# International Workshop on motives in Tokyo, 2011

### Organizing Committee:

Thomas Geisser (Nagoya University), Tomohida Terasoma (Tokyo University), Shuji Saito (TIT)### Speakers

Aravind Asok (University of Southern California)

Joseph Ayoub (University of Zuerich)

Mathias Flach (Caltech)

Dan Grayson (Univ Illionois at Urbana-Champaign)

Christian Haesemeyer (USLA)

Lars Hesselholt (Nagoya)

Uwe Jannsen (Regensburg)

Marc Levine (Duisburg-Essen)

Stephan Mueller-Stach (Mainz)

Oliver Roendigs (Osnabruck)

Alena Pirutka (Strassbourg)

Marco Schlichting (Warwick)

Tamas Szamuely (Budapest)

J"org Wildeshaus (Universite' Paris 13)

Takao Yamazaki (Tohoku University)

### This workshop is supported by

Interactive Research Center of Science, Graduate School of Science and Engineering, Tokyo Institute of Technology,

Graduate School of Mathemtaical Sciences, the University of Tokyo, Global COE Program, "The research and traning center for new developement of mathematics",

JSPS Grant-in-aid (A) #23244002 representative Makoto Matsumoto,

JSPS Grant-in-aid (B) #23340001 representative Tomohide Terasoma,

JSPS Grant-in-aid (B) #30571963 representative Thomas Geisser,

JSPS Grant-in-aid (S) #19104001@representative Toshiyuki Katusra

JSPS Grant-in-aid (B) #22340003 representative Shuji Saito,

### Program

### 12(Mon)/Dec:

9:30-10:30 Grayson, "Daniel Quillen and his work on algebraic K-theory."

10:45-11:45 Mueller-Stach, "Introduction to periods and Nori motives I."

14:00-15:00 Yamazaki, "Voevodsky's motif and Weil reciprocity."

15:30-16:30 Haesemeyer, "Obstructions to embeddings in algebraic geometry I."

16:45-17:45 Schlichting, "Introduction to higher Grothendieck-Witt groups."

### 13(Tue)/Dec:

9:30-10:30 Mueller-Stach, "Introduction to periods and Nori motives II."

10:45-11:45 Grayson, "The motivic spectral sequence via acyclic binary complexes."

14:00-15:00 Asok, "Obstructions to embeddings in algebraic geometry II."

15:30-16:30 Flach, "Weil-etale cohomology and Zeta functions of arithmetic schemes I."

16:45-17:45 Schlichting, "Geometric representability of hermitian K-theory in A1 homotopy theory."

18:30-20:30: Reception

### 14(Wed)/Dec:

9:30-10:30 Wildeshaus, "Motivic intersection complex for Shimura varieties."

10:45-11:45 Roendigs, "Voevodsky's slice filtration."

Free afternoon

### 15(Thu)/Dec:

9:30-10:30 Roendigs, "The slice filtration on hermitian K-theory."

10:45-11:45 Levine, "Connections between motivic and classical homotopy theory."

14:00-15:00 Pirutka, "On some apects of unramified cohomology."

15:30-16:30 Jannsen, "Higher dimensional class field theory over local field I."

16:45-17:45 Ayoub, "Relative version of the Kontsevich-Zagier conjecture on periods."

### 16(Fri)/Dec:

9:30-10:30 Hesselholt, "Real algebraic K-theory."

10:45-11:45 Szamuely, "1-motives and arithmetic."

14:00-15:00 Jannsen, "Higher dimensional class field theory over local fields II."

15:30-16:30 Flach, "Weil-etale cohomology and Zeta functions of arithmetic schemes II."

### Abstracts

Ayoub. Relative version of the Kontsevich-Zagier conjecture on periods, We will explain the proof of a relative version of the K-Z conjecture where numbers (and rational numbers) will be replaced by Laurent series (and rational functions) over the field of complex numbers.

Grayson I. Quillen and his work on algebraic K-theory, Daniel Quillen, the founder of the field of algebraic K-theory, died last Spring. We will give an overview of his foundational work in algebraic K-theory with an eye toward the genesis of motivic cohomology.

Grayson II. The motivic spectral sequence via acyclic binary complexes, A recently constructed operation on exact categories amounts to "loop space" on K-theory. Combining it with direct sum K-theory seems to lead to a simpler proof of the spectral sequence connecting motivic cohomology to K-theory. We will report on the work, currently in progress.

Jannsen. Higher dimensional class field theory over local field, The aim is to give an overview on the present status of the theory. The reciprocity map for smooth projective varieties over local fields is known to be an isomorphism after completion if the variety has good reduction (work of S. Saito and myself), but has a non-trivial cokernel in general (described by work of S. Saito and myself in the case of semi-stable reduction).The kernel is non-trivial in general as well (work of S. Sato and R. Sugiyama), but is a direct sum of a finite group and a group which is $\ell$-divisible for all primes $\ell$ different from the residue characteristic of the local field (work of S. Saito and myself for surfaces, and my student P. Forr? in general).

Levine. Connections between motivic and classical homotopy theory, Voevodsky's slice filtration in the motivic stable homotopy category can be thought of as a weighted version of the classical Postnikov tower in stable homotopy theory, with G_m replacing S^1. Following Pelaez's method, one can construct a two-variable Postnikov tower in the motivic stable homotopy category, which takes into account both G_m and S^1 connectivity. Relying on this and some other constructions, we prove a number of connectedness properties for Voevodsky's slice filtration and its Betti realization, and use this to show that, for k an algebraically closed field of characteristic zero, the motivic stable homotopy category contains the classical stable homotopy category as a full subcategory. This gives a ``motivic" structure to classical stable homotopy theory. As an example, one can use the slice filtration to filter the stable homotopy groups of spheres by ``weights".

Mueller-Stach. Introduction to periods and Nori motives, We report on recent joint work with Annette Huber about periods and Nori motives following ideas of Kontsevich and Nori. As preparation we need to introduce the various notions of periods and the abelian category of mixed motives after Nori.

Pirtuka. On some aspects of unramified cohomology, We will explain some interactions of the properties of unramified cohomology groups with other arithmetical questions, such as the study of Chow groups, the integral version of the Tate conjecture and some local-global principles.

Roendigs I. Voevodsky's slice filtration, Several interesting cohomology theories on varieties, such as motivic cohomology, various flavours of algebraic K-theory, and algebraic cobordism, are representable in Voevodsky's motivic stable homotopy category. Voevodsky introduced a certain filtration on this category which leads to filtrations on the cohomology theories listed above. After presenting the construction of the slice filtration, several examples are used to illustrate the applicability of the filtration.

Roendigs II. The slice filtration on hermitian K-theory, This talk describes the slice filtration on hermitian K-theory, and its relation to Milnor's conjecture on quadratic forms.