Tomoyuki Abe (Kavli IPMU)
Title : Ramification theory from homotopical point of view
Abstract : T. Saito defined the characteristic cycle for étale sheaves using the existence of singular support proved by Beilinson, which is a milestone of the ramification theory. He characterized characteristic cycle so that the "Milnor formula" holds for any isolated characteristic function. Beilinson's theorem is used to ensure that there are "enough" supply of isolated characteristic functions for the characterization. However, if we take non-isolated characteristic functions in the Milnor formula, the behavior is not so clear. From 𝔸1-homotopical point of view, considering non-isolated characteristic functions is natural. We consider multiple functions to "localize" constructible sheaves to zero-cycles, which I hope to give a new perspective on the theory of characteristic cycles.
Alexander Beilinson (University of Chicago)
Title : Height pairing and the Picard-Lefschetz-Illusie formula
Abstract : Suppose we have a family of smooth projective varieties over ℚ with singular fiber having isolated singularity such that its blowup is smooth. Spencer Bloch conjectured that the height pairing between algebraic cycles supported on the exceptional divisor of the blowup can be read from the limiting Hodge structure. I will sketch a proof.
Bhargav Bhatt (University of Michigan)
Title : p-adic cohomology theories via stacks
Abstract : I'll explain stacky approaches to various cohomology theories in algebraic geometry: de Rham cohomology in characteristic 0, crystalline cohomology in characteristic p, and (relative as well as absolute) prismatic cohomology in mixed characteristic. In each case, I'll explain how important structural features of the cohomology theory are quite natural from the stacky approach. At the end, I'll discuss some parallels to Saito's recent work on Frobenius-Witt differentials. (This is joint work with Lurie and independently due to Drinfeld.)
Ana Caraiani (Imperial College London)
Title : A comparison theorem for ordinary p-adic modular forms
Abstract : I will discuss joint work with Elena Mantovan and James Newton, where we compare ordinary completed cohomology with (higher) Hida theory, in the special case of the modular curve. Both these notions go back to Hida, though the former can be reinterpreted using Emerton's functor of ordinary parts applied to completed cohomology, and the latter has been redeveloped and expanded recently by Boxer and Pilloni to incorporate higher coherent cohomology. Our work gives a new proof to a theorem of Ohta, that is perhaps more amenable to generalisation. The key ingredients are the Bruhat stratification on the Hodge-Tate period domain, and the integral comparison results pioneered by Bhatt, Morrow and Scholze.
Kęstutis Česnavičius (CNRS-Université Paris-Saclay),
Title : Adic continuity and descent for flat cohomology
Abstract : I 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.
Ofer Gabber (CNRS-IHÉS)
Title : Properties of oriented products and nearby cycles over general bases
Quentin Guignard (MPIM Bonn)
Title : Etale covers with bounded ramification
Abstract : Deligne refined Fontaine-Winterberger's equivalence by giving a description of the category of finite separable extensions with bounded ramification of a local field. I will discuss higher dimensional variants of Deligne's result.
Luc Illusie (Université Paris-Saclay)
Title : Revisiting Deligne-Illusie
Abstract : I will revisit my 1987 paper with Deligne on mod p2 liftings and decompositions of de Rham complexes at the light of recent developments in p-adic Hodge theory. I will discuss new inputs and results coming from work by various authors on derived de Rham complexes and prismatic cohomology.
Kazuya Kato (University of Chicago)
Title : Upper ramification groups of arbitrary valuation rings
Abstract : This is a joint work with Vaidehee Thatte. In the classical ramification theory, for a complete discrete valuation ring A with perfect residue field, the upper ramification groups are defined as subgroups of the absolute Galois group of the field of fractions of A. Abbes and Saito generalized this theory to arbitrary complete discrete valuation rings. We generalize it to arbitrary valuation rings (whose value groups can be of arbitrary type).
Moritz Kerz (Universität Regensburg)
Title : Density of local systems with quasi-unipotent monodromy at infinity
Abstract : The famous monodromy theorem tells us that local systems on algebraic varieties which are of geometric origin have quasi-unipotent monodromy at infinity. A deep conjecture says that these local systems of geometric origin are Zariski dense in all local systems. I will explain why the density holds for local systems with quasi-unipotent monodromy at infinity. This is joint work with H. Esnault.
Shinichi Kobayashi (Kyushu University)
Title : Local units and root numbers of Hecke L-functions
in anticyclotomic extensions at inert primes
Abstract : In this talk, I explain recent works with A. Burungale and K. Ota on Iwasawa theory for Hecke characters of imaginary quadratic fields in anticyclotomic extensions at inert primes. In our inert setting, root numbers of Hecke characters vary interestingly depending on the conductor at p, and special arithmetic phenomena arise. In the late 1980's K. Rubin envisioned an Iwasawa theory reflecting such phenomena and conjectured a structure of the Iwasawa module of local units over anticyclotomic extensions of the unramified quadratic extension of ℚp, which plays a key role in his theory. Recently we proved the conjecture and the proof also led us new developments such as an application to Kato's local epsilon conjecture in this setting.
Teruhisa Koshikawa (RIMS-Kyoto University)
Title : Logarithmic prismatic site
Abstract : The logarithmic prismatic site is an analogue of the prismatic site of Bhatt and Scholze for log p-adic formal schemes. Several results have been extended to this log setting. I will explain the definition of the site and discuss some of the results.
Wiesława Niziol (CNRS-Sorbonne Université)
Title : Cohomology of p-adic analytic spaces
Abstract : I will discuss recent results on p-adic cohomologies of rigid analytic spaces obtained in a joint work with Pierre Colmez.
Shuji Saito (The University of Tokyo)
Title : The Abbes-Saito formula for motivic ramification filtrations
Abstract : This is a joint work with Kay Rülling and is an attempt to give a motivic interpretation on a small part of Takeshi Saito's spectacular achievements in ramification theory. Abbes-Saito introduced a condition that the ramification of a torsor under a finite group over a smooth scheme U is bounded by an effective divisor supported on the boundary of a partial compactification of U by using a geometric construction called dilatations. Using this, T. Saito defined the characteristic forms of torsors. In this talk we first explain for every Nisnevich sheaf F of abelian groups on the category of smooth schemes satisfying a property called ``reciprocity'', how one can define a ``motivic filtration'' on the group F(U) of sections over a smooth scheme U which are parametrized by pairs of partial compactifications of U and effective divisors supported on the boundaries. It turns out that the motivic filtrations recover classically known conductors such as Kato-Matsuda's Swan conductor in case F(U) is the group of torsors over U under a finite abelian group, and the irregularity in case F(U) is the group of connections of rank one on U, and the Rosenlicht-Serre conductor in case F(U) is the group of sections over U of a commutative algebraic group. One can also consider the group of torsors under a finite flat group schemes, which gives rise to new conductors. The main theorem affirms that the motivic filtration on F agrees with another filtration on F which is defined in the same manner as Abbes-Saito's construction using dilatations. This gives rise to the characteristic form associated to a section of a reciprocity sheaf, which determines whether the section vanishes or not in a graded quotient of the motivic filtration. If time permits, we compute the characteristic forms for some examples of reciprocity sheaves.
Daichi Takeuchi (RIKEN)
Title : Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic
Abstract : For a function on a smooth variety with an isolated singular point, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (étale). In this talk, I will give a formula which expresses the local epsilon factor of the vanishing cycles complex in terms of the bilinear form. In particular, the sign of the local epsilon factor is determined by the discriminant of the bilinear form. This can be regarded as a refinement of Milnor formula in SGA 7, which compares the total dimension of the vanishing cycles and the rank of the bilinear form.
Akio Tamagawa (RIMS-Kyoto University)
Title : The m-step solvable anabelian geometry of global fields and finitely generated fields
(joint work with Mohamed Saïdi)
Abstract : The (birational) anabelian geometry of global fields and fields finitely generated over the prime field has been developed and established by Neukirch, Uchida, Pop, and others. In this talk, I will present some recent progress on the m-step solvable anabelian geometry of global fields and finitely generated fields, where the absolute Galois groups are replaced by their maximal m-step solvable quotients for suitable m (hopefully as small as possible).
Takeshi Tsuji (The University of Tokyo)
Title : Integral cohomologies in the p-adic Simpson correspondence
Abstract : I will discuss how much the cohomology of a small Higgs module (defined by the Dolbeault complex) recovers the cohomology of the associated small generalized representation in the integral p-adic Simpson correspondence by Faltings for a proper smooth scheme over the ring of integers of a p-adic field. For the latter cohomology, suggested by Matthew Morrow through personal discussions on this topic, I consider the cohomology defined similarly to the Ainf cohomology introduced by Bhatt, Morrow, and Scholze. It should coincide with the prismatic cohomology of Bhatt and Scholze with coefficients in a certain kind of crystal naturally associated to the small Higgs module. I will also explain how the cohomology of a small Higgs module could be related directly to the prismatic cohomology despite the local nature of delta ring structures introduced in the theory of prisms.
Liang Xiao (Peking University)
Title : Several questions regarding slopes of modular forms
Abstract : In classical theory, slopes of modular forms are p-adic valuations of the eigenvalues of the Up-operator. On the Galois side, they correspond to the p-adic valuations of eigenvalues of the crystalline Frobenius on the Kisin's crystabelian deformations space. I will report on some recent progress on the study of slopes, hopefully overview the scope of the theory, and possibly explain some of its arithmetic applications. This includes some joint projects with collaborators including Yongquan Hu, Ruochuan Liu, Nha Truong, Bin Zhao.
Yuri Yatagawa (Tokyo Institute of Technology)
Title : Singular support and characteristic cycle of a rank one sheaf in codimension two
Abstract : The singular support and the characteristic cycle of a constructible sheaf on a smooth variety are defined by Beilinson and Saito and are expected to be computed using ramification theory. We focus on a rank one sheaf whose logarithmic ramification along the boundary is "clean". We compute the singular support and the characteristic cycle of such a rank one sheaf by admitting a blow-up along a sub scheme of the boundary and to remove a closed sub scheme of the variety in codimension ≥3. The computation is given in terms of the ramification theories introduced by Brylinski, Kato, Matsuda, Abbes, and Saito.
Weizhe Zheng (MCM-Chinese Academy of Sciences)
Title : Ultraproduct cohomology and the decomposition theorem
Abstract : Ultraproducts of étale cohomology provide a large family of Weil cohomology theories for algebraic varieties. Their properties are closely related to questions of l-independence and torsion-freeness of l-adic cohomology. I will present recent progress in ultraproduct cohomology with coefficients, such as the decomposition theorem. This talk is based on joint work with Anna Cadoret.
Ahmed Abbes (CNRS-IHÉS), Kenichi Bannai (Keio University-RIKEN),
Naoki Imai (The University of Tokyo), Tadashi Ochiai (Osaka University),
Atsushi Shiho (The University of Tokyo)