Updated on 2024/02/01

写真a

 
YAMAZAKI Takao
 
Organization
Faculty of Science and Engineering Professor
Other responsible organization
Mathematics Course of Graduate School of Science and Engineering, Master's Program
Mathematics Course of Graduate School of Science and Engineering, Doctoral Program
External link

Degree

  • 博士（数理科学） ( 東京大学 )

  • 修士（理学） ( 東京工業大学 )

Education

  • 1999.3
     

    The University of Tokyo   doctor course   completed

  • 1996.3
     

    Tokyo Institute of Technology   master course   completed

  • 1994.3
     

    Tokyo Institute of Technology   others   withdrawn before completion

Research History

  • 2022.4 - 　

    中央大学理工学部教授

  • 2012.4 - 2022.3

    東北大学大学院理学研究科数学専攻教授

  • 2007.4 - 2012.3

    東北大学大学院理学研究科数学専攻准教授

  • 2006.4 - 2007.3

    東北大学大学院理学研究科数学専攻助教授

  • 2004.4 - 2006.3

    筑波大学大学院数理物質研究科数学専攻講師

  • 2004.1 - 2004.3

    筑波大学数学系講師

  • 1999.6 - 2003.12

    筑波大学数学系助手

  • 1999.4 - 1999.5

    日本学術振興会特別研究員 (PD)

  • 1996.4 - 1999.3

    日本学術振興会特別研究員 (DC1)

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Professional Memberships

  • 日本数学会

Research Interests

  • "number theory, algebraic geometry, arithmetic geometry"

Research Areas

  • Natural Science / Algebra  / Algebra

Papers

  • Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

    Rei Inoue, Takao Yamazaki

    Algebras and Representation Theory   26 ( 6 )   3167 - 3183   2023.5

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    Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    DOI： 10.1007/s10468-023-10205-1

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    Other Link： https://link.springer.com/article/10.1007/s10468-023-10205-1/fulltext.html

  • Motives with modulus, III: The categories of motives Reviewed

    Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, Takao Yamazaki

    Annals of K-Theory   7 ( 1 )   119 - 178   2022.6

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    Publishing type：Research paper (scientific journal)   Publisher：Mathematical Sciences Publishers  

    DOI： 10.2140/akt.2022.7.119

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  • Reciprocity sheaves, II Reviewed

    Bruno Kahn, Shuji Saito, Takao Yamazaki

    Homology, Homotopy and Applications   24 ( 1 )   71 - 91   2022.3

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  • Unramified logarithmic Hodge–Witt cohomology and -invariance Reviewed

    Wataru Kai, Shusuke Otabe, Takao Yamazaki

    Forum of Mathematics, Sigma   10   2022

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    Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

    Abstract

    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 $G(\operatorname {Spec} k) \to G(X)$ is an isomorphism for any $\mathbb {P}^1$-invariant Nisnevich sheaf with transfers G. This generalises a result of Binda, Rülling and 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.

    DOI： 10.1017/fms.2022.6

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  • Tensor structures in the theory of modulus presheaves with transfers Reviewed

    Kay Rülling, Rin Sugiyama, Takao Yamazaki

    Mathematische Zeitschrift   300 ( 1 )   929 - 977   2021.7

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    Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    DOI： 10.1007/s00209-021-02819-2

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    Other Link： https://link.springer.com/article/10.1007/s00209-021-02819-2/fulltext.html

  • Motives with modulus, I: Modulus sheaves with transfers for non-proper modulus pairs Reviewed

    Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, Takao Yamazaki

    Épijournal de Géométrie Algébrique   Volume 5   2021.1

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    Publishing type：Research paper (scientific journal)   Publisher：Centre pour la Communication Scientifique Directe (CCSD)  

    We develop a theory of modulus sheaves with transfers, which generalizes
    Voevodsky's theory of sheaves with transfers. This paper and its sequel are
    foundational for the theory of motives with modulus, which is developed in
    [KMSY20].



    Comment: 64 pages

    DOI： 10.46298/epiga.2021.volume5.5979

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  • Motives with modulus, II: Modulus sheaves with transfers for proper modulus pairs Reviewed

    Bruno Kahn, Hiroyasu Miyazaki, Shuji Saito, Takao Yamazaki

    Épijournal de Géométrie Algébrique   Volume 5   2021.1

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    Publishing type：Research paper (scientific journal)   Publisher：Centre pour la Communication Scientifique Directe (CCSD)  

    We develop a theory of sheaves and cohomology on the category of proper
    modulus pairs. This complements [KMSY21], where a theory of sheaves and
    cohomology on the category of non-proper modulus pairs has been developed.



    Comment: 31 pages

    DOI： 10.46298/epiga.2021.volume5.5980

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  • Maximal abelian extension of $$X_0(p)$$ unramified outside cusps Reviewed

    Takao Yamazaki, Yifan Yang

    manuscripta mathematica   162 ( 3-4 )   441 - 455   2020.7

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    Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

    DOI： 10.1007/s00229-019-01136-7

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    Other Link： http://link.springer.com/article/10.1007/s00229-019-01136-7/fulltext.html

  • MIXED HODGE STRUCTURES WITH MODULUS Reviewed

    Florian Ivorra, Takao Yamazaki

    Journal of the Institute of Mathematics of Jussieu   1 - 35   2020.3

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    Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

    We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension, we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon 1-motives to generalize Kato–Russell’s Albanese varieties with modulus to 1-motives.

    DOI： 10.1017/s1474748020000043

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  • Rational torsion of generalized Jacobians of modular and Drinfeld modular curves. Reviewed

    Fu-Tsun Wei, Takao Yamazaki

    Forum Mathematicum   31   647 - 659   2019

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  • Nori Motives of Curves With Modulus and Laumon 1-motives Reviewed

    Florian Ivorra, Takao Yamazaki

    Canadian Journal of Mathematics   70 ( 4 )   868 - 897   2018.8

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    Publishing type：Research paper (scientific journal)   Publisher：Canadian Mathematical Society  

    <title>Abstract</title>Let <italic>k</italic> be a number field. We describe the category of Laumon 1-isomotives over <italic>k</italic> as the universal category in the sense of M. Nori associated with a quiver representation built out of smooth proper <italic>k</italic>-curves with two disjoint effective divisors and a notion of <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0008414X00055607_inline01" /> for such “curves with modulus”. This result extends and relies on a theorem of J. Ayoub and L. Barbieri-Viale that describes Deligne's category of 1-isomotives in terms of Nori's Abelian category of motives.

    DOI： 10.4153/cjm-2017-037-x

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  • Reciprocity sheaves Reviewed

    Bruno Kahn, Shuji Saito, Takao Yamazaki, Kay Ruelling

    COMPOSITIO MATHEMATICA   152 ( 9 )   1851 - 1898   2016.9

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

    We start developing a notion of reciprocity sheaves, generalizing Voevodsky's homotopy invariant presheaves with transfers which were used in the construction of his triangulated categories of motives. We hope that reciprocity sheaves will eventually lead to the definition of larger triangulated categories of motivic nature, encompassing non-homotopy invariant phenomena.

    DOI： 10.1112/S0010437X16007466

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  • Suslin homology of relative curves with modulus Reviewed

    Kay Ruelling, Takao Yamazaki

    JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES   93   567 - 589   2016.6

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：OXFORD UNIV PRESS  

    We compute the Suslin homology of relative curves with modulus. This result may be regarded as a modulus version of the computation of motives for curves, due to Suslin and Voevodsky.

    DOI： 10.1112/jlms/jdw006

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  • TORSION POINTS ON JACOBIAN VARIETIES VIA ANDERSON'S p-ADIC SOLITON THEORY Reviewed

    Shinichi Kobayashi, Takao Yamazaki

    ASIAN JOURNAL OF MATHEMATICS   20 ( 2 )   323 - 352   2016.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：INT PRESS BOSTON, INC  

    Anderson introduced a p-adic version of soliton theory. He then applied it to the Jacobian variety of a cyclic quotient of a Fermat curve and showed that torsion points of certain prime order lay outside of the theta divisor. In this paper, we evolve his theory further. As an application, we get a stronger result on the intersection of the theta divisor and torsion points on the Jacobian variety for more general curves. New examples are discussed as well. A key new ingredient is a map connecting the p-adic loop group and the formal group.

    DOI： 10.4310/AJM.2016.v20.n2.a5

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  • RATIONAL TORSION ON THE GENERALIZED JACOBIAN OF A MODULAR CURVE WITH CUSPIDAL MODULUS Reviewed

    Takao Yamazaki, Yifan Yang

    DOCUMENTA MATHEMATICA   21   1669 - 1690   2016

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：UNIV BIELEFELD  

    We consider the generalized Jacobian (J) over tilde (0) (N) of a modular curve X-0 (N) with respect to a reduced divisor given by the sum of all cusps on it. When N is a power of a prime &gt;= 5, we exhibit that the group of rational torsion points (J) over tilde (0) (N) (Q)(Tor) tends to be much smaller than the classical Jacobian.

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  • Algebraic integrable systems related to spectral curves with automorphisms Reviewed

    Rei Inoue, Pol Vanhaecke, Takao Yamazaki

    JOURNAL OF GEOMETRY AND PHYSICS   87   198 - 216   2015.1

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

    We apply a reduction to the Beauville systems to obtain a family of new algebraic completely integrable systems, related to curves with a cyclic automorphism. (C) 2014 Elsevier B.V. All rights reserved.

    DOI： 10.1016/j.geomphys.2014.07.008

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  • p-進ソリトン理論入門 Reviewed

    山崎隆雄

    数理解析研究所講究録別冊   53   325 - 342   2015

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  • K-groups of reciprocity functors for G(a) and abelian varieties Reviewed

    Kay Ruelling, Takao Yamazaki

    JOURNAL OF K-THEORY   14 ( 3 )   556 - 569   2014.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

    We prove that the K-group of reciprocity functors, defined by F. Ivorra and the first author, vanishes over a perfect field as soon as one of the reciprocity functors is G(a) and one is an abelian variety.

    DOI： 10.1017/is014009014jkt279

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  • Torsion Points on Hyperelliptic Jacobians via Anderson's p-adic Soliton Theory Reviewed

    Yuken Miyasaka, Takao Yamazaki

    TOKYO JOURNAL OF MATHEMATICS   36 ( 2 )   387 - 403   2013.12

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：TOKYO JOURNAL MATHEMATICS EDITORIAL OFFICE ACAD CENTER  

    We show that torsion points of certain orders are not on a theta divisor in the Jacobian variety of a hyperelliptic curve given by the equation y(2) = x(2g+1) x with g &gt;= 2. The proof employs a method of Anderson who proved an analogous result for a cyclic quotient of a Fermat curve of prime degree.

    DOI： 10.3836/tjm/1391177978

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  • VOEVODSKY'S MOTIVES AND WEIL RECIPROCITY Reviewed

    Bruno Kahn, Takao Yamazaki

    DUKE MATHEMATICAL JOURNAL   162 ( 14 )   2751 - 2796   2013.11

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

    We describe Somekawa's K-group associated to a finite collection of semiabelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky's category of motives. While Somekawa's definition is based on Weil reciprocity, Voevodsky's category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

    DOI： 10.1215/00127094-2381379

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  • The Brauer–Manin pairing, class field theory, and motivic homology Reviewed

    Takao Yamazaki

    Nagoya Mathematical Journal   210   29 - 58   2013.6

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    Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

    <title>Abstract</title>For a smooth proper variety over a <italic>p</italic>-adic field, its Brauer group and abelian fundamental group are related to higher Chow groups by the Brauer–Manin pairing and class field theory. We generalize this relation to smooth (possibly nonproper) varieties, using motivic homology and a variant of Wiesend’s ideal class group. Several examples are discussed.

    DOI： 10.1215/00277630-2077008

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  • ヴォエヴォドスキーのモチーフとヴェイユ相互律 Reviewed

    山崎隆雄

    数理解析研究所講究録別冊   44   165 - 181   2013

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  • Singular Fiber of the Mumford System and Rational Solutions to the KdV Hierarchy Reviewed

    Rei Inoue, Pol Vanhaecke, Takao Yamazaki

    COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS   63 ( 4 )   508 - 532   2010.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：WILEY  

    We study the singular isolevel manifold M-g (0) of the genus g Mumford system associated to the spectral curve y(2) = x(2g+1). We show that M-g (0) is stratified by g + 1 open subvarieties of additive algebraic groups of dimension 0, 1, .... , g, and we give an explicit description of M-g(0) in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus g Mumford system, which is closely related to rational solutions of the KdV hierarchy. (C) 2009 Wiley Periodicals, Inc.

    DOI： 10.1002/cpa.20310

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  • MILNOR K-GROUP ATTACHED TO A TORUS AND BIRCH-TATE CONJECTURE Reviewed

    Takao Yamazaki

    INTERNATIONAL JOURNAL OF MATHEMATICS   20 ( 7 )   841 - 857   2009.7

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：WORLD SCIENTIFIC PUBL CO PTE LTD  

    We formulate ( and prove under a certain assumption) a conjecture relating the order of Somekawa&apos;s Milnor K-group attached to a torus T and the value of the Artin L-function attached to the cocharacter group of T ( regarded as an Artin representation) at s = -1. The case T = G(m) reduces to the classical Birch-Tate conjecture.

    DOI： 10.1142/S0129167X09005571

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  • Class field theory for a product of curves over a local field Reviewed

    Takao Yamazaki

    MATHEMATISCHE ZEITSCHRIFT   261 ( 1 )   109 - 121   2009.1

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

    We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K-1 of a product of curves over a number field.

    DOI： 10.1007/s00209-008-0315-8

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  • A counterexample to generalizations of the Milnor-Bloch-Kato conjecture Reviewed

    Michael Spiess, Takao Yamazaki

    JOURNAL OF K-THEORY   4 ( 1 )   77 - 90   2009

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：CAMBRIDGE UNIV PRESS  

    We construct an example of a torus T over a field K for which the Galois symbol K(K;T,T)/nK(K;TT) -&gt; H(2)(K,T[n] circle times T[n]) is not injective for some n. Here K(K; T, T) is the Milnor K-group attached to T introduced by Somekawa. We show also that the motive M(T x T) gives a counterexample to another generalization of the Milnor-Bloch-Kato conjecture (proposed by Beilinson).

    DOI： 10.1017/is008008014jkt066

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  • Jacobian variety and integrable system - after Mumford, Beauville and Vanhaecke Reviewed

    Rei Inoue, Yukiko Konishi, Takao Yamazaki

    JOURNAL OF GEOMETRY AND PHYSICS   57 ( 3 )   815 - 831   2007.2

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ELSEVIER SCIENCE BV  

    Beauville [A. Beauville, Jacobiennes des courbes spectrales et systemes hamiltoniens completement integrables, Acta. Math. 164 (1990) 211-235] introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system [D. Mumford, Tata Lectures on Theta II, Birkhauser, 1984]. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the intersection of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke [P. Vanhaecke, Linearising two-dimensional integrable systems and the construction of action-angle variables, Math. Z. 211 (1992) 265-313; P. Vanhaecke, Integrable systems in the realm of algebraic geometry, in: Lecture Notes in Mathematics, vol. 1638, 2001]. (c) 2006 Elsevier B.V. All rights reserved.

    DOI： 10.1016/j.geomphys.2006.06.004

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  • Cohomological study on variants of the Mumford system, and integrability of the Noumi-Yamada system Reviewed

    R Inoue, T Yamazaki

    COMMUNICATIONS IN MATHEMATICAL PHYSICS   265 ( 3 )   699 - 719   2006.8

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

    The purpose of this paper is twofold. The first is to apply the method introduced in the works of Nakayashiki and Smirnov [11, 12] on the Mumford system to its variants. The other is to establish a relation between the Mumford system and the isospectral limit Q(g)((I)) and Q((II)) of the Noumi-Yamada system [15]. As a consequence, we prove the algebraically completely integrability of the systems Q(g)((I)) and Q(g)((II)), and get explicit descriptions of their solutions.

    DOI： 10.1007/s00220-006-0028-y

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  • On Chow and Brauer groups of a product of Mumford curves Reviewed

    T Yamazaki

    MATHEMATISCHE ANNALEN   333 ( 3 )   549 - 567   2005.11

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：SPRINGER  

    Let C-1, (...) , C-d be Mumford curves defined over a finite extension of Q(p), and let X = C-1 x (...) x C-d. We shall show the following: (1) The cycle map CH0 (X)/n -&gt; H-2d (X, mu(n)circle times(d) ) is injective for any non-zero integer n. (2) The kernel of the canonical map CH0 (X). Hom (Br(X), Q/Z) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH0 (X).

    DOI： 10.1007/s00208-005-0675-x

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  • Torsion zero-cycles on a product of canonical lifts of elliptic curves Reviewed

    T Yamazaki

    K-THEORY   31 ( 4 )   289 - 306   2004.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

    Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T (X) and T (Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T (X)/T(X)(div) is conjectured to be finite, where T (X) div is the maximal divisible subgroup of T (X). Furthermore, F(X) is conjectured to be isomorphic to T (Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.

    DOI： 10.1023/B:KTHE.0000031391.23103.0a

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  • On Tate duality for Jacobian varieties Reviewed

    T Yamazaki

    JOURNAL OF NUMBER THEORY   99 ( 2 )   298 - 306   2003.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC ELSEVIER SCIENCE  

    Let A be an abelian variety over a p-adic field k and A(t) its dual. The group of k-rational point A(k) has a p-adic decreasing filtration U A(k). When A = J is a Jacobian variety, we give a precise description of the exact annihilator of U-n A(k) with respect to the Tate pairing A(k) x H-l (k, A(t)) --&gt; Q/Z. As an application. we give another proof of the result of McCallum in the special case A = J, which says that U-n,4(k) annihilates ker(H-l(k, A(t)) --&gt; H-l(k(t),A(t))) whenever k(t)/k is a finite extension of conductor less than or equal to n. (C) 2002 Elsevier Science (USA). All rights reserved.

    DOI： 10.1016/S0022-314X(02)00067-7

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  • Tate duality and ramification of division algebras Invited Reviewed

    Takao Yamazaki

    Acta Mathematica et Informatica Universitatis Ostraviensis   10   153 - 160   2002

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  • The exponential homomorphism for the second syntomic cohomology Reviewed

    T Yamazaki

    JOURNAL OF ALGEBRA   240 ( 1 )   103 - 119   2001.6

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：ACADEMIC PRESS INC  

    Let O-K be a complete discrete valuation ring with perfect residue field F of characteristic p greater than or equal to 3 and M a free filtered Dieudonne module. The second relative syntomic cohomology H-2((O-K, F), S(M, 2)) with coefficients in M is an object related to arithmetic geometry, such as the albanese kernel. In this article, we study H-2((O-K, F), S(M, 2)) by constructing the exponential homomorphism. We determine the structure of H-2((O-K, F), S(M, 2)), under the assumption that M is of Hodge-Witt type and that the absolute ramification index of O-K is prime to p. (C) 2001 Academic Press.

    DOI： 10.1006/jabr.2000.8717

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  • Formal chow groups, p-divisible groups, and syntomic cohomology Reviewed

    T Yamazaki

    DUKE MATHEMATICAL JOURNAL   102 ( 2 )   359 - 390   2000.4

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：DUKE UNIV PRESS  

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  • On Swan conductors for Brauer groups of curves over local fields Reviewed

    T Yamazaki

    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY   127 ( 5 )   1269 - 1274   1999.5

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：AMER MATHEMATICAL SOC  

    For an element w of the Brauer group of a curve over a local field, we define the "Swan conductor" sw(w) of w, which measures the wildness of the ramification of w. We give a relation between sw(w) and Swan conductors for Brauer groups of henselian discrete valuation fields defined by Kato.

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  • Reduced norm map of division algebras over complete discrete valuation fields of certain type Reviewed

    T Yamazaki

    COMPOSITIO MATHEMATICA   112 ( 2 )   127 - 145   1998.6

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    Language：English   Publishing type：Research paper (scientific journal)   Publisher：KLUWER ACADEMIC PUBL  

    We study a ramification theory for a division algebra D of the following type: The center of D is a complete discrete valuation field K with the imperfect residue field F of certain type, and the residue algebra of D is commutative and purely inseparable over F. Using Swan conductors of the corresponding element of Br(K), we define Herbrand's psi-function of D/K, and it describes the action of the reduced norm map on the filtration of D*. We also gives a relation among the Swan conductors and the 'ramification number' of D, which is defined by the commutator group of D*.

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Books

  • Motives

    山崎隆雄（ Role： Sole author）

    岩波書店  2022.3  （ ISBN:9784000298292 ）

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    Total pages：334頁   Language：Japanese   Book type：Scholarly book

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  • 初等整数論―数論幾何への誘い

    山崎隆雄（ Role： Sole author）

    共立出版  2015 

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    Language：Japanese   Book type：Scholarly book

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MISC

  • モジュラー曲線のアーベル被覆と非合同部分群 Invited Reviewed

    山崎隆雄

    2021年度(第28回)整数論サマースクール報告集「モジュラー曲線と数論」   485 - 506   2023.3

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  • 未解決問題から眺める数論 Invited

    山崎隆雄

    数理科学   59 ( 6 )   16 - 22   2021.6

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  • 『モーデル・ファルティングスの定理』の書評 Invited

    山崎 隆雄

    数理科学   2018.2

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  • 平方剰余の相互法則の函数体類似

    山崎隆雄

    数学セミナー   55 ( 2 )   28 - 32   2016.2

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

    CiNii Books

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  • 数論幾何の誕生--Weil 予想から混合モチーフの三角圏へ

    山崎隆雄

    数理科学   53 ( 2 )   61 - 67   2015.2

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

    CiNii Books

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  • J.-P. Serre 著 『Lectures on N_X(p)』の書評

    山崎隆雄

    数学   66   330 - 335   2014

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)  

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  • フェルマー予想とabc予想

    山崎隆雄

    数学セミナー   2010.12

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

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  • トーラス係数の Milnor $K$ 群と Birchi-Tate 予想

    山崎隆雄

    第５２回代数学シンポジウム報告集   2007

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (other)  

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  • 有理数と実数

    山崎隆雄

    数学セミナー   45 ( 9 )   12 - 15   2006.9

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (trade magazine, newspaper, online media)  

    CiNii Books

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  • 局所体上の曲面の Chow 群.

    山崎隆雄

    数理解析研究所講究録   1376   117 - 128   2004

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    Language：Japanese   Publishing type：Article, review, commentary, editorial, etc. (scientific journal)   Publisher：Kyoto University  

    CiNii Books

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    Other Link： http://hdl.handle.net/2433/25623

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Presentations

  • Torsion birational motives of surfaces and unramified cohomology

    Seminar on Arithmetic Geometry and Algebraic Groups (online)  2023.11 

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  • Torsion birational motives of surfaces and unramified cohomology.

    Number theory seminar. Tokyo University  2023.11 

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  • Torsion birational motives of surfaces and unramified cohomology

    Niigata algebra seminar (hybrid)  2023.10 

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  • Torsion birational motives of surfaces and unramified cohomology. International conference

    Takao Yamazaki

    Japan-Taiwan Joint Conference on Number Theory 2023  2023.8 

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    Country：Taiwan, Province of China  

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  • Torsion birational motives of surfaces and unramified cohomology

    Takao Yamazaki

    Waseda number theory seminar  2023.6 

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  • Unramified cohomology and P^1-invariance Invited International conference

    Takao Yamazaki

    Motives, quadratic forms and arithmetic  2022.10 

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    Country：France  

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  • Unramified cohomology and P^1-invariance Invited

    Colloquium, Tohoku university  2022.6 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • Abelian covering of modular curves and non-congruence subgroups Invited

    Number theory summer school  2021.8 

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    Language：Japanese   Presentation type：Oral presentation (general)  

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  • Lax tensor products for reciprocity sheaves International conference

    YAMAZAKI Takao

    p-adic cohomology and arithmetic geometry 2019  2019.11 

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    Language：English  

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  • Arithmetic of generalized Jacobians of modular curves, I and II International conference

    YAMAZAKI Takao

    HKU Number Theory Days 2019  2019.7 

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    Language：English  

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  • Eisenstein ideal and Iwasawa theory International conference

    YAMAZAKI Takao

    Eisenstein ideal and Iwasawa theory  2019.6 

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    Language：English  

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  • Arithmetic of generalized Jacobians of modular curves International conference

    YAMAZAKI Takao

    Asia-Australia Algebra Conference  2019.1 

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    Language：English  

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  • Motives and mixed Hodge structures with modulus Invited

    YAMAZAKI Takao

    Mathematical Society of Japan meeting, algebra session, special lecture  2018.9 

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    Language：Japanese  

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  • Generalized Jacobians of modular and Drinfeld modular curves Invited

    YAMAZAKI Takao

    Japan-Taiwan Joint conference on Number theory  2018.9 

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    Language：English  

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  • Mixed Hodge structures with modulus and 1-motives Invited

    YAMAZAKI Takao

    Hodge theory and algebraic geometry  2018.8 

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    Language：Japanese  

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  • Mixed Hodge structures with modulus Invited

    YAMAZAKI Takao

    Pan Asia Number Theory Conference  2018.6 

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  • Mixed Hodge structures with modulus Invited

    YAMAZAKI Takao

    Motives in Tokyo  2018.3 

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    Language：English  

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  • Generalized Jacobians of modular and Drinfeld modular curves Invited

    YAMAZAKI Takao

    Hokuriku number theory conference  2017.12 

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    Language：Japanese  

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  • Nori motives of curves with modulus and Laumon 1-motives

    Regulators in Niseko 2017  ( JAPAN )   2017.9 

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    Presentation type：Oral presentation (invited, special)  

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  • Nori motives of curves with modulus and Laumon 1-motives International conference

    Differential forms in algebraic geometry  ( Germany )   2016.9 

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    Presentation type：Oral presentation (invited, special)  

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  • Nori motives of curves with modulus and Laumon 1-motives International conference

    Japan-Taiwan Joint conference on Number theory 2016  ( Taiwan )   2016.9 

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    Presentation type：Oral presentation (invited, special)  

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  • Motives with modulus International conference

    Algebraic K-theory and Motivic Cohomology  ( Germany )   2016.6 

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    Presentation type：Oral presentation (invited, special)  

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  • Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus International conference

    Workshop on arithmetic geometry in Hakodate  ( JAPAN )   2016.5 

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    Presentation type：Oral presentation (invited, special)  

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  • Motives with modulus International conference

    Generalizations of A^1-Homotopy Invariance in Algebraic Geometry and Homotopy Theory  ( Germany )   2016.4 

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    Presentation type：Oral presentation (invited, special)  

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  • Motives with modulus International conference

    Motives in Tokyo, 2016  ( JAPAN )   2016.2 

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    Presentation type：Oral presentation (invited, special)  

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  • Motives with modulus International conference

    Algebraic geometry seminar, Rennes university  ( France )   2015.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Non-homotopy invariant motive theory

    Workshop on arithmetic geometry at Tambara 2015  ( JAPAN )   2015.6 

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    Presentation type：Oral presentation (invited, special)  

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  • p-adic soliton theory and arithmetic geometry

    Novel visage of arithmetic and derived geometry  ( JAPAN )   2014.10 

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    Presentation type：Oral presentation (invited, special)  

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  • Reciprocity sheaves

    Workshop on arithmetic geometry in Hakodate  ( JAPAN )   2014.5 

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    Presentation type：Oral presentation (invited, special)  

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  • Voevodsky's motif and Weil reciprocity International conference

    NCTS Mini-Workshop on Number Theory  ( Taiwan )   2014.3 

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    Presentation type：Oral presentation (invited, special)  

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  • p-adic soliton theory and torsion points on Jacobian varieties. International conference

    East Asia Number Theory Conference 2014  ( JAPAN )   2014.1 

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    Presentation type：Oral presentation (invited, special)  

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  • p-adic soliton theory and arithmetic geometry International conference

    Arithmetic and Algebraic Geoemtry 2014  ( JAPAN )   2014.1 

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    Presentation type：Oral presentation (invited, special)  

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  • p-adic soliton theory and torsion points on Jacobian varieties.

    代数的整数論とその周辺  ( JAPAN )   2013.12 

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    Presentation type：Oral presentation (general)  

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  • Reciprocity sheaves: towards non-homotopy invariant motive theory. International conference

    Motives in Tokyo, 2013  ( JAPAN )   2013.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Reciprocity sheaves: towards non-homotopy invariant motive theory. International conference

    Arithmetic, differential and geometry  ( JAPAN )   2013.10 

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    Presentation type：Oral presentation (invited, special)  

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  • Reciprocity sheaves International conference

    Workshop on reciprocity sheaves  ( JAPAN )   2013.7 

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    Presentation type：Oral presentation (invited, special)  

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  • p-adic soliton theory and arithmetic geometry

    北海道大学談話会  2013.5 

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    Presentation type：Oral presentation (general)  

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  • Torsion points on Jacobian varieties via Anderson's p-adic soliton theory International conference

    Seminaire de theorie des nombres  ( France )   2012.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Algebraic cycles on a product of curves over a p-adic field International conference

    Seminaire autour des cycles algebriques  ( France )   2012.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Voevodsky's category of motives and Milnor K-groups attached to semi-abelian varieties International conference

    Number Theory Day in Keio  ( JAPAN )   2012.9 

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    Presentation type：Oral presentation (invited, special)  

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  • Torsion points on Jacobian varieties via Anderson's p-adic soliton theory International conference

    Tohoku-Fudan Workshop  ( JAPAN )   2012.5 

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  • Somekawa's K-groups and Voevodsky's Hom groups.

    北海道大学数論幾何セミナー  2012.4 

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    Presentation type：Oral presentation (general)  

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  • Torsion points on the quotient of a Fermat Jacobian via Anderson's p-adic soliton theory, I. International conference

    Mini-workshop on number theory  ( Taiwan )   2012.3 

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    Presentation type：Oral presentation (invited, special)  

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  • On Theta divisor, Jacobian, KdV and KP equations International conference

    TIMS Seminar on Arithmetic Geometry  ( Taiwan )   2012.3 

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    Presentation type：Oral presentation (invited, special)  

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  • Voevodsky's motif and Weil reciprocity International conference

    International workshop on motives in Tokyo  ( JAPAN )   2011.12 

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    Presentation type：Oral presentation (invited, special)  

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  • Voevodsky's motif and Weil reciprocity International conference

    International workshop on motives in Tokyo  ( JAPAN )   2011.12 

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    Presentation type：Oral presentation (invited, special)  

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  • Torsion on theta divisors of hyperelliptic Jacobians and p-adic tau functions International conference

    Géométrie arithmétique et motivique  ( France )   2011.9 

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    Presentation type：Oral presentation (general)  

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  • Class field theory for open varieties over local fields. International conference

    NCTS Seminar in Number Theory  ( Taiwan )   2011.4 

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    Presentation type：Oral presentation (general)  

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  • Class field theory for open varieties over local fields International conference

    International workshop on motives in Tokyo, part 6  ( JAPAN )   2010.12 

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    Presentation type：Oral presentation (general)  

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  • Krichever 対応/p-進タウ関数とその数論幾何への応用

    p-進佐藤理論と数論幾何  2010.9 

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    Presentation type：Oral presentation (general)  

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  • 局所体上の開多様体のブラウアー群と0-サイクル

    日本数学会  ( 名古屋 )   2010.9 

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    Presentation type：Oral presentation (invited, special)  

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  • Brauer-Manin pairing for open varieties over local fields International conference

    Seminar in Number Theory  ( Taiwan Taipei )   2010.7 

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    Presentation type：Oral presentation (general)  

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  • p-adic Sato theory and arithmetic geomety

    Kyushu Algebraic Number Theory 2010  2010.3 

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    Presentation type：Oral presentation (general)  

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  • Zero-cycles on a variety over a (two dimensional) local field

    Workshop on period integral and motif  2010.2 

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    Presentation type：Oral presentation (general)  

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  • Higher dimensional class field theory for a product of curves International conference

    Japan-Korea Number Theory 2010  ( Republic of Korea )   2010.1 

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    Presentation type：Oral presentation (general)  

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  • Application of p-adic tau functions to the Manin-Mumford conjecture (after Anderson)

    P-adic Special Functions and Arithmetic Geometry  2009.10 

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    Presentation type：Oral presentation (general)  

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  • Milnor K-theory attached to semi-abelian varieties and class field theory International conference

    Workshop on arithmetic geometry in Tambara  ( JAPAN )   2009.5 

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    Presentation type：Oral presentation (invited, special)  

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  • Counter examples to variants of the Milnor-Bloch-Kato conjecture International conference

    Industrious Number Theory  ( Republic of Korea )   2009.3 

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    Presentation type：Oral presentation (invited, special)  

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  • 2-adic arithmetic-geometric mean and elliptic curves International conference

    Kensaku Kinjo, Yuken Miyasaka

    Japan-Korea Joint Seminar on Number Theory and Related Topics 2008  2008.11 

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    Presentation type：Oral presentation (general)  

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  • Counter examples to variants of the Milnor-Bloch-Kato conjecture International conference

    P-adic method and its applications in arithmetic geometry at Sendai  2008.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Degenerate fibers of the Mumford system and rational solutions to the KdV hierarchy International conference

    Number Theory and Physics at the Crossroads  ( Canada Banff )   2008.9 

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    Presentation type：Oral presentation (general)  

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  • Generalized Jacobians and rational solutions for the Mumford system International conference

    Journées integrables  ( France Poitiers )   2008.6 

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    Presentation type：Oral presentation (invited, special)  

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  • Degenerate fiber of the Mumford system and rational solutions

    Pol Vanhaecke, Rei Inoue

    日本数学会  2008.3 

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    Presentation type：Oral presentation (general)  

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  • Recent development on the even Mumford system International conference

    Séminaire groupes, algèbres, et géométrie  ( France )   2007.11 

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    Presentation type：Oral presentation (invited, special)  

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  • Milnor K-group attached to a torus and Birch-Tate conjecture

    第５２回代数学シンポジウ  2007.8 

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    Presentation type：Oral presentation (invited, special)  

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  • Class field theory for a product of curves over local fields International conference

    Number Theory and Algebraic Geometry Seminar  ( Belgium )   2007.3 

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    Presentation type：Oral presentation (invited, special)  

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  • Jacobian variety and integrable system --- after Mumford, Beauville and Vanhaecke

    Rei Inoue, Yukiko Konishi

    日本数学会  2006.9 

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    Presentation type：Oral presentation (general)  

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  • Class field theory for a product of curves over local fields International conference

    Mini-workshop on modular forms, Calabi-Yau varieties and string duality  ( JAPAN )   2006.8 

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    Presentation type：Oral presentation (general)  

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  • Arithmetic of a product of curves over local fields

    Takao Yamazaki

    Seminar National Center for Theoretical Sciences (South)  2006.5 

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  • Arithmetic of a product of curves over local fields

    Takao Yamazaki

    Workshop on algebraic geometry. Kagoshima University.  2006.2 

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  • Chow motive of a product of curves and Milnor K-groups

    Takao Yamazaki

    Workshop on Motives. Tokyo University  2005.12 

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  • Mumford curves, p-adic periods and algebraic cycles

    Takao Yamazaki

    Algebra seminar. Tohoku University.  2005.10 

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  • Cohomology of affine Jacobi variety and integrable system (after Nakayashiki-Smirnov)

    Takao Yamazaki

    Mini-workshop on modular forms, Calabi-Yau varieties and string duality. Tsuda College.  2005.6 

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  • Brauer-Manin pairing for a product of Mumford curves

    Takao Yamazaki

    Algebra seminar. Kyushu University  2005.6 

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  • Brauer-Manin pairing for a product of Mumford curve

    Takao Yamazaki

    Mini-workshop with Yves André. Hiroshima University.  2005.2 

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  • On Chow and Brauer groups of surfaces over local fields

    Takao Yamazaki

    Canadian Number Theory Association VIII Meeting  2004.7 

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  • On Chow and Brauer groups of varieties over local fields

    Takao Yamazaki

    Workshop on Calabi-Yau Varieties;Mirror Symmetry;Related Topics;Tsuda College  2004.6 

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  • Tate duality and ramification of division algebras

    Takao Yamazaki

    Seminar on Number Theory and Automorphic forms. Osaka University.  2004.2 

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  • On Chow groups of surfaces over local fields

    Takao Yamazaki

    Algebraic Number Theory and Related Topics  2003.12 

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  • On Chow groups of surfaces over local fields

    Takao Yamazaki

    Symposium on Arithmetic Geometry. Hokkaido University.  2003.11 

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  • On Chow and Brauer groups of surfaces over local fields

    Takao Yamazaki

    Algebra seminar. Hiroshima University.  2003.10 

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  • Formal Chow group and relative syntomic cohomology

    Takao Yamazaki

    Algebra colloquium. Bielefeld University.  2002.12 

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  • On the structure of Chow groups of surfaces over local fields

    Takao Yamazaki

    London number theory seminar. London University, King's college.  2002.10 

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  • Tate Duality and Ramification of Division Algebras

    Takao Yamazaki

    Algemeen colloquium. Utrecht University.  2001.9 

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  • On Tate duality for Jacobian varieties

    Takao Yamazaki

    The 15th Czech and Slovak international conference on number theory  2001.9 

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  • Formal Chow groups, p-divisible groups, and syntomic cohomology

    Takao Yamazaki

    Number theory semiar. Tokyo Metropolitan University.  1999.10 

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  • Formal Chow groups, p-divisible groups, and syntomic cohomology

    Takao Yamazaki

    Conference on Hodge theory and algebraic geometry. Kagoshima University.  1999.10 

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  • Formal Chow groups, p-divisible groups, and syntomic cohomology

    Takao Yamazaki

    Colloquium. Tsukuba University  1999.7 

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  • Formal Chow groups, p-divisible groups, and syntomic cohomology

    Takao Yamazaki

    Algebraic and Complex Geometry Seminar. Osaka University.  1999.4 

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  • Formal completion of Chow groups and syntomic cohomology

    Takao Yamazaki

    Number theory seminar. Tokyo University.  1999.1 

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  • Ramification of Brauer groups of curves over local fields

    Takao Yamazaki

    Number Theorey Seminar. Kanazawa University.  1997.11 

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  • Ramification theory for division algebras of certain type

    Takao Yamazaki

    Number Theory Symposium  1996.9 

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  • Reduced norm map of division algebras over complete valuation fields of certain type

    Takao Yamazaki

    Number theory seminar. Tokyo University.  1996.7 

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Research Projects

  • Modular curves and 1-motives

    Grant number：21K03153  2021.4 - 2025.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Tohoku University

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    Grant amount： \4030000 （ Direct Cost: \3100000 、 Indirect Cost： \930000 ）

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  • Motives and modulus

    Grant number：18K03232  2018.4 - 2022.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Tohoku University

    Yamazaki Takao

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    Grant amount： \4420000 （ Direct Cost: \3400000 、 Indirect Cost： \1020000 ）

    A category of motives with modulus is constructed. This has been considered as a fundamental goal in this research subject for a decade. The results are published or accepted as a series of three papers. The first is devoted to the theory of modulus pairs and sheaves on them, the second to the sheaf theory on proper modulus pairs, and the third to the construction of the triangulated category of motives with modulus. We have also studied related subjects such as reciprocity sheaves and their tensor product, mixed Hodge structure with modulus, P1-invariant sheaves with transfers, and arithmetic of 1-motives of modular curves.

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  • Motive theory and algebraic cycles based on Weil reciprocity

    Grant number：15K04773  2015.4 - 2018.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Tohoku University

    Yamazaki Takao

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    Grant amount： \5850000 （ Direct Cost: \4500000 、 Indirect Cost： \1350000 ）

    A triangulated category of motives with modulus is constructed in such a way that it enables us to handle non-homotopy invariant phenomena. This is a generalization of Voevodsky's triangulated category of motives. Our category has intimate relationship with reciprocity sheaves that was introduced in our previous work. We also studied the mixed Hodge structures with modulus, which is a counter-part of motives with modulus in the Hodge theory. We proved that the Nori category arising from curves with modulus is equivalent to Laumon 1-motives. This is an extension of a result of Ayoub and Barbieri-Viale that considers the case without modulus.

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  • Motive theory based on Weil reciprocity

    Grant number：24654001  2012.4 - 2015.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Challenging Exploratory Research  Tohoku University

    YAMAZAKI Takao

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    Grant amount： \3900000 （ Direct Cost: \3000000 、 Indirect Cost： \900000 ）

    Celebrated theory of mixed motives, due to Voevodsky, is yet to be extended in order to take into account non-homotopy invariant objects. Indeed, there are fundamental objects, such as relative Picard group of an algebraic curve, that are not homotopy invariant. By taking Weil reciprocity in the place of homotopy invariance, we established a theory of `reciprocity presheaves', which is expected to become a foundation for an extended motive theory encompassing non-homotopy invariant objects.

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  • Reseach on algebraic cycles by cohomology

    Grant number：23340003  2011.4 - 2015.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Chuo University

    SATO Kanetomo, ASAKURA Masanori, KIMURA Shun-ichi, SAITO Shuji, YAMAZAKI Takao

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    Grant amount： \18200000 （ Direct Cost: \14000000 、 Indirect Cost： \4200000 ）

    As a tool to study vector bundles on algebraic varieties, we have the notion of Chern class, which measures `how a vector bundle is twisted', in a linear space called cohomology. In this research under report, we start with a naive quiestion concernning `in what kind of cohomology the Chern classes are defined', and formulated a minimal set of axioms for such cohomology. We also proved the Riemann-Roch theorem holds in such cohomology theories.

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  • Higher Chow groups and higher dimensional class field theory

    Grant number：22684001  2010.4 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Young Scientists (A)  Tohoku University

    TAKAO Yamazaki

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    Grant amount： \12610000 （ Direct Cost: \9700000 、 Indirect Cost： \2910000 ）

    Higher Chow groups of varieties are important objects in arithmetic geometry in general. When the base field is of arithmetic nature, higher Chow groups are known to be related with arithmetic objects such as the fundamental group and Brauer group. Over a local field, such relation has been studied for smooth projective varieties. We extended it to non-projective varieties. Besides, in a joint work with Bruno Kahn, we described the tensor product in the triangulated category of motives in terms of Somekawa K-groups, which provides us with a new method to compute the higher Chow groups of a product of varieties.

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  • A study of arithemetic geometry by p-aidc methods

    Grant number：22340001  2010.4 - 2014.3

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B)  Tohoku University

    TSUZUKI NOBUO, KATO Fumiharu, SHIHO Atsushi, YAMAZAKI Takao, NAKAJIMA Yukiyoshi, YAMAUCHI Takuya, KAWAMURA Hisa-aki, ABE Tomoyuki, SUWA Noriyuki

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    Grant amount： \16770000 （ Direct Cost: \12900000 、 Indirect Cost： \3870000 ）

    We investigated the foundation of p-adic methods in arithmetic geometry, rigid analytic technique and cohomology theory arising from differential forms (i.e., p-adic cohomology), and applied them to study arithmetic varieties. We construct p-adic Clemens-Schmid exact sequence for semistable families over algebraic curves of positive characteristic, which is a p-adic analogue of exact sequence describing kernel and cokernel of monodromy operations for semistable families over the complex unit disk. We studied the purity theorem for isocrystals and established the full faithfulness of restriction functors to open subschemes for isocrystals on geometrically unibranch varieties of positive characteristic. As a consequence, we proved pure of weight 1 for first rigid cohomology of proper and geometrically unibranch varieties. We also developed the weight theory in p-adic cohomology and the theory of arithmetic D-modules.

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  • p-adic Sato theory and arithmetic geometry

    Grant number：22654001  2010 - 2011

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Challenging Exploratory Research  Tohoku University

    YAMAZAKI Takao, KOBAYASHI Shinichi, IKEDA Takeshi, YAMAZAKI Rei, KONDO Satoshi

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    Grant amount： \3190000 （ Direct Cost: \2800000 、 Indirect Cost： \390000 ）

    In order to clarify the integrable structure of non-linear PDEs such as KdV and KP equations, Mikio Sato introduced the tau function. Anderson introduced its p-adic analogue and applied it to arithmetic problem : a torsion point of"prime order" on the theta divisor of a Jacobian variety seldom exists. We extended this result to"prime power order". In the course of proving this result, we established many fundamental results on p-adic Sato theory, which are expected to be useful for other future research as well.

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  • Higher Chow groups of varieties over local fields

    Grant number：19740002  2007 - 2009

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Young Scientists (B)  Tohoku University

    YAMAZAKI Takao

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    Grant amount： \3960000 （ Direct Cost: \3300000 、 Indirect Cost： \660000 ）

    Higher Chow groups of varieties is a wide generalization of the ideal class group in the algebraic number theory, and are regarded as a fundamental research theme. When varieties are defined over a local field, it naturally appears in arithmetic context, such as the higher dimensional class field theory and Brauer-Manin pairing. We developed the class field theory for a product of curves, and the Brauer-Manin pairing for open varieties (which was previously considered only for complete varieties).

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  • A study of arithmetic varieties by p-adic methods

    Grant number：17340008  2005 - 2008

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (B) 

    TSUZUKI Nobuo, KATO Fumiharu, SHIHO Atsushi, NAKAJIMA Yoshiyuki, MATSUDA Shigeki, ITO Hiroyuki, KIMURA Shunichi, TAGUCHI Yuichiro

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    Grant amount： \17020000 （ Direct Cost: \14800000 、 Indirect Cost： \2220000 ）

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  • Finite dimensionality of Motives

    Grant number：18540033  2006 - 2007

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research  Grant-in-Aid for Scientific Research (C)  Hiroshima University

    KIMURA Shun-ichi, KIMURA Kenichiro, YAMAZAKI Takao, IKEDA Atsushi

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    Grant amount： \4010000 （ Direct Cost: \3500000 、 Indirect Cost： \510000 ）

    There are 6 major results during this period.
    1. Consider a rigid tensor category, and let V be a Schur finite object. Then there exists a smallest Young diagram which kills V, say λ. namely μ kills V, then μ contains λ. In this way, we can define the Schur dimension of V to be λ.
    2. Generalizing the notion of finite dimensionality of an object in a tensor category, we defined the notion of finite dimensional morphism. An object V is finite dimensional if and only if the identity morphism of V is finite dimensional. In this way, we obtain a new and convenient criterion for finite dimensionality.
    3. When the motive of X is finite dimensional, then its motivic zeta, formal sum of the Chow motives of the symmetric products of X, becomes rational. In this joint work with Professor Javier Elizondo, we studied what happens if we replace the symmetric products by Chow varieties. To our surprise, the formal sum is not rational, even when X is the projective plane.
    4. In a joint work with Prefessor Jacob Murre, we proved the uniqueness of the Picard Functor, assuming that the Chow motive Pic(X) is finite dimensional.
    5. Professor Uwe Jannsen prove that if the cycle map is injective, then it is bijective. We generalized this result to Chow motives.
    6. In this joint work with Professor Nobuyoshi Takahashi and Professor Kenichiro Kimura, we studied, if the motivic zeta of Schur finite object V becomes rational. The answer depends on the Schur dimension of V. If the Schur dimension is a hook, then the motivic zeta is rational. In general, the motivic zeta is always determinantally ration], but not necessarily uniformly rational.

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  • 局所体上の多様体のChow群とBrauer群

    Grant number：16740005  2004 - 2006

    日本学術振興会  科学研究費助成事業  若手研究(B) 

    山崎 隆雄

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    Grant amount： \3300000 （ Direct Cost: \3300000 ）

    今年度は、代数体上の多様体の高次Chow群に関する研究を行った。具体的な成果として、代数体上の曲線の積の高次Chow群の構造に関するKahnの予想を部分的に(可除な群を法として)証明することができた。 Kahnの予想は、曲線の高次Chow群に関するBlochの予想を曲線の積に拡張したものである。 Blochの予想は、 Raskindによって可除な群を法として証明されている。我々の結果とKahnの予想の関係は、Raskindの結果とBlochの予想の関係に並行的である。
    この結果の証明は、曲線の積の高次Chow群に関するRaskind-Spiessの研究を利用した上で、次の二つの(曲線に関する)結果を組み合わせることで行われる。
    第一のものは、昨年度までに得られた局所体上の多様体に関する次の研究結果である:局所体上のgoodまたはsemi-stable reductionを持つ曲線を考える。すると、基礎体の有限次拡大に付随する高次Chow群(の主要部)のノルム写像は全射になる。(なお、この結果は、昨年度行った類体論に関する研究でも中心的な役割を果たした。)
    第二のものは、次の「ハッセのノルム定理の高次Chow群における類似」である:代数体上の曲線を考える。その高次Chow群の元が、基礎体の有限次拡大に付随する高次Chow群のノルム写像の像に入るための必要十分条件は、全ての完備化でノルム写像の像に入ることである。(上記した咋年度の結果から、局所的な条件はほとんど全ての素点で成立することに注意。なお、ハッセのノルム定理は体の乗法群に関する類似の定理である。)
    最後に、井上玲氏・小西由紀子氏との共同研究で、代数曲線のヤコビ多様体を等レベル面に持つような代数的完全可積分系で新しいタイプのものを構成することが出来た。

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  • 局所体上の多様体のChow群の研究

    Grant number：12740006  2000 - 2001

    日本学術振興会  科学研究費助成事業  奨励研究(A)  筑波大学

    山崎 隆雄

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    Grant amount： \2300000 （ Direct Cost: \2300000 ）

    局所体上の代数多様体のChow群の構造は、おおむね三層構造をなすことが知られている。そして、その第一層はNeron-Severi群などの離散的な対象で捉えられる。第二層は、ある場合にはPicard多様体・albanese多様体などの幾何的な対象で捉えられている。第三層は、どのように捉えていいのか現在では全く明らかになっていない。
    私は論文′Formal Chow groups,p-divisible groups,and syntomic cohomology′において、これまで知られていたよりずっと一般的な状況下で、第二層を幾何的な対象で捉える方法論を提出した。ここでは「幾何的な対象」は、アーベル多様体から一般化されてp-divisible groupとして現れてきている。
    さらに、同論文では第三層を二次のsyntomic cohomologyと結びつけるという結果も得ていた。それを踏まえて、私は論文′The exponential homomorphism for the second syntomic cohomology groups′において、二次syntomic cohomologyに関するより深い分析を行い、その構造をある程度あきらかにした。この結果は、Chow群の第三層の構造に対して重要な示唆を与えている。
    さらに、p-divisible groupと一次syntomic cohomologyの間に明示的な結びつきを与える結果を得ている。この結果は局所類体論の明示公式(explicit reciprocity law)に応用できることが期待できる。しかもp-divisible groupに対する完全な明示公式を与えることまで期待できるため、この点に関しては今後も研究を続けるつもりである。

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  • p-進佐藤理論と数論幾何

    2010.4 　 　

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    佐藤理論のp-進化を展開し、数論幾何への応用を狙う。

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  • 滝の原教養講座

    2016.9 - 　

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    滝の原教養講座、於宇都宮高校

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    仙台数学セミナー

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    2012.12 - 　

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    現代数学講演会、於宮城第一高等学校

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  • 公開講座

    2010.10 - 　

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    「フェルマーの最終定理とabc予想」のタイトルで講義をした

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  • 科学者の卵養成講座

    2010.6 - 　

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    「フェルマーの最終定理とabc予想」のタイトルで講義をした

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    「整数と多項式、フェルマー予想とabc予想」のタイトルで講義をした

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    現代数学講演会、於仙台第一高等学校

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