Tsuzuki in the étale case and to Tsuzuki in the proper case (see also Zureick-Brown) and effective descent is due to Lazda. We present here a new proof of these results that uses the overconvergent site (as Zureick-Brown did). This allows us to treat both types of descent simultaneously, to do the étale and proper case together and to generalize the results to constructible (and not merely overconvergent) crystals and to formal schemes (and not only algebraic varieties).
François Loeser :
Tropical functions on skeleta
Skeleta are subspaces of non-archimedean spaces endowed with a tropical-like structure. They are encountered in a wide array of situations, for instance as images of non-archimedean spaces under retractions. The aim of this talk is to present some new finiteness results for tropical functions on skeleta. This is joint work with Antoine Ducros, Ehud Hrushovski and Jinhe Ye.
Wiesława Nizioł :
Duality for p-adic pro-étale cohomology of analytic curves
I will discuss duality theorems, both arithmetic and geometric, for p-adic pro-étale cohomology of rigid analytic curves. This is joint work with Pierre Colmez and Sally Gilles.
Shun Ohkubo :
A note on the convergence Newton polygons of p-adic differential equations in the regular singular case
Convergence Newton polygons are fundamental invariants defined for p-adic differential equations on curves. In this talk, we study that the convergence Newton polygons of p-adic differential equations on p-adic discs in the regular singular case, which include some p-adic hypergeometric differential equations.
Andrea Pulita :
Exponents for irregular differential modules. A Tannakian approach to the theory of exponents
(slides)
Since more than one century, exponents are numerical invariants of regular singular differential equations. They admit a $p$-adic analogue introduced by Christol and Mebkhout in 1997 to deal with the problem of the finite dimensionality of de Rham cohomology of p-adic differential equations. The aim of the talk is twofold : simplify the existing definition of exponents in the p-adic case and on the other hand generalize it to possibly irregular differential equations and unify the non-archimedean and archimedean definitions. This is a work in progress with C.Lazda and A.Pál.
Tsuji Takeshi :
Generalized Coleman power series and Iwasawa cohomology for
Lubin-Tate extensions
For Lubin-Tate extensions of a finite (not necessarily unramified)
extension K of the local field F defining the Lubin-Tate formal group,
we discuss generalized Colman series, their images under the
logarithmic derivation map, and a generalized explicit reciprocity law
for the multiplicative group. We use fields of norms and Cartier
isomorphisms. The case of cyclotomic extensions was studied by P. Colmez
before. Thanks to the determination of the image of the log derivation
map for an arbitrary finite extension K (which was inspired by the
computation of L. Berger in the case K=F), we obtain another proof of
the description of the first Iwasawa cohomology H^1 of a local Galois
O_F-representation by Schneider-Venjakob in terms of the Lubin-Tate
phi-Gamma module. I will also mention some results on lifting
Lubin-Tate phi-Gamma modules (suggested by L. Berger) and the above
description to the universal deformation of the Lubin-Tate formal
group, towards a multivariable theory.
Masha Vlasenko :
Cohomology and congruences
We consider the module of differential forms on the complement of a toric hypersurface. For a positive integer k we define the submodule of so-called kth formal derivatives, which can be characterized by certain divisibility properties of their expansion coefficients. It turns out that under a certain condition, which we call the kth Hasse-Witt condition, the quotient of all differential forms by the kth formal derivatives is a free module of finite rank. This result is a p-adic phenomenon, and for k=1 we recover an analog of the explicit construction of unit-root crystals given by Nickolas Katz in the 1980s. We will also discuss several applications of our result. For k=1 one obtains Dwork's congruences and Gauss' congruences for expansion coefficients of rational functions. With higher k our theorem can be used to explain some supercongruences, to show existence and derive some p-adic properties of the excellent Frobenius lifts and to prove integrality of the mirror map and integrality of instanton numbers in some key examples of mirror symmetry. This is joint work with Frits Beukers.