# International Workshop on motives in Tokyo, 2012

### Organizing Committee:

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

Masanori Asakura (Hokkaido)

Francis Brown (CNRS-IHES)

Henri Gillet (UI Chicago)

Lars Hesselholt (Nagoya)

Shane Kelly (Universitaet Duisburg-Essen)

Moritz Kerz (Regensburg)

Alexander Merkurjev (UCLA)

Kay Ruelling (FU-Berlin)

Niranjan Ramachandran (Maryland)

Kanetomo Sato (Chuou Univ. Tokyo)

Rin Sugiyama (Nagoya)

Goncalo Tabuada (MIT)

Takehiko Yasuda (Osaka Univ.)

Jo"rg Wildeshaus (Universite' Paris 13)

Olivier Wittenberg (Ecole normale superieure, Paris)

Changlong Zhong (Ottawa)

### 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",

Graduate School of Mathemtaical Sciences, the University of Tokyo, Leading Graduate Course for Frontiers of Mathematical Sciences and Physics

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,

### Program

### 10(Mon)/Dec:

9:30-10:30 Merkurjev, "Generalized Rost motives I."

10:45-11:45 Zhong, "On the torsion of Chow group of complete spin flags."

14:00-15:00 Ramachandran, "The Lichtenbaum-Milne conjecture on special values I."

15:30-16:30 Wildeshaus, "Compactifications and purity."

16:45-17:45 Ruelling, "K-groups of reciprocity functors I."

### 11(Tue)/Dec:

9:30-10:30 Ramachandran, "The Lichtenbaum-Milne conjecture on special values II."

10:45-11:45 Merkurjev, "Generalized Rost motives II."

14:00-15:00 Tabuada, "Noncommutative motives."

15:30-16:30 Ruelling, "K-groups of reciprocity functors II."

16:45-17:45 Yasuda, "p-cyclic quotient singularities and motivic integration."

18:30-20:30: Reception

### 12(Wed)/Dec:

9:30-10:30 Sato, "Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance."

10:45-11:45 Gillet, "Rational Points on Varieties I."

Free afternoon

### 13(Thu)/Dec:

9:30-10:30 Gillet, "Rational Points on Varieties II."

10:45-11:45 Kerz, "Skeleton sheaves over finite fields."

14:00-15:00 Kelly, "ldh cohomological descent for motives."

15:30-16:30 Sugiyama, "Remarks on Lefschetz classes on simple CM abelian varieties."

16:45-17:45 Wittenberg, "Degrees of zero-cycles and Euler characteristics over Henselian fields I."

### 14(Fri)/Dec:

9:30-10:30 Asakura, "Real regulator on K_1 of elliptic surfaces."

10:45-11:45 Hesselholt, "Algebraic K-theory, the failure of excision, and what to do about it."

14:00-15:00 Wittenberg, "Degrees of zero-cycles and Euler characteristics over Henselian fields II."

15:30-16:30 Brown, "The motivic Lie algebra of the category of mixed Tate motives over Z."

### Abstracts

### Asakura:

In this talk I will explain a certain method for computation of real regulator of K_1 of elliptic surface by using the Picard-Fuchs operator.

### Brown:

The category of mixed Tate motives over Z is a Tannakian category whose Lie algebra is freely generated by one element in every odd degree >1. An important open problem in Grothendieck-Teichmuller theory is to construct these generators explicitly. A very closely related problem, due to Drinfeld, is to give an explicit construction of all rational associators. This would have applications in knot theory, deformation quantization, the Kashiwara-Vergne problem, and the theory of multiple zeta values. In this talk I will describe some elementary recipes for solving both these problems, modulo some standard conjectures. If time permits, I will explain a connection with modular forms which answers a question posed by Y. Ihara and M. Matsumoto in their study of the stable derivation algebra.

### Gillet:

In the first lecture I will give an over view of some of the questions and results in the area, including recent work using motivic techniques. A central example is: Conjecture (Bombieri-Lang): Let V be a variety of general type defined over a number field K then the set V(K) of rational points in V is not Zariski dense in V. In the second lecture I will discuss criteria for descent, and some recent work on a special case of the analog of the Bombieri-Lang conjecture for varieties over function fields of characteristic p.

### Kelly:

We discuss how a theorem of Gabber on alterations can be used to replace the hypothesis of resolution of singularities in Voevodsky's work on motives, if we are willing to work with Z[1/p] coefficients (where p is the characteristic of the perfect base field).

### Kerz:

Recently Deligne and Drinfeld studied compatible systems of lisse l-adic sheaves on the subcurves of a variety over a finite field. It is expected that any such system should be induced from a lisse l-adic sheaf on the variety if a certain ramification condition is verified. In this talk I present joint work with Shuji Saito on the rank one case of this problem.

### Merkurjev:

We will discuss properties of varieties possessing a generalized Rost motive and applications.

### Ramachandran:

This talk will be about the conjecture of Lichtenbaum-Milne on special values of zeta functions of varieties over finite fields. We will begin with a gentle introduction and then move to the recent results about the p-part.

### Ruelling:

Given semi abelian varieties over a field G_1,..., G_r, Somekawa introduced the K-group K(G_1,..., G_r), which is a functor on field extensions of k, having traces for finite field extensions. Taking G_i to be G_m one recovers Milnor K-theory, taking G_i to be the Jacobians of smooth projective curves C_i one can describe the kernel of the Albanese map for the product of the C_i's via K(G_1,..., G_r), by work of Spiess and Raskind. Recently Kahn and Yamazaki generalized this definition to the case where the G_i's are allowed to be homotopy invariant Nisnevich sheaves with transfer (HINST) and assuming the ground field to be perfect relate this K-group to the tensor product of the G_i's in the category of HINST. In my talk I want to introduce reciprocity functors following Serre's treatment of commutative algebraic groups. These functors are defined on finitely generated field extensions over a given ground field, they have traces and they have symbols indexed by closed points in regular projective curves over some field, which satisfy a reciprocity law. Examples of reciprocity functors are smooth commutative algebraic groups, homotopy invariant Nisnevich sheaves with transfers, Rost cycle modules and Ka"hler differentials. Then I introduce a Somekawa-type K-group of a finite family of reciprocity functors, which is itself a reciprocity functor and satifies a certain universal property. We compute these K-groups in various cases. In particular we can describe Milnor K-theory, the tensor product of HINST and - in characteristic zero- the Ka"hler differentials in terms of these K-groups. If time permits we discuss some open questions and problems at the end.

### Sato:

I will formulate axioms of cohomology theory on a big site which include neither homotopy invariance nor purity. I will also talk about a Riemann-Roch theorem without denominators for Chern class maps to such cohomology theory.

### Sugiyama:

In this talk, I will talk about a necessary and sufficient condition for all Hodge (resp. Tate) classes to be are generated by Lefschetz classes on a simple CM abelian variety over a number field (resp. finite field). Here Lefschetz classes are Hodge (resp. Tate) classes of degree one. For certain simple CM abelian variety $A$ over a number field, I explain the relationship between the condition for $A$ and for a reduction of $A$.

### Tabuada:

I will describe the recent developments in the theory of noncommutative motives.

### Wittenberg:

The index of a variety is the smallest positive degree of a zero-cycle lying on it. I will explain a relation between the index of a variety defined over a strictly Henselian local field, Euler characteristics of coherent sheaves, and the cobordism class of the variety. As an application, rationally connected varieties over the maximal unramified extension of a p-adic field have p-power index. A conjecture of Kato and Kuzumaki about Fano hypersurfaces over p-adic fields will also be discussed. (Joint work with H. Esnault and M. Levine.)

### Yasuda:

This talk is concerned with quotient singularities associated with modular representations of a cyclic group of prime order. I will present a way to compute stringy invariants of such singularities by using the motivic integration. The computed invariants provide information on resolution of singularities if any.

### Zhong:

In this talk I will talk about an upper bound of the annihilator of the torsion part of the Chow group CH^d(X) where $X$ is the variety of complete flags of a linear algebraic group $G$ of type B_n or D_n. Such upper bound does not depend on the rank of the group $G$ but only on the codimension d. It is obtained by studying the characteristic maps of Chow groups and K_0 of flag varieties. This is joint work with S. Baek and K. Zainoulline.