# International Workshop on motives in Tokyo, 2019

### Organizing Committee:

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

Masanori Asakura (Hokkaido)

Tom Bachman (MIT)

Federico Binda (Regenburg)

Thomas Geisser (Rikkyo)

Hiroyasu Miyazaki (Riken)

Ryomei Iwasa (Copenhagen)

Wataru Kai (Tohoku)

Shane Kelly (TIT)

Hakon Kolderup (Oslo)

Stephen Lichtenbaum (Brown)

Niranjan Ramachandran (Maryland)

Kanetomo Sato (Chuo)

Anand Sawant (Muenchen)

Takashi Suzuki (Chuo)

### This workshop is supported by

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

JSPS Grant-in-aid (A) #15H02048 representative Tomohide Terasoma,

JSPS Grant-in-aid (B) 15H03607 representative Kazuhiro Fujiwara,

JSPS Grant-in-aid (C) #18K03258 representative Thomas Geisser,

### Program

### 12(Tue)/Feb:

10:00-11:00 Lichtenbaum, ": Derived exterior powers and the constant in the functional equation."

11:30-12:30 Ramachandran, "Cup products and Heisenberg groups."

14:00-15:00 Binda, "Semi-purity for cycles with modulus."

### 13(Wed)/Feb:

10:00-11:00 Suzuki, "Weil-etale cohomology for local fields and curves with coefficients in Neron models."

11:30-12:30 Geisser, "A Weil-etale version of the Birch and Swinnerton-Dyer conjecture over function fields."

14:00-15:00 Asakura, "New p-adic hypergeometric functions concerning with syntomic regulators."

15:30-16:30 Kai, "Albanese map and the Neron-Severi group over p-adic fields."

### 14(Thu)/Feb:

10:00-11:00 Miyazaki, "Mayer-Vietoris squares for motives with modulus."

11:30-12:30 Bachmann, "p-complete etale motivic stable homotopy theory."

14:00-15:00 Iwasa, "K-theory and Modulus Condition."

15:30-16:30 Kolderup, "Correspondences and motives arising from cohomology theories."

18:00-20:00: Reception

### 15(Fri)/Feb:

10:00-11:00 Sawant, "Strict A^1-homology and applications."

11:30-12:30 Sato, "Etale cohomology and Selmer groups of arithmetic schemes."

14:00-15:00 Kelly, "K-theory of valuation rings."

### Abstract

### Asakura:

We introduce a new p-adic function, which we call the p-adic hypergeometric function of logarithmic type (this is different from Dwork's). The main result is that the special values give the syntomic regulators of K_2 of hypergeometric curves. We also expect that they agree with the special values of p-adic L-functions of elliptic curves in some cases, according to the conjecture of Perrin-Riou. The preprint is available at arXiv.1811.03770.

### Bachmann:

I will explain a proof of the following theorem: for suitable schemes S, the functor SH(S_et)_p^\wedge -> SH_et(S)_p^\wedge, from the p-complete stable homotopy category of the small etale site of S to the p-complete stable etale motivic homotopy category over S, is an equivalence. As a consequence we deduce that weight zero etale motivic homotopy groups coincide with etale cohomology of the sphere spectrum.

### Binda:

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### Geisser:

(joint with T.Suzuki) We give a version of the Birch and Swinnerton-Dyer conjecture for abelian varieties A over function fields in terms of the Weil-etale cohomology with coefficients in the Neron-model and the Lie algebra of A. We use the work of Kato-Trihan to show that the conjecture holds if the Tate-Shafarevich group is finite.

### Iwasa:

Let $X$ be a regular separated noetherian scheme and $D$ an effective Cartier divisor on $X$. In this talk, I present a construction of a homotopy-coniveau type filtration of the relative $K$-theory spectrum $K(X,D)$ by using algebraic cycles with modulus. Assume further that $D$ admits an affine open neighbourhood in $X$. Then I show that the induced filtration on $\pi_0$ deserves to be said gmotivicff. More precisely, the Adams operation $\psi^k$ acts on the $p$-th graded piece of $K_0(X,D)$ by the multiplication by $k^p$ and the graded pieces are isomorphic to Chow groups with modulus up to bounded torsion. This is a joint work in progress with Wataru Kai.

### Kai:

We consider the Albanese map of zero-cycles for smooth projective varieties over p-adic fields. For general reasons, there is a pairing between its cokernel and the Galois-fixed part of the Neron-Severi group modulo rational classes. Under certain good reduction hypotheses, we show that this pairing is perfect. The proof uses an existence theorem of Azumaya algebras (Gabber-de Jong) and a duality theorem (Saito-Sato) which generalizes the Lichtenbaum duality for curves to higher dimensions.

### Kelly:

I will discuss several results showing that the algebraic K-theory of valuation rings behaves as though such rings were regular Noetherian, and some new proofs of known results concerning cdh descent of algebraic K-theory. This is joint work with Matthew Morrow.

### Kolderup:

Since Suslin and Voevodskyfs introduction of finite correspondences, several alternate correspondence categories have been constructed in order to provide different linear approximations to the motivic stable homotopy category. In joint work with Andrei Druzhinin, we provide an axiomatic approach to a class of correspondence categories that are defined by an underlying cohomology theory. For such cohomological correspondence categories, one can prove strict homotopy invariance and cancellation properties, resulting in a well behaved associated derived category of motives.

### Lichtenbaum:

The notion of k-th exterior power of a coherent sheaf is only well-behaved for locally free sheaves. If E is an arbitrary coherent sheaf, the better notion is its derived exterior power, which lives in the derived category. We will give a conjecture which computes the constant term in the functional equation for the zeta-function of a regular scheme X projective over Z in terms of Euler characteristics of derived exterior powers of sheaves of differentials on X. This conjecture is true if the dimension of X is 1 or 2.

### Miyazaki:

To study a smooth variety U, it is often useful to consider a pair (X,D) such that X is proper and U=X-D, where D is an effective Cartier divisor on X. For such a pair (which is called a modulus pair), Kahn-Saito-Yamazaki constructed the motive with modulus M(X,D) as an object in the category of motives with modulus MDM. However, the construction is more or less indirect, and they used auxiliary pairs such that X is not necessarily proper. In this talk we provide a direct construction of MDM without using non-proper varieties. This is a joint work with Bruno Kahn.

### Ramachandran:

(joint work with E. Aldrovandi) Recall the classical correspondence between divisors and line bundles over smooth varieties; a basic question is how to generalize this to cycles of higher codimension. The talk will discuss results on this question with special attention to codimension two cycles. It will also indicate the connection between the Heisenberg group and cup-products.

### Sato:

Selmer groups and Tate-Shafarevich groups of Galois representaions, defined by Bloch-Kato, are expeced to be related with analytic behaviors of L-functions (Tamagawa number conjecture). In this talk, I would like to explain that etale cohomology of a d-dimensional proper regular arithmetic scheme with `Q_p(d)f-coefficients is isomorphic to Selmer groups. I will further state a formula relating the order of Tate-shafarevich groups with p-adic Abel-Jacobi mappings assuming d=2 (i.e., in the case of arithmetic surfaces).

### Sawant:

We will introduce a new version of A^1-homology, which is better behaved and more computable compared to the usual A^1-homology and describe some applications. The talk is based on joint work with Fabien Morel.

### Suzuki:

I will discuss finiteness and duality properties of Weil-etale cohomology of local fields and curves with coefficients in abelian varieties, tori, lattices and their Neron models. The key step is to give a certain group scheme structure to the corresponding etale cohomology, using the so-called rational etale site (or its pro-category version). The resulting group schemes satisfy a certain finiteness and duality. Taking the cohomology of the Weil group of the finite base field, we obtain the desired statement about Weil-etale cohomology ( including its profinite group structure in the local case). This result has applications to values of L-functions. Joint work with Thomas Geisser.