• HCM Workshop: Recent developments in integral p-adic cohomology theories
• Schedule

• Schedule
• Participants

I will revisit the theory of Hodge-Tate local systems in the light of the p-adic Simpson correspondence. This is a joint work with Michel Gros.

In this series, I will discuss recent joint work with M. Morrow and P. Scholze on the construction of a new cohomology theory for a proper smooth scheme over a p-adic ring. This theory interpolates between the existing ones (such as etale, de Rham, crystalline) integrally, and thus sheds some new light on the behavior of torsion in cohomology as one specializes an algebraic variety from characteristic 0 to characteristic p.

Fontaine has defined filtered modules with Frobenius which define (or should define) p-adic Galois representations. This theory has been extended to families over a smooth base. We give an extension to semistable schemes. This is more difficult because the commutative algebra poses interesting new problems. As an application we show that certain Shimura varieties admit a semistable model.

I will explain the classification of certain modifications of vector bundles on the fundamental curve of p-adic Hodge theory in terms of phi-modules over Fontaine's ring A_inf. I will try to explain in more details the following two points:

- the reason why this is not an exact equivalence, a fact linked to the discrepancy between torsion in p-adic étale cohomology and crystalline cohomology.

- the analogy with archimedean Hodge theory with the hope to understand the link between the Bhatt-Morrow-Scholze construction and Deligne's construction of the modification of twistors in terms of λ-connexions.

In this talk, which is strongly connected to my joint work with Uwe Jannsen, after recalling the definition of -gauges, I'll explain how they can be used to get additional structures on p-torsion cohomologies and to build p-torsion sheaves on various sites (over syntomic schemes, perfectoid spaces,...

Topological Hochschild homology is Hochschild homology relative to the initial ring S of higher algebra. Descending from Z to S, as long advocated by Waldhausen, has several important consequences. First, denominators tend to disappear. Namely, Bökstedt's periodicity theorem shows that the topological Hochschild homology of a perfect field k of positive characteristic is a polynomial algebra over k on a generator in degree 2, the divided Bott element. Second, topological Hochschild homology comes equipped with a "cyclotomic" structure, which gives rise to an inverse Frobenius operator on its equivariant homotopy groups.

Periodic cyclic homology was introduced by Connes as the appropriate extension of de Rham cohomology to noncommutative rings and may be defined as the Tate cohomology of the circle group with coefficients in the Hochschild homology spectrum. By analogy, we define periodic topological cyclic homology to the be the Tate cohomology of the circle group with coefficients in the topological Hochschild spectrum. We will show that for a scheme smooth and proper over a finite field, the resulting infinite dimensional cohomology theory naturally produces Deninger's cohomological interpretation of the Hasse-Weil zeta function by regularized determinants.

We report on ongoing joint work with Ruochuan Liu. Previously, we defined a category of (phi, Gamma)-modules associated to an arbitrary adic space over Q_p and a fully faithful functor into this category from Q_p-local systems. This category is exact but not abelian. Now restrict attention to rigid analytic spaces; in this case, we define a larger category of coherent (phi, Gamma)-modules and establish certain functoriality properties, particularly for higher direct images along smooth proper morphisms. This implies the finite-dimensionality of cohomology of a Q_p-local system on a smooth proper variety over a finite extension of Q_p.

I will review the theory of so called "Breuil-Kisin" modules, which gives a description of semi-stable representations in terms of certain Frobenius modules over W[[u]], and explain some of the applications of this theory, both old and new.

We construct a functor from the category of p-adic etale local systems on a smooth rigid analytic variety X over a p-adic field to the category of B_dR-vector bundles over X with an integrable connection, which can be regarded as a first step towards the sought-after p-adic Riemann-Hilbert correspondence. As a consequence, we obtain the following rigidity theorem for p-adic local systems on a connected rigid analytic variety: if the stalk of such a local system at one point, regarded as a p-adic Galois representation, is de Rham in the sense of Fontaine, then the stalk at every point is de Rham. Along the way, we also establish some basic properties of the p-adic Simpson correspondence. Finally, we give an application of our results to Shimura varieties. Joint work with Xinwen Zhu.

For a semistable scheme over a mixed characteristic local ring I will present a proof of a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. I will also explain how to combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finitness of \'etale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction. This is a joint work with Pierre Colmez.