Details of the Researcher

PHOTO

Wataru Kai
Section
Graduate School of Science
Job title
Assistant Professor
Degree

  • Mathematical Sciences (The University of Tokyo)

Research History 3

  • 2017/04 - Present
    Tohoku University　Mathematical Institute　Assistant Professor

  • 2016/10 - 2017/03
    Universität Duisburg-Essen　Fakultät für Mathematik　Wissenschaftlicher Mitarbeiter

  • 2016/04 - 2016/09
    The University of Tokyo

Education 1

  • The University of Tokyo

    2013/04 - 2016/03

Professional Memberships 1

  • The Mathematical Society of Japan

Research Interests 3

  • additive number theory

  • algebraic cycles

  • algebraic geometry

Research Areas 1

  • Natural sciences / Algebra /

Papers 8

  1. ISOMORPHISMS up to BOUNDED TORSION between RELATIVE K0-GROUPS and CHOW GROUPS with MODULUS

    Ryomei Iwasa, Wataru Kai

    Journal of the Institute of Mathematics of Jussieu　20　(6)　1947-1968　2021/11/18

    Publisher: Cambridge University Press

    DOI： 10.1017/S1474748020000055 　

    ISSN： 1475-3030 1474-7480

  2. A Moving Lemma for Algebraic Cycles With Modulus and Contravariance Peer-reviewed

    Wataru Kai

    International Mathematics Research Notices　2021　(1)　475-522　2020/12/28

    Publisher: Oxford University Press (OUP)

    DOI： 10.1093/imrn/rnz018 　

    ISSN： 1073-7928

    eISSN： 1687-0247

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    <title>Abstract</title> We prove a moving lemma that implies the contravariance of Bloch–Esnault’s additive higher Chow group in smooth affine schemes and that of Binda–Saito’s higher Chow group (taken in the Nisnevich topology).

  3. CHERN CLASSES WITH MODULUS Invited Peer-reviewed

    RYOMEI IWASA, WATARU KAI

    Nagoya Mathematical Journal; Shuji Saito's 60th birthday volume　236　84-133　2019/12

    Publisher: Cambridge University Press (CUP)

    DOI： 10.1017/nmj.2018.52 　

    ISSN： 0027-7630

    eISSN： 2152-6842

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    In this paper, we construct Chern classes from the relative <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763018000521_inline1" /><tex-math>$K$</tex-math></alternatives></inline-formula>-theory of modulus pairs to the relative motivic cohomology defined by Binda–Saito. An application to relative motivic cohomology of henselian dvr is given.

  4. Isomorphisms up to bounded torsion between relative $K_0$-groups and Chow groups with modulus Peer-reviewed

    Ryomei Iwasa, Wataru Kai

    Journal of the Institute of Mathematics of Jussieu　2018/11/12

    DOI： 10.1017/S1474748020000055 　

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    In this note, we establish isomorphisms up to bounded torsion between relative $K_0$-groups and Chow groups with modulus as defined by Binda-Saito.

  5. Suslin’s moving lemma with modulus Peer-reviewed

    Wataru Kai, Hiroyasu Miyazaki

    Annals of K-Theory　3　(1)　55-70　2018/01/01

    Publisher: Mathematical Sciences Publishers

    DOI： 10.2140/akt.2018.3.55 　

    ISSN： 2379-1683

    eISSN： 2379-1691

  6. A higher dimensional generalization of Lichtenbaum duality in terms of the Albanese map Peer-reviewed

    Wataru Kai

    Compositio Mathematica　152　(9)　1915-1934　2015/12/07

    DOI： 10.1112/S0010437X16007600 　

    ISSN： 1570-5846

    eISSN： 1570-5846

  7. Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus Peer-reviewed

    Federico Binda, Jin Cao, Wataru Kai, Rin Sugiyama

    Journal of Algebra　469　437-463　2015/03/07

    DOI： 10.1016/j.jalgebra.2016.07.036 　

    ISSN： 0021-8693

    eISSN： 1090-266X

  8. Notes on a p-adic exponential map for the Picard group Peer-reviewed

    Wataru Kai

    TOKYO J. MATH　2013/09/24

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    Part of these notes was written as the author's 2013 master thesis. For proper flat schemes over a complete discrete valuation ring of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard scheme is available and smooth, it recovers the isomorphism coming from its formal completion. A reinterpretation of an old theorem of Mattuck is given.

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Misc. 6

  1. Notes on Mitsui's Prime Number Theorem with Siegel zeros

    Wataru Kai

    arXiv　2022/09

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    In these notes, we refine Mitsui's Prime Number Theorem, which for a number field $K$ predicts how many prime elements there are in bounded convex sets in $K\otimes \mathbf R$, by incorporating potential Siegel zeros of Hecke L-functions. This allows the norm of the modulus to grow at a pseudopolynomial rate with respect to the size $X$ of the convex set as opposed to powers of $\log X$. The extra flexibility will be essential in our future application to the study of linear patterns of prime elements. We also hope that our updated exposition will make Mitsui's work accessible to a wider mathematical audience.

  2. Unramified logarithmic Hodge-Witt cohomology and $\mathbb{P}^1$-invariance

    Wataru Kai, Shusuke Otabe, Takao Yamazaki

    Forum of Mathematics, Sigma　10　2021/05/16

    Publisher: Cambridge University Press (CUP)

    DOI： 10.1017/fms.2022.6 　

    eISSN： 2050-5094

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    Let $X$ be a smooth proper variety over a field $k$ and suppose that the degree map $\mathrm{CH}_0(X \otimes_k K) \to \mathbb{Z}$ is isomorphic for any field extension $K/k$. We show that $G(\mathrm{Spec} k) \to G(X)$ is an isomorphism for any $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers $G$. This generalize a result of Binda-R\"ulling-Saito that proves the same conclusion for reciprocity sheaves. We also give a direct proof of the fact that the unramified logarithmic Hodge-Witt cohomology is a $\mathbb{P}^1$-invariant Nisnevich sheaf with transfers.

  3. The Green-Tao theorem for affine curves over F_q

    Wataru Kai

    arXiv　2021/01/04

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    Green and Tao famously proved in a 2008 paper that there are arithmetic progressions of prime numbers of arbitrary lengths. Soon after, analogous statements were proved by Tao for the ring of Gaussian integers and by L\^e for the polynomial rings over finite fields. In 2020 this was extented to orders of arbitrary number fields by Kai-Mimura-Munemasa-Seki-Yoshino. We settle the case of the coordinate rings of affine curves over finite fields. The main contribution of this paper is subtle choice of a polynomial subring of the given ring which plays the role of $\mathbb Z$ in the number field case. This is enabled by the Riemann-Roch formula.

  4. Constellations in prime elements of number fields

    Wataru Kai, Masato Mimura, Akihiro Munemasa, Shin-ichiro Seki, Kiyoto Yoshino

    arXiv　2020/12/31

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    Given any number field, we prove that there exist arbitrarily shaped constellations consisting of pairwise non-associate prime elements of the ring of integers. This result extends the celebrated Green-Tao theorem on arithmetic progressions of rational primes and Tao's theorem on constellations of Gaussian primes. Furthermore, we prove a constellation theorem on prime representations of binary quadratic forms with integer coefficients. More precisely, for a non-degenerate primitive binary quadratic form $F$ which is not negative definite, there exist arbitrarily shaped constellations consisting of pairs of integers $(x,y)$ for which $F(x,y)$ is a rational prime. The latter theorem is obtained by extending the framework from the ring of integers to the pair of an order and its invertible fractional ideal.

  5. On the $p$-adic exponential map for the Picard group (Algebraic Number Theory and Related Topics 2013)

    Kai Wataru

    RIMS Kokyuroku Bessatsu　53　87-94　2015/09

    Publisher: Kyoto University

    ISSN： 1881-6193

  6. Chow's moving lemma with modulus Invited Peer-reviewed

    KAI Wataru

    Proceedings of International Colloquium on K-theory, 2016 TIFR

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    To appear.

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Presentations 9

  1. 数体に対するGreen-Taoの定理 Invited

    甲斐亘

    代数的整数論とその周辺2021　2021/12/16

  2. Albanese map and the Neron-Severi group over p-adic fields International-presentation

    Wataru Kai

    International Workshop on motives in Tokyo, 2019　2019/02/13

  3. Chern classes with modulus International-presentation

    International Workshop on Motives in Tokyo 2017　2017/02

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    岩佐亮明との共同研究にもとづく。

  4. Motivic cohomology relative to a divisor, and relative Chern classes International-presentation

    Japan-Taiwan Joint Conference on Number Theory 2016　2016/09

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    岩佐亮明との共同研究にもとづく。

  5. Toward Chern classes with modulus International-presentation

    International Conference in K-theory　2016/08

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    岩佐亮明との共同研究にもとづく。３０分間。

  6. Chow's moving lemma with modulus International-presentation

    International Workshop on Motives in Tokyo 2016　2016/02

  7. A moving lemma for algebraic cycles with modulus and contravariance International-presentation

    International Colloquium on K-theory　2016/01

  8. 代数的サイクルのモジュラス付き移動補題と引き戻し写像

    Regulators in Niseko 2015　2015/09

  9. アフィン空間におけるモジュラス付き移動補題

    代数的整数論とその周辺2015　2015/01

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    45分間。

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Research Projects 3

  1. Additive number theory in number fields

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Early-Career Scientists

    Institution: Tohoku University

    2022/04/01 - 2027/03/31

  2. Chern classes with modulus and higher structures of algebraic cycles

    Offer Organization: Japan Society for the Promotion of Science

    System: Grants-in-Aid for Scientific Research

    Category: Grant-in-Aid for Early-Career Scientists

    Institution: Tohoku University

    2018/04/01 - 2023/03/31

  3. 数体またはｐ進体上定義された多様体の代数的サイクル理論

    甲斐 亘

    Offer Organization: 日本学術振興会

    System: 科学研究費助成事業

    Category: 特別研究員奨励費

    Institution: 東京大学

    2015/04/24 - 2017/03/31

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    当初目標としていたアルバネーゼ写像の分析では見るほどの成果が挙げられなかったが、モジュラス付き代数的サイクルについての研究で成果が出せた。これは代数多様体の閉部分集合の情報である代数的サイクルの無限遠でのふるまいを考慮に入れるもので、名前は類対論に使われる概念「モジュラス」を由来としている。したがって数体やp進体上の多様体の研究にも将来役に立つと期待される。 モジュラス付き代数的サイクルを用いて従来の（モジュラス無しの）方法に倣いコホモロジー理論を作ると、考察している多様体と無限遠部分のK理論の差である相対K理論と並行した性質を持つものになるはずであるというBloch氏とEsnault氏が１０年余り前に提出したドグマがあった。このドグマには具体的な場合の計算により状況証拠はあったが、理論的にしっかりした像は描けていなかった。 ２８年度はこのドグマに裏付けを与えるべく、代数的K群から当該コホモロジー理論への非常に自然な比較写像の構成を岩佐亮明氏との協業により行なった。Bloch・Esnault両氏の提唱したプログラムの進行に弾みをつけるものであり満足している。写像の構成には代数的サイクル特有のテクニックを追究することに加え、従来の一般的枠組みに無限遠の付加情報を組み込む必要があったが、これによってできた無限遠情報の入った新たな枠組みは代数的サイクル以外の（無限遠を考慮する）コホモロジー理論にも適用できるはずであり、その面でも価値がある。