Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Profile and Settings

• Name (Japanese)

  Asakura

• Name (Kana)

  Masanori

• Name

  200901071674253187

Achievement

Research Interests

• ホッジ理論   L functions   p-adic special functions   periods of integrals   regulators   

Research Areas

• Natural sciences / Algebra / Arithmetic Geometry

Research Experience

• 2008/04 - Today Hokkaido university

Awards

• 2021/03 The Mathematical Society of Japan The MSJ Algebra Prize
   Regulators of algebraic K-groups and algebraic cycles

Published Papers

• Frobenius Action on a Hypergeometric Curve and an Algorithm for Computing Values of Dwork’s p-adic Hypergeometric Functions

  Masanori Asakura

  Transcendence in Algebra, Combinatorics, Geometry and Number Theory 373 1 - 45 2194-1009 2021 [Refereed]
  
• Hypergeometric functions and L-functions.

  Masanori Asakura

  RIMS Kôkyûroku Bessatsu B86 3 - 20 2021 [Refereed]
  
• Regulators of K_1 of Hypergeometric Fibrations

  ASAKURA Masanori, OTSUBO Noriyuki

  Arithmetic L-functions and differential geometric methods, Progr. Math. 338 1 - 30 2021 [Refereed][Not invited]
  
• Explicit logarithmic formulas of special values of hypergeometric functions 3F2

  Masanori Asakura, Toshifumi Yabu

  Communications in Contemporary Mathematics 22 (05) 1950040 - 1950040 0219-1997 2020/08 [Refereed]
   

  In [M. Asakura, N. Otsubo and T. Terasoma, An algebro-geometric study of special values of hypergeometric functions [Formula: see text], to appear in Nagoya Math. J.; https://doi.org/10.1017/nmj.2018.36 ], we proved that the value of [Formula: see text] of the generalized hypergeometric function is a [Formula: see text]-linear combination of log of algebraic numbers if rational numbers [Formula: see text] satisfy a certain condition. In this paper, we present a method to obtain an explicit description of it.
• Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance.

  Masanori Asakura, Kanetomo Sato

  J. Math. Sci. Univ. Tokyo 26 (3) 249 - 334 2019 [Refereed]
  
• An Algebro-geometric study of special values of hypergeometric functions 3F2

  ASAKURA Masanori, OTSUBO Noriyuki, TERASOMA Tomohide

  Nagoya Math. J. 236 47 - 62 2019 [Refereed][Not invited]
  
• A functional logarithmic formula for the hypergeometric function 3F2

  ASAKURA Masanori, OTSUBO Noriyuki

  Nagoya Math. J. 236 29 - 46 2019 [Refereed][Not invited]
  
• Regulators of K_2 of hypergeometric fibrations

  ASAKURA Masanori

  Res. Number Theory 4 (2) 2018 [Refereed][Not invited]
  
• CM regulators and hypergeometric functions, II

  ASAKURA Masanori, OTSUBO Noriyuki

  Math. Z. 289 (3-4) 1325 - 1355 2018 [Refereed][Not invited]
  
• CM periods, CM regulators and hypergeometric functions, I.

  ASAKURA Masanori, OTSUBO Noriyuki

  Canad. J. Math. 70 (3) 481 - 514 2018 [Refereed][Not invited]
  
• A simple construction of regulator indecomposable higher Chow cycles in elliptic surfaces

  Masanori Asakura

  Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles, and Arithmetic 427 231 - 240 2016/02/04 [Refereed][Not invited]
  
• Beilinson's Hodge conjecture for smooth varieties

  Rob De Jeu, James D. Lewis, Masanori Asakura

  Journal of K-Theory 11 (2) 243 - 282 1865-2433 2013/04 [Refereed][Not invited]
   

  Let U/C be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, and cl r,m: CHr (U,m) âŠ-â"š → homMHS (â"š(0), H 2r-m (U, â"š(r))) the cycle class map. Beilinson once conjectured cl r,m to be surjective [Be] however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of cl r,m in more detail (as well as at the generic point of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of cl m,m at the generic point is the same for integral or rational coefficients. © 2013 ISOPP.
• Syntomic cohomology and Beilinson's Tate conjecture for K_2

  ASAKURA Masanori, SATO Kanetomo

  J. Algebraic Geom. 22 (3) 481 - 547 2013 [Refereed][Not invited]
  
• Quintic surface over p-adic local fields with infinite p-primary torsion in the Chow group of 0-cycles

  Masanori Asakura

  REGULATORS 571 1 - 17 0271-4132 2012 [Refereed][Not invited]
   

  We construct a quintic surface over p-adic local fields such that there is infinite p-primary torsion in the Chow group of 0-cycles.
• Beilinson's Tate conjecture for K_2 of elliptic surface:survey and examples

  ASAKURA Masanori, SATO Kanetomo

  Cycles, Motives and Shimura Varieties 35 - 58 2010 [Refereed][Not invited]
  
• Local units are generated by certain cyclotomic units

  ASAKURA Masanori

  RIMS Kokyuroku 京都大学 B12 183 - 191 1881-6193 2009 [Refereed][Not invited]
  
• Maximal components of Noether-Lefschetz locus for Beilinson-Hodge cycles

  Masanori Asakura, Shuji Saito

  MATHEMATISCHE ANNALEN 341 (1) 169 - 199 0025-5831 2008/05 [Refereed][Not invited]
   

  We discuss the Noether-Lefschetz locus for Beilinson's Hodge cycles on the complement of two or three hyperplane sections in a smooth projective surface in P-3. The main theorem gives an explicit description of maximal components of the Noether-Lefschetz locus.
• Surjectivity of p-adic regulators on K-2 of Tate curves (vol 165, page 267, 2006)

  Masanori Asakura

  INVENTIONES MATHEMATICAE 172 (1) 213 - 229 0020-9910 2008/04 [Refereed][Not invited]
  
• Surfaces over a p-adic field with infinite torsion in the Chow group of 0-cycles

  Masanori Asakura, Shuji Saito

  ALGEBRA & NUMBER THEORY 1 (2) 163 - 181 1937-0652 2007 [Refereed][Not invited]
   

  We give an example of a projective smooth surface X over a p-adic field K such that for any prime l different from p, the l-primary torsion subgroup of CH(0)(X), the Chow group of 0-cycles on X, is infinite. A key step in the proof is disproving a variant of the Bloch-Kato conjecture which characterizes the image of an l-adic regulator map from a higher Chow group to a continuous etale cohomology of X by using p-adic Hodge theory. With the aid of the theory of mixed Hodge modules, we reduce the problem to showing the exactness of the de Rham complex associated to a variation of Hodge structure, which is proved by the infinitesimal method in Hodge theory. Another key ingredient is the injectivity result on the cycle class map for Chow group of 1-cycles on a proper smooth model of X over the ring of integers in K, due to K. Sato and the second author.
• Beilinson's Hodge Conjecture with Coefficients

  ASAKURA Masanori, SAITO Shuji

  London Math. Soc. Lecture Note Ser. 344 3 - 37 2007 [Refereed][Not invited]
  
• Surjectivity of p-adic regulators on K-2 of Tate curves

  Masanori Asakura

  INVENTIONES MATHEMATICAE 165 (2) 267 - 324 0020-9910 2006/08 [Refereed][Not invited]
  
• Noether-Lefschetz locus for Beilinson-Hodge cycles I

  M Asakura, S Saito

  MATHEMATISCHE ZEITSCHRIFT 252 (2) 251 - 273 0025-5874 2006/02 [Refereed][Not invited]
   

  We discuss the Noether-Lefschetz locus for Beilinson's Hodge cycles. Here ``Beilinson's Hodge cycles'' mean the cohomology class of Bloch's higher Chow cycles on smooth varieties. The main result is to give an estimate of the dimension of the maximal component of the Noether-Lefschetz locus for the universal family of open complete intersections.
• Generalized Jacobian rings for open complete intersections

  M Asakura, S Saito

  MATHEMATISCHE NACHRICHTEN 279 (1-2) 5 - 37 0025-584X 2006 [Refereed][Not invited]
   

  In this paper, we develop the theory of Jacobian rings of open complete intersections, which are pairs (X,Z) where X is a smooth complete intersection in the projective space and Z is a simple normal crossing divisor in X whose irreducible components are smooth hypersurface sections on X. Our Jacobian rings give an algebraic description of the cohomology of the open complement X - Z and it is a natural generalization of the Poincare residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize Macaulay's duality theorem and Donagi's symmetrizer lemma for usual Jacobian rings for hyper-surfaces. A feature that distinguishes our generalized Jacobian rings from usual ones is that there are instances where duality fails to be perfect while the defect can be controlled explicitly by using the defining equations of Z in X. Two applications of the main results are given: One is the infinitesimal Torelli problem for open complete intersections. Another is an explicit bound for Nori's connectivity in case of complete intersections. The results have been applied also to study of algebraic cycles in several other works. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
• A criterion of exactness of the Clemens-Schmid sequences arising from semi-stable families of open curves

  M Asakura

  OSAKA JOURNAL OF MATHEMATICS 40 (4) 977 - 980 0030-6126 2003/12 [Refereed][Not invited]
  
• Arithmetic Hodge structure and nonvanishing of the cycle class of 0-cycles

  M Asakura

  K-THEORY 27 (3) 273 - 280 0920-3036 2002/11 [Refereed][Not invited]
   

  We construct 0-cycles on a self-product of a curve whose cycle classes are linearly independent in the extension group of an arithmetic Hodge structure. This is an extended version of Theorem 4.5 of Asakura's contribution to CMR Lecture Notes Ser. 24 (( 2000), pp. 133-154).
• On the K-1-groups of algebraic curves

  M Asakura

  INVENTIONES MATHEMATICAE 149 (3) 661 - 685 0020-9910 2002/09 [Refereed][Not invited]
  
• Motives and algebraic de Rham cohomology

  Masanori Asakura

  CRM proceedings and Lecture Notes 24 133 - 154 2000 [Refereed][Not invited]
  
• On the kernel of the reciprocity map of normal surfaces over finite fields

  K Matsumi, K Sato, M Asakura

  K-THEORY 18 (3) 203 - 234 0920-3036 1999/11 [Refereed][Not invited]
   

  The reciprocity map of a smooth proper variety over a finite field is known to have a trivial kernel and dense image. In this paper, we investigate the reciprocity map of a normal surface proper over a finite field and give two examples of normal projective surfaces whose reciprocity maps are not injective.

Presentations

• Regulators and special values of L-functions  [Invited]

  Masanori Asakura

  日本数学会 代数学分科会  2021/03
• p-adic Beilinson conjecture for elliptic curves over Q  [Invited]

  Masanori Asakura

  SNU-HU Joint Symposium  2020/11
• A generalization of Ross symbol in higher K groups and p-adic hypergeometric functions  [Invited]

  Masanori Asakura

  p-adic cohomology and arithmetic geometry  2019/11
• An algorithm of computing special values of Dwork’s p-adic hypergeometric functions in polynomial time  [Invited]

  Masanori Asakura

  Hypergeometric Series, Mahler Measures, and Multiple Zeta Values  2019/10
• New p-adic hypergeometric function concerning with syntomic regulators  [Invited]

  Masanori Asakura

  Motives in Tokyo  2019/02
• New p-adic hypergeometric function concerning with syntomic regulators  [Invited]

  Masanori Asakura

  The Asia-Australia Algebra Conference  2019/01
• Hypergeometric functions and L-functions  [Invited]

  Masanori Asakura

  Algebraic Number Theory and related topics  2018/11
• Log formula on generalized hypergeometric function 3F2  [Invited]

  Masanori Asakura

  大阪大学理学研究院 談話会  2018/11
• Log formula on generalized hypergeometric function 3F2  [Invited]

  Masanori Asakura

  東北大学理学研究院 談話会  2018/05
• F-isocrystal and p-adic regulators via hypergeometric functions  [Invited]

  ASAKURA Masanori

  Hakodate workshop in arithmetic geometry 2017  2017/05  函館
• F-isocrystal and p-adic regulators via hypergeometric functions  [Invited]

  ASAKURA Masanori

  代数・解析・幾何学セミナー  2017/02  鹿児島大学
• Regulagtors on hypergeometric fibrations  [Invited]

  ASAKURA Masanori

  Motives and Complex Multiplication  2016/08  アスコナ(スイス)
• Regulators of hypergeometric fibrations  [Invited]

  ASAKURA Masanori

  Arithmetic L-functions and Differential Geometric Methds  2016/05  パリ大学
• The period conjecture of Gross-Deligne for fibrations  [Invited]

  ASAKURA Masanori

  Arithmetic and Algebraic Geometry 2015  2015/01
• Period and regulator for fibrations with CM structure  [Invited]

  ASAKURA Masanori

  Motives in Tokyo  2014/12  東京大学
• Period and regulator for fibrations with CM structure  [Invited]

  ASAKURA Masanori

  Cohomological Realization of Motives  2014/12  バンフ(カナダ)
• Real regulator on K_1 of a fibration of curves  [Invited]

  ASAKURA Masanori

  Recent advances in Hodge theory: period domains, algebraic cycles, and arithmetic  2013/06  ブリティッシュコロンビア大学
• Syntomic regulator on elliptic fibrations  [Invited]

  ASAKURA Masanori

  Algebraic K-theory and Its Applications  2011/03  南京大学
• K_2 のTate 予想とp進レギュレーター  [Invited]

  朝倉 政典

  日本数学会秋季総合会  2009/09  大阪大学

Research Projects

• Arithmetic study of regulators using special functions

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2019/04 -2023/03 

  Author : 朝倉 政典

   

  当該年度においては、p進レギュレーターとp進周期を特殊関数を用いて研究を行い、一定の成果をあげた。より具体的には、超幾何モチーフというコホモロジーに付加構造を与えた対象を独自に導入し、そのp進周期とp進レギュレーターについての一般的な結果を証明した。 (1) 超幾何モチーフ RobertsとRodriguez-Villegasが2021年のプレプリントで超幾何モチーフを導入した。それは、Beukers, Cohen, Mellitによる代数多様体(=BCM多様体)のコホモロジーのある部分商として与えられており、文字通り、周期などといった重要な幾何的不変量が超幾何関数ないしその有限体類似によって表示されるという特徴をもつ。これはおそらく最も一般的な対象をカバーしていると思われるが、しかし一方で、ルジャンドル型楕円曲線やDworkのK3曲面などといった、慣れ親しんだ対象が彼らのいう超幾何モチーフかどうかわからない(または証明が困難)という短所がある。そこで、代表者は、コホモロジー群ではなくコホモロジーの族を対象として、超幾何モチーフを独自に定義た。代表者の定義の特徴は、BCM多様体といった特殊な多様体を用いない内在的な定義となっていることである。これにより、例えばルジャンドル型楕円曲線が超幾何モチーフになっていることが容易にわかる、といった特徴がある。 (2) 超幾何モチーフのp進周期とp進レギュレーターの研究 上記で代表者が導入した超幾何モチーフに関して、２つの研究成果を得た。まず、定義体が有限体のときのフロベニウス固有値をp進超幾何関数によって記述した。応用として、DworkのK3曲面のフロベニウス固有多項式のより精密な公式を得ている。２つ目に、超幾何モチーフの拡大データを考察し、それを記述する一般的な結果を得た。これは昨年度の高次Rossシンボルの研究の一般化とみなせる。
• Arithmetic study of motivic cohomology, periods and regulators

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2015/04 -2018/03 

  Author : Asakura Masanori, OTSUBO Noriyuki

   

  The thema of this research is periods and regulators of algebraic varieties, and I studied them from the viewpoiunt of arithmetic. In particular, the hypergeometric functions play an important role. There are 3 knods of the results which I obtained from 2015--2017. One is about the periods of algebraic varieties, in particular we studied the Gross-Deligne conjecture. This is the joint work with Fresan at the university of Paris. The second is about the Beilinson regulators on K1 of hypergeometric fibrations introduced by Otsubo and myself. This is the joint work with Noroyuki Otsubo at Chiba university. The third is about p-adic regulators for syntomic cohomology groups. This is the joint work with Kazuaki Miyatani at Hiroshima university. All the works are written up in preprints. Some of them are already published, and we are preparing for publishing the rest.
• Motivic cohomology over discrete valuation rings

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011/04 -2016/03 

  Author : Geisser Thomas, Hesselholt Lars, Saito Shuji, Sato Kanetomo, Asakura Masanori

   

  Arithmetic geometry is the study of integral or rational solutions of systems of polynomial equations. For this, it is often useful to study the solutions in other domains, like complex number, real numbers, finite fields, or p-adic fields. An important invariant of such solution sets are motivic cohomology, higher Chow groups, and Suslin homology. During this project, I studied these invariants, and proved several interesting results about them.
• New developments in number theoretic geometry, topology, and algorithm

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011/04 -2016/03 

  Author : Matsumoto Makoto, TAMAGAWA AKIO, Mochizuki Shinichi, Hoshi Yuichiro, Tsuzuki Nobuo, Terasoma Tomohide, Saito Shuji, Tsuji Takeshi, Shiho Atsushi, Morita Shigeyuki, Shimada Ichiro, Kimura Shun-ichi, Kamada Seiichi, Sakuma Makoto, Ishii Akira, Takahashi Nobuyoshi, Hiranouchi Toshiro, Haramoto Hiroshi, Kaneko Masanobu, Taguchi Yuichiro, Furusho Hidekazu, Nishimura Takuji, Hagita Mariko, Yamauchi Takuya, Asakura Masanori, Mizusawa Yasushi

   

  We studied pure mathematics such as number theory, algebra, geometry, in an interdisciplinary manner. In addition, we studied there application in other branch of science and engineering. In pure mathematics side, we constructed a mixed elliptic motif obtained from universal family of elliptic curves. Also, given an l-adic linear representation of arithmetic fundamental group of an algebraic curve, we compared the image of the representation and the image of the Galois group of k-rational point of curves. As for applicational research, we developped a fast numerical integration algorithm based on quasi-Monte Carlo. The method depends on a point set (called Niederreiter-Xing point sets) whose basis is in the theory of rational points of algebraic curves). We introduced a new criteria for uniformity of point set named WAFOM, and our algorithm uses point sets obtained by scrambling Niederreiter-Xing point sets whose WAFOM value is small. Its effectiveness is empirically confirmed.
• Geometric moduli theory and its theoretical applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011/04 -2016/03 

  Author : Nakamura Iku, IWASAKI KATSUNORI, ONO KAORU, TERAO HIROAKI, WENG LIN, ASAKURA MASANORI, ISHII AKIRA, OOMOTO Toru, KATSURA TOSHIYUKI, KATSURADA HIDENORI, SAITO MASAHIKO, ABE NORIYUKI, TANABE KENICHIRO, NAKAMURA KENTARO, HARASHITA SHUSHI, YOSHINAGA MASAHIKO

   

  In this project we aimed at studying global structures of certain geometric spaces so that we may apply them to the related mathematical theories. The main results of our studies are 1) construction of the second compactifications of moduli spaces of abelian varieties, and study of the relation with the other important compactifications, 2) proof of Riemann hypothesis for some of zeta functions of the moduli spaces of semi-stable vector bundles over an algebraic curve, 3) a characterization of one of Painleve differential equations through the study of stable vector bundles of rank two, 4) proof of the isomorphism between the quantum cohomology ring and the Jacobi ring of a potential in mirror symmetry through the study of the moduli space of Lagrangian submanifolds of a toric manifold, 5) generalization and further study of Arrow's impossibility theorem in statistical economics in terms of hyperplane arrangement.
• Research on Mixed motives and regulator on algebraic K-theory

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2012/04 -2015/03 

  Author : ASAKURA Masanori

   

  We studied motives, which are the objects arising from cohomology groups of algebraic varieties, and got some nice results. More precisely, we studied the periods of motives with complex multiplication and the regulator as its extension intensively. In particular, we got nice description in terms of special values of generalizaed hypergeometric functions. We hope that our results brings interesting progress on the Beilinson conjecture on special values of L-functions. This is a joint work with Noriyuki Otsubo at Chiba university. We also got a result on the conjecture of Gross and Deligne on the periods of motives with complex multiplication. This is a joint work with Fresan at ETHZ.
• Reseach on algebraic cycles by cohomology

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011/04 -2015/03 

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

   

  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.
• Mixed motives, algebraic K-theory and regulator

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2009 -2011 

  Author : ASAKURA Masanori

   

  I studied regulator which is a main topic in this research project. In particular, I had a stimulating research on syntomic regulator on K-groups of elliptic surfaces, and Got results, for example, construction of new elements in the Selmer group of Bloch-Kato. I also got a result on torsion subgroup of Chow group of 0-cycles.
• Research on rationally connected varieties

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2007 -2009 

  Author : SATO Eiichi, FURUSHIMA Mikio, YOKOYAMA Kazuhiro, TAKAYAMA Shigeharu

   

  For the study of higher dimensional algebraic varieity X we take a hyperplane section A of X for the use of Lefschetz Theorem and try to find the structure of X by the one of A.This time we studied whether the bundle structure of A is preserved to X and next the structure of blowing-up is also so. Moreover generalizing the method,we investigate the preservation of the extremal ray.As applications we get the following : Theorem. Let us consider a sequence {X_n} of smooth projective varieties so that X_n s an ample divisor in X_{n+1} for each n. Here n runs over each positive integer. Assume X_1 has an elementary contraction f : X_1 -> Y with dim X_1 - dim Y > 1 and dim X_1 > 2.Then for each n there is an inductively extended morphism f_n : X_n -> Y with f_{n-1}=i_{n-1}f_n where i_{n-1} : X_{n-1} -> X_{n} is a natural embedding. For a very general point y of Y a smooth fiber of f_n is a weighted complete intersection for large enough n.The above theorem says that a variety enjoying a sequence {X_n} of smooth projective varieties has the structure of property "symmetry".
• Hodge theory, mixed motives and regulator

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2006 -2008 

  Author : ASAKURA Masanori

   

  P進局所体上の楕円曲面のK_2についてのテイト予想について実質的な成果をあげることができた。 また、その結果、K_1のp進レギュレーターについて、非消滅のための数値的な条件を得ることができた。 これらの結果を応用することで、0サイクルのなすチャウ群のねじれ部分群が有限になるような曲面の例を構成することに成功した。
• An research of algebraic K-theory of arithmetic varieti

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2005 -2008 

  Author : TAKEDA Yuichiro, YUICHIRO Taguchi, EIICHI Sato, MICHIAKI Inaba, ASAKURA Masanori, NAKASHIMA Tohru

   

  本研究の目的は、キューブや代数サイクルといった幾何的な対象を用いて、代数的K理論の元を構成する方法を確立することであった。得られた結果は次のとおりである。(1)楕円曲面上の一次や二次のキューブで、そのBott-Chern形式がKronecker-Eisenstein級数を用いて表されるものを構成した。(2)Goncharovにより定義された代数的サイクル上の積分が、レギュレーター写像に一致することの証明を考案した。(3)Goncharovによる代数的サイクル上の積分をBlochのポリログサイクルに対して計算して、それがポリログ関数を用いて表わされることを示した。
• Research of modular and quasimodular forms arising in various areas of mathematics and their application

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2003 -2006 

  Author : KANEKO Masanobu, KOIKE Masao, NAGATOMO Kiyokazu, TAKATA Toshie, ASAKURA Masanori

   

  Modular and quasimodular solutions of a differential equation that arose in our work with Don Zagier has been investigated. Of particular interest are modular solutions of weight fifth of integers, which are closely connected to the famous Rogers-Ramanujan functions, and quasimodular forms which turned out to be "extremal" in the sense we defined anew. The latter exteremal quasimodular forms were further studied in a joint work with Koike. We have given explicit formulas for them in case of depth one and two and found the differential equations they satisfy. We have made several interesting observations on the Fourier coefficients of extremal quasimodular forms of depth less than five, but could not give a proof. Also, as an application of quasimodular forms, we gave a condition for Fourier coefficients of cusp forms on the modular group being "ordinary" for a prime in terms of certain polynomials. A connection of this and the supersingular polynomials may be of some interest. Our study also concerns so called multiple zeta values. In particular, when we look closely into the double shuffle relations of the double zeta values, we are naturally led to the period polynomials of modular forms on the full modular group. To understand the connection, we have defined and studied the double Eisenstein series and computed their Fourier coefficients. As an application, we have found several formulas for the Fouries coefficients of the Ramanujan tau function, the coefficients of weight 12 cusp form known as the discriminant function or Jacobi's delta function.
• ホッジ理論と代数的サイクルの研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2002 -2004 

  Author : 朝倉 政典

   

  代数的サイクルと混合モチーフについて研究している。 混合モチーフは数論的代数幾何学における壮大な構想であり、理論として確立されたあかつきには、代数幾何学のみならず整数論へも数多くの深い応用をもつことが期待されている重要な分野である。 しかし多くの優れた研究者の努力にも関わらず、混合モチーフはいまだ定義すらない極めて研究の困難な分野でもある。 私は特に複素数体上の混合モチーフの理論を確立することを目的として研究してきた。これまでに、数論的ホッジ構造という概念を導入し、代数曲面上の0-サイクルや、代数曲線のK群についてのブロック予想について研究してきた。 本年度の研究では、代数曲線のK群に関して新しい方向へ踏み出していった。より詳しく説明すると、これまで研究によって代数曲線のK群の研究にはベイリンソン・ホッジ予想が鍵となることが分かっているが、その予想を管状近傍型多様体に対して一般化することを試み、肯定的な結果を得ることができた。但し、予想そのものは未だ解決されておらず今後の研究の進展が待たれる。更にこの研究から派生する問題として、クレメンス・シュミット完全列に関する研究結果を得た。これは既に投稿済みであり掲載が決まっている。
• Many-sided researches on rationally connected projective varieties-Fano varieties are unirational ?

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2002 -2004 

  Author : SATO Eiichi, CHO Koji, INABA Michi-aki, TAKAYAMA Sigeharu, FURUSHIMA Mikio, MORIWAKI Atsushi

   

  The aim of the scientific research is to study rationally connected projective varieties and problems related with them and to develope them. 1.Reseach results. ・The following results were obtained as for families of rational curves : When a projective variety X contains an ample divisor A which has fiber structure whose general fiber is a projective line, the fiber structure of A is extended to X. This yields that X inherits the property of blowing-down from A. Furthermore it has an application to the procedure for constructing conclete minimal model.(Title : Hyperplane section principle of Lefschetz about ${bf P}^1$-fiber space and blowing-down). ・We studied the structure of infinite sequence of rationally connected varieties. These varieties are closely related with unirational, toric and abelian varieties. Particularly we determined the structure of such varieties with Picard number 2 「Title : Tower theorem on smooth projective varieties」。 ・We are now studying the birational group of cubic 3-folds and the unirationality of quartic 3-folds. ・I had a talk on the behavior of rational curves on infinite dimensional projective varieties in 2003 at Kochi Univ. 2.Meeting: We organized a meeting for algebraic geometry "higher dimensional varieties" at Kyushu Univ in 2003 and published the report. 3.Discussions. Takagi(in Rims of Kyoto Univ) had lectures of 3-dimensional Q-primary Fano varieties -and discussion with us and Nakayama did the one on endmorphisms of varieties. Abe and Aoki (Kyoto Univ) had discussions about the constructions of vector bundles and algebraic stacks respectively. 4.An investigator Takayama published "Iitaka fibrations via multiplier ideals", Furushima "non-normal Del Pesso surfaces" "compactification of $C^3$" and Hanamura "relative Chow-Kuneth projectors for modular varieties" and so on respectively.
• ホッジ理論と代数的サイクルの研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2000 -2001 

  Author : 朝倉 政典

   

  昨年度より代数的サイクルと混合モチーフについて研究している。混合モチーフは数論的代数幾何学における壮大な構想であり、理論として確立されたあかつきには、代数幾何学のみならず整数論へも数多くの深い応用をもつことが期待されている重要な分野である。しかし多くの優れた研究者の努力にも関わらず、混合モチーフはいまだ定義すらない極めて研究の困難な分野でもある。私は特に複素数体上の混合モチーフの理論を確立することを目的として研究してきた。これまでに、数論的ホッジ構造という概念を導入し、代数曲面上の0-サイクルや、代数曲線のK群についてのブロック予想について研究してきた。 本年度の研究では、代数曲線のK群に関して更なる研究結果を得ることに成功した。より詳しく説明すると、これまでK群の元を扱うときにその元のサポートに条件がついていたのであるが、その条件を弱めることができた。鍵となるのはベイリンソン予想であるが、これについてネーター・レフシェッツ型の定理を、斎藤秀司氏と共同で証明することができた。これらの研究結果は、論文として執筆中である。また多くの研究集会、セミナー等においても講演した。特に本年度は、フランスのフーリエ研究所における研究集会において講演する機会を得た。