Motives in Tokyo 2026

This is joint work with Achim Krause and Baptiste Morin. We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to study the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to give a new proof of the Beilinson fibre square and to prove conjecture C_EP(Q_p(n)) of Fontaine and Perrin-Riou. We also use our techniques to compute the correction factor C(X,n) defined by Flach and Morin in their reformulation of the Tamagawa number conjecture for smooth proper schemes over number rings.

Lars Hesselholt

My purpose with this talk is to make some propaganda for the theory of Gestalten, which is currently being developed by Scholze and Stefanich with parts of the technical underpinning provided by Aoki. In particular, following Scholze, I will explain how the theory gives a natural explanation of the fact that the assignment

(R, R^{+}) \mapsto D^{◻} (R, R^{+})

promotes to a sheaf on Huber's adic space

Spa (R, R^{+})

Annette Huber

The Period Conjecture makes a qualitative prediction about all linear relations between the periods of motives over number fields. It is a theorem in the case of 1-motives, e.g. for numbers like logarithms or algebraic numbers or periods of elliptic curves over number fields.

In joint work with Martin Kalck, we explain how to deduce dimension formulas via the structure theory of finite dimensional algebras over perfect fields.

Marc Levine

We give a report on versions of classical Atiyah-Bott localization in the setting of generalized motivic cohomology theories living in the minus part of the motivic stable homotopy category, with the main examples being Witt sheaf cohomology and Witt theory. The main thrust is to replace the classical approach relying on a G_m-action with one that uses instead an action by SL_2 or the normalizer of the torus in SL_2. We will also discuss recent work involving other groups. The new input involves a description of the real points of BG, from work of Ambrosi-de Gaay Fortnam and Mantovani-Matszangosz-Wendt, together with Bachmann’s description of the real realisation as \rho localisation.

Hiroyasu Miyazaki

There have been several attempts to generalize the classical theory of motives to capture non-A^1 invariant phenomena. Among such is the theory of motives with modulus, which can be considered as a combination of the theory of motives and some ideas from ramification theory. In this talk, I will report on some recent developments of the theory of motives with modulus, with emphasis on motivic homotopy theory. I will give a quick overview of the theory and discuss an attempt towards a motivic duality theorem in this context. This talk is based on an ongoing joint work with Doosung Park and a prior joint work with Bruno Kahn.

Baptiste Morin

We give a conjectural description of Zeta-values of arithmetic schemes in terms of two perfect complexes of abelian groups and a canonical trivialization. Then we state a (proven) archimedean analogue of this conjecture. In order to show compatibility with the classical Bloch-Kato conjecture, we study the Beilinson fiber square from a prismatic viewpoint, which provides precise integral information. This implies new cases of the Bloch-Kato conjecture. This is joint with Matthias Flach and Achim Krause.

Thomas Nikolaus

The main result of the present talk is an identification of the K-theory of the category of G-equivariant sheaves on a locally compact Hausdorff space equipped with an action of a finite group G. This extends a result of Efimov. To formulate this result, we introduce a new theory, which we call Bredon sheaf cohomology. This is a sheaf cohomology version of equivariant Bredon cohomology and, for nice G-CW complexes, agrees with Bredon cohomology. The main result is a consequence of a very strong uniqueness result for Bredon sheaf cohomology, generalizing a result of Clausen. We will present a number of consequences. If time permits, we will explain how this fits into the general theory of noncommutative motives and how this can be used to devise proof strategies to tackle fundamental questions in geometric topology and construct Chern characters in topological K-homology.

Noriyuki Otsubo

We define Chow motives whose realizations give Gaussian hypergeometric functions over the complex numbers and over finite fields, and lift known formulas on both sides to isomorphisms of motives. Over a finite field, the Frobenius endomorphisms of these motives are expressed in terms of so-called hypergeometric algebraic correspondences. This generalizes Coleman’s theorem for Fermat curves, and resembles the Eichler-Shimura congruence relation for modular curves. Assuming a conjecture partially proved, we define the adelic Gaussian hypergeometric function, which generalizes the adelic beta function of Anderson-Ihara.

Jinhyun Park

Title: Big de Rham-Witt-valued log and Geisser-Hesselholt-Costeanu dlog for certain non-reduced algebras

In this talk, I will first sketch how one can relate the relative Milnor K-theory of the truncated polynomial in one variable over a regular algebra R to the big de Rham-Witt forms over R. We can regard it as the big de Rham-Witt-valued logarithm. We sketch the story via a theory of algebraic cycles. Using this logarithm, we sketch how we can generalize the p-tyical de Rham-Witt-valued dlog on the Milnor K-theory of R by Geisser-Hesselholt-Costeanu (cf. Krishna-Park) to to the Milnor K-theory of the non-reduced algebra R[t]/(t^{m+1}). This is a work in progress with a student of mine, Jaehong Kim.

Oliver Röndigs

Consider the special linear group SL_n over a field. Murthy proved that every vector bundle over SL_2 is trivial. Swan constructed a nontrivial vector bundle of rank two over SL_4 for the field of complex numbers. This led Nakamoto and Torii to investigate vector bundles over SL_3, and to conjecture that they are all trivial. The talk will discuss this conjecture using unstable A^1-homotopy theory.

Shuji Saito

A philosophical implication of Grothendieck’s standard conjecture is the existence of an intrinsic integral structure on Weil cohomology theories. It leads us to the following question: Let K be a complete discrete valuation field with the ring R of integers. Is there a canonical integral (i.e. an R-lattice) structure on the de Rham cohomology of proper smooth K-schemes, which is functorial for K-morphisms? I will report on a joint work in progress with A. Merici and K. Ruelling aiming to answer to this question by aid of a variant of the tame cohomology theory introduce by Huebner-Schmidt together with rigid analytic geometry. A main result obtained so far gives a positive answer to the question by replacing de Rham cohomology by cohomology of the structure sheaf.

Takeshi Saito

The definitions of singular supports and characteristic cycles of constructible sheaves are indirect involving the local acyclicity and vanishing cycles. In the transcendental context, there is a direct construction using microlocalization. I will discuss an attempt toward its analogue in the algebraic context.

Alexander Schmidt

In 1983, A. Grothendieck coined the word "anabelian" (read as: "far from being abelian") and formulated several anabelian conjectures. One of them claimed that two hyperbolic curves defined over a number field are isomorphic if and only if their étale fundamental groups are. This has been proven by H. Nakamura, A. Tamagawa and, in greater generality, by S. Mochizuki. In this talk we report on some work in progress towards higher dimensional anabelian results.

Brian Shin

Serre's classical GAGA principle says that coherent sheaves on a proper complex variety are the same as those on its analytification. Consequently, one finds that the algebraic and analytic Picard groups agree in this case. In other words, the degree-two weight-one motivic cohomology of a proper complex variety is controlled by its analytification. This raises a question: what other parts of motivic cohomology satisfy a similar sort of GAGA principle? In this talk, I will report on an answer to this question in mixed characteristic geometry: there is a GAGA principle for motivic zero-cycles on proper varieties over algebraically closed non-archimedean fields. This is joint work-in-progress with Toni Annala, Tess Bouis, Elden Elmanto, and Mahdi Rafiei.

Vova Sosnilo

The category of localizing motives introduced by Blumberg--Gepner--Tabuada serves as a natural place for studying such cohomological invariants as K-theory, Hochschild homology, Blanc's topological K-theory. On the one hand, many phenomena of cohomology theories of this kind can be encoded in properties of this category. On the other hand, the construction of the category is very abstract, making it inaccessible for calculations. We provide a new construction of the category by proving that the functor of taking the motive Cat^st ----> Motloc is a Dwyer-Kan localization at a certain class of motivic equivalences. We apply this result to various categorification question. In particular, we show that any ring can be represented as the K-theory of a monoidal stable category. This talk is based on joint work with Maxime Ramzi and Christoph Winges as well as an ongoing work with Maxime Ramzi, Stefan Schwede and Christoph Winges.

Kennichi Sugiyama

A positive integer

n

is called a congruent number if it is the area of a right triangle with rational sides. It is well known that

n

is a congruent number if and only if the Mordell-Weil group of an elliptic curve

E_{n} : y^{2} = x^{3} - n^{2} x

has positive rank and, according to the BSD conjecture, it is also equivalent to

L (E_{n}, 1) = 0

. Tunnel shows that these special values appear as Fourier coefficients of a theta series of weight

3 / 2

which correspond to

f_{E_{1}}

of by the Shimura correspondence. Here

f_{E_{1}}

is a cusp form of weight

2

with level

32

E_{1}

. Suppose that

N

is a prime and Gross gives a geometric interpretation of these phenomena for

f \in S_{2} (Γ_{0} (N))

. Later, by representation theory, Böcherer and Schulze-Pillot generalized Gross's theorem for square free

N

. Following Gross's argument, we will generalize their results for

N = p M

, where

p

is an odd prime and

M

is an odd integer prime to

p

(

M

may not be square free).

Rin Sugiyama

A field

F

is a

B_{s}

-field if, for every finite extension

E^{'} / E

F

, the norm map

K_{s}^{M} (E^{'}) \to K_{s}^{M} (E)

of the Milnor

K

-groups is surjective. In particular, finite fields (resp. local fields) satisfy this condition for

s = 1

(resp.

s = 2

). In this talk, for such a field

F

and a

d

-dimensional variety

X

over

F

, we discuss the structure of the higher Chow group

C H^{d + n} (X, n)

and prove that

C H^{d + s} (X, s)

is isomorphic to the direct sum of the Milnor

K

-group

K_{s}^{M} (F)

and a divisible group. This is a joint work with T. Hiranouchi.

Takashi Suzuki

I will report on my ongoing joint work with Antoine Galet (Jussieu), which generalizes Kato-Parshin's class field theory for higher local fields (including the existence theorem) in the style of local Tate duality and which takes care of ind-pro-...-ind-pro algebraic structures on cohomology. The setting is over ind-pro-...-ind-pro completions of coherent sites (modeled after pro-etale sites). The arithmetic inputs are the duality over relatively perfect sites by Kato and its mixed characteristic (nearby cycle) version by Kato and the speaker. These allow us to inductively perform a relative duality for each intermediate i-local field/(i-1)-local field step, all the way down to the final (finite) residue field.

Poster:

Photos:

Travel:

There are two international airports in Tokyo: Haneda and Narita. Haneda is more convenient but Narita has more flights.

Haneda: It takes about 60 minutes to get from Haneda Airport to Komaba campus via the subway system with two transfers and costs between 600 JPY and 1,000 JPY. Routes can be found on google maps. For example, one can take the monorail to Hamamatsucho Station and then transfer to the JR Yamanote line.

Narita: It takes about 90 minutes to get from Narita Airport to Komaba campus via the subway system with one or two transfers and costs around 3,000 JPY. Routes can be found on google maps. For example, one can take the Keisei Skyliner via Nippori Station, or the Narita Express via Shibuya Station. Another option is the Airport Limousine Bus.

Shibuya station: Almost all routes involve a transfer at Shibuya station, famous for its bustling energy. Coming from the airport you will arrive on the east side of Shibuya station, and have to cross it to get to the line on the west (i.e., opposite) side of Shibuya station. Everything is clearly sign-posted. Follow the Inokashira line logos:

.

⚠️ The main trap on the Inokashira line is that both express (急行) and local (各駅停車) trains run on this line, and only the local (各駅停車) trains will stop at Station.

Train ticket payment: You can pay for your train ticket with cash at the station. Some credit cards may work in some ticket machines. If you have an iPhone you can add a Suica to the Apple Wallet and pay by swiping your phone at the gate. You can also use Suica to pay for stuff in most stores. It's better to Suica up before leaving your home country.

Local navigation: Google maps works pretty well for navigating the trains and streets in Tokyo.

SIM cards: You can order a data eSIM / physical SIM before leaving your home country and activate it in advance, so you’ll have internet access as soon as you land. There are various options such as Ubigi or ESIMJAPAN.com or World eSIM or ... Alternatively, you can purchase a SIM card (physical or eSIM) in Tokyo or at Tokyo airport. If you want to use data from your home provider make sure to enable international roaming.

Power: Japan and the US use the same shape power plugs. If your devices are US you don't need a power adapter. If they are not but you have a US adapter, that should work.

Cash: You can use credit card and/or suica almost everywhere in Japan these days, and you should be able to withdrawal cash from any convenience store with a credit card, so you shouldn't need so much cash. 15,000 JPY ≈ 90 USD ≈ 80 EUR is probably sufficient "safety" cash, unless you're trying to regularly pay for stuff with cash.

Motives in Tokyo 2026 on the occasion of Thomas Geisser's 60th Birthday

Information:

Speakers:

Program:

Titles & Abstracts:

Matthias Flach