Mar. 2021, The Mathematical Society of Japan, The MSJ Algebra Prize　　　　　　　　　　　　　　　
Regulators of algebraic K-groups and algebraic cycles

Zeta functions of certain K3 families: application of the formula of Clausen　　　　　　　　　　　　　　　
Masanori Asakura
Tohoku Math. J., 77, 1, 33, 53, 2025, [Peer-reviewed]

Milnor K -theory, F -isocrystals and syntomic regulators　　　　　　　　　　　　　　　
Masanori Asakura, Kazuaki Miyatani
J. Inst. Math. Jussieu, 23, 3, 1357, 1415, 2024, [Peer-reviewed]

New p-adic hypergeometric functions and syntomic regulators.　　　　　　　　　　　　　　　
Masanori Asakura
J. Theor. Nombres Bordeaux, 35, 393, 451, 2023, [Peer-reviewed]

A numerical approach toward the p-adic Beilinson conjecture for elliptic curves over Q
Masanori Asakura, Masataka Chida
Research in the Mathematical Sciences, 10, 1, 58, 2023, [Peer-reviewed]
English

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, Springer International Publishing, 2021, [Peer-reviewed]
English, In book

Hypergeometric functions and L-functions.　　　　　　　　　　　　　　　
Masanori Asakura
RIMS Kôkyûroku Bessatsu, B86, 3, 20, 2021, [Peer-reviewed]
Japanese

Regulators of K_1 of Hypergeometric Fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori, OTSUBO Noriyuki
Arithmetic L-functions and differential geometric methods, Progr. Math., 338, 1, 30, 2021, [Peer-reviewed]
English

Explicit logarithmic formulas of special values of hypergeometric functions 3F2
Masanori Asakura, Toshifumi Yabu
Communications in Contemporary Mathematics, 22, 05, 1950040, 1950040, World Scientific Pub Co Pte Lt, Aug. 2020, [Peer-reviewed]
Scientific journal, 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, [Peer-reviewed]
English

An Algebro-geometric study of special values of hypergeometric functions 3F2　　　　　　　　　　　　　　　
ASAKURA Masanori, OTSUBO Noriyuki, TERASOMA Tomohide
Nagoya Math. J., 236, 47, 62, 2019, [Peer-reviewed]
English

A functional logarithmic formula for the hypergeometric function 3F2　　　　　　　　　　　　　　　
ASAKURA Masanori, OTSUBO Noriyuki
Nagoya Math. J., 236, 29, 46, 2019, [Peer-reviewed]

Regulators of K_2 of hypergeometric fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori
Res. Number Theory, 4, 2, 2018, [Peer-reviewed]

CM regulators and hypergeometric functions, II　　　　　　　　　　　　　　　
ASAKURA Masanori, OTSUBO Noriyuki
Math. Z., 289, 3-4, 1325, 1355, 2018, [Peer-reviewed]

CM periods, CM regulators and hypergeometric functions, I.　　　　　　　　　　　　　　　
ASAKURA Masanori, OTSUBO Noriyuki
Canad. J. Math., 70, 3, 481, 514, 2018, [Peer-reviewed]

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, Cambridge University Press, 04 Feb. 2016, [Peer-reviewed]
English, In book

Beilinson's Hodge conjecture for smooth varieties
Rob De Jeu, James D. Lewis, Masanori Asakura
Journal of K-Theory, 11, 2, 243, 282, Apr. 2013, [Peer-reviewed]
English, Scientific journal

Syntomic cohomology and Beilinson's Tate conjecture for K_2
ASAKURA Masanori, SATO Kanetomo
J. Algebraic Geom., 22, 3, 481, 547, 2013, [Peer-reviewed]

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, 2012, [Peer-reviewed]
English, International conference proceedings

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, [Peer-reviewed]

Local units are generated by certain cyclotomic units
ASAKURA Masanori
RIMS Kokyuroku, B12, 183, 191, Kyoto University, 2009, [Peer-reviewed]
English

Maximal components of Noether-Lefschetz locus for Beilinson-Hodge cycles
Masanori Asakura, Shuji Saito
MATHEMATISCHE ANNALEN, 341, 1, 169, 199, May 2008, [Peer-reviewed]
English, Scientific journal

Surjectivity of p-adic regulators on K-2 of Tate curves (vol 165, page 267, 2006)
Masanori Asakura
INVENTIONES MATHEMATICAE, 172, 1, 213, 229, Apr. 2008, [Peer-reviewed]
English

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, 2007, [Peer-reviewed]
English, Scientific journal

Beilinson's Hodge Conjecture with Coefficients　　　　　　　　　　　　　　　
ASAKURA Masanori, SAITO Shuji
London Math. Soc. Lecture Note Ser., 344, 3, 37, 2007, [Peer-reviewed]

Surjectivity of p-adic regulators on K-2 of Tate curves
Masanori Asakura
INVENTIONES MATHEMATICAE, 165, 2, 267, 324, Aug. 2006, [Peer-reviewed]
English, Scientific journal

Noether-Lefschetz locus for Beilinson-Hodge cycles I
M Asakura, S Saito
MATHEMATISCHE ZEITSCHRIFT, 252, 2, 251, 273, Feb. 2006, [Peer-reviewed]
English, Scientific journal

Generalized Jacobian rings for open complete intersections
M Asakura, S Saito
MATHEMATISCHE NACHRICHTEN, 279, 1-2, 5, 37, 2006, [Peer-reviewed]
English, Scientific journal

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, Dec. 2003, [Peer-reviewed]
English, Scientific journal

Arithmetic Hodge structure and nonvanishing of the cycle class of 0-cycles
M Asakura
K-THEORY, 27, 3, 273, 280, Nov. 2002, [Peer-reviewed]
English, Scientific journal

On the K-1-groups of algebraic curves
M Asakura
INVENTIONES MATHEMATICAE, 149, 3, 661, 685, Sep. 2002, [Peer-reviewed]
English, Scientific journal

Motives and algebraic de Rham cohomology　　　　　　　　　　　　　　　
Masanori Asakura
CRM proceedings and Lecture Notes, 24, 133, 154, 2000, [Peer-reviewed]

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, Nov. 1999, [Peer-reviewed]
English, Scientific journal

Regulators and special values of L-functions　　　　　　　　　　　　　　　
Masanori Asakura
日本数学会 代数学分科会, 17 Mar. 2021, Japanese, Keynote oral presentation
16 Mar. 2021 - 19 Mar. 2021, [Invited]

p-adic Beilinson conjecture for elliptic curves over Q　　　　　　　　　　　　　　　
Masanori Asakura
SNU-HU Joint Symposium, 13 Nov. 2020, English, Oral presentation
13 Nov. 2020 - 13 Nov. 2020, [Invited]

A generalization of Ross symbol in higher K groups and p-adic hypergeometric functions　　　　　　　　　　　　　　　
Masanori Asakura
p-adic cohomology and arithmetic geometry, 12 Nov. 2019, English, Invited oral presentation
11 Nov. 2019 - 15 Nov. 2019, [Invited]

An algorithm of computing special values of Dwork’s p-adic hypergeometric functions in polynomial time　　　　　　　　　　　　　　　
Masanori Asakura
Hypergeometric Series, Mahler Measures, and Multiple Zeta Values, 25 Oct. 2019, English, Oral presentation
23 Oct. 2019 - 26 Oct. 2019, [Invited]

New p-adic hypergeometric function concerning with syntomic regulators　　　　　　　　　　　　　　　
Masanori Asakura
Motives in Tokyo, 13 Feb. 2019, English, Oral presentation
12 Feb. 2019 - 15 Feb. 2019, [Invited]

New p-adic hypergeometric function concerning with syntomic regulators　　　　　　　　　　　　　　　
Masanori Asakura
The Asia-Australia Algebra Conference, 22 Jan. 2019, English, Oral presentation
21 Jan. 2019 - 25 Jan. 2019, [Invited]

Hypergeometric functions and L-functions　　　　　　　　　　　　　　　
Masanori Asakura
Algebraic Number Theory and related topics, 30 Nov. 2018, English, Oral presentation
26 Nov. 2018 - 30 Nov. 2018, [Invited]

Log formula on generalized hypergeometric function 3F2　　　　　　　　　　　　　　　
Masanori Asakura
大阪大学理学研究院 談話会, 12 Nov. 2018, Japanese, Invited oral presentation
12 Nov. 2018 - 12 Nov. 2018, [Invited]

Log formula on generalized hypergeometric function 3F2　　　　　　　　　　　　　　　
Masanori Asakura
東北大学理学研究院 談話会, 21 May 2018, Japanese, Invited oral presentation
21 May 2018 - 21 May 2018, [Invited]

F-isocrystal and p-adic regulators via hypergeometric functions　　　　　　　　　　　　　　　
ASAKURA Masanori
Hakodate workshop in arithmetic geometry 2017, 29 May 2017, English, Oral presentation
函館, [Invited], [International presentation]

F-isocrystal and p-adic regulators via hypergeometric functions　　　　　　　　　　　　　　　
ASAKURA Masanori
代数・解析・幾何学セミナー, 16 Feb. 2017, English
鹿児島大学, [Invited]

Regulagtors on hypergeometric fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori
Motives and Complex Multiplication, 17 Aug. 2016, English, Oral presentation
アスコナ(スイス), [Invited], [International presentation]

Regulators of hypergeometric fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori
Arithmetic L-functions and Differential Geometric Methds, 28 May 2016
パリ大学, [Invited]

The period conjecture of Gross-Deligne for fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori
Arithmetic and Algebraic Geometry 2015, 30 Jan. 2015, English
[Invited], [International presentation]

Period and regulator for fibrations with CM structure　　　　　　　　　　　　　　　
ASAKURA Masanori
Motives in Tokyo, 19 Dec. 2014, English
東京大学, [Invited], [International presentation]

Period and regulator for fibrations with CM structure　　　　　　　　　　　　　　　
ASAKURA Masanori
Cohomological Realization of Motives, 08 Dec. 2014, English
バンフ(カナダ), [Invited], [International presentation]

Real regulator on K_1 of a fibration of curves　　　　　　　　　　　　　　　
ASAKURA Masanori
Recent advances in Hodge theory: period domains, algebraic cycles, and arithmetic, 11 Jun. 2013, English
ブリティッシュコロンビア大学, [Invited], [International presentation]

Syntomic regulator on elliptic fibrations　　　　　　　　　　　　　　　
ASAKURA Masanori
Algebraic K-theory and Its Applications, 15 Mar. 2011, English
南京大学, [Invited], [International presentation]

K_2 のTate 予想とp進レギュレーター　　　　　　　　　　　　　　　
朝倉 政典
日本数学会秋季総合会, 21 Sep. 2009, Japanese, Invited oral presentation
大阪大学, [Invited], [Domestic Conference]

代数多様体のレギュレーターとL関数について 複素幾何およびp進幾何の両方からの研究
科学研究費助成事業
01 Apr. 2023 - 31 Mar. 2027
朝倉 政典
２０２３年度においては、次の２つの課題について研究を行った。
（１）超幾何関数のフロベニウス構造の研究。代数多様体の変形族に対し、ドラムコホモロジーに作用するピカール・フックス方程式の研究は、周期積分の研究において基本的かつ有用である。代数多様体の基礎体をp進体にし、コホモロジーをリジッドコホモロジーにすることで、ピカール・フックス方程式にフロベニウス構造が定まる。このフロベニウス構造は代数多様体の数論的不変量を表しており、重要である。本研究では、ピカール・フックス方程式が超幾何関数の満たす微分方程式（超幾何微分方程式）になる場合を詳しく研究した。２０２１年にKiran Kedlayaは、超幾何微分方程式のフロベニウス構造に現れるある定数項がp進ガンマ関数の特殊値になっていることを示した。本研究では、萩原啓氏（慶応大学）と共同で、Kedlayaの定理の一般化を行い、プレプリントとして書き上げた。Kedlayaの定理に現れる定数項はp進ガンマ関数であったが、パラメーターの条件を変えると、p進L関数の特殊値が現れることを証明した。これは、代数多様体の退化族の対数的クリスタリンコホモロジーのフロベニウス構造の研究に応用がある。萩原氏との共同研究の成果は、投稿予定である。
（２）アデリック超幾何関数の研究について。伊原康隆およびAndersonは、フェルマー曲線のタワーのガロア作用を考えることで、アデリック・ベータ関数を定義した。伊原・アンダーソン理論と呼ばれる。本研究では、大坪紀之氏（千葉大学）と共同で、フェルマー曲線ではなく、超幾何曲線のタワーを考えることで、伊原・アンダーソン理論の一般化を行った。超幾何曲線とは、ガウス超幾何が周期に現れる代数曲線で、退化ファイバーにフェルマー曲線がある。これについては、現在も研究が進行中である。
日本学術振興会, 基盤研究(C), 北海道大学, 23K03025

Arithmetic study of regulators using special functions
Grants-in-Aid for Scientific Research
01 Apr. 2019 - 31 Mar. 2023
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。
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 19K03391

Arithmetic study of motivic cohomology, periods and regulators
Grants-in-Aid for Scientific Research
01 Apr. 2015 - 31 Mar. 2018
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 15K04769

Motivic cohomology over discrete valuation rings
Grants-in-Aid for Scientific Research
01 Apr. 2011 - 31 Mar. 2016
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), 23340004

New developments in number theoretic geometry, topology, and algorithm
Grants-in-Aid for Scientific Research
01 Apr. 2011 - 31 Mar. 2016
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (A), 23244002

Geometric moduli theory and its theoretical applications
Grants-in-Aid for Scientific Research
01 Apr. 2011 - 31 Mar. 2016
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (S), Hokkaido University, 23224001

Research on Mixed motives and regulator on algebraic K-theory
Grants-in-Aid for Scientific Research
01 Apr. 2012 - 31 Mar. 2015
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Hokkaido University, 24540001

Reseach on algebraic cycles by cohomology
Grants-in-Aid for Scientific Research
01 Apr. 2011 - 31 Mar. 2015
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Chuo University, 23340003

Mixed motives, algebraic K-theory and regulator
Grants-in-Aid for Scientific Research
2009 - 2011
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), Hokkaido University, 21740001

Research on rationally connected varieties
Grants-in-Aid for Scientific Research
2007 - 2009
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".
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), Kyushu University, 19540037

Hodge theory, mixed motives and regulator
Grants-in-Aid for Scientific Research
2006 - 2008
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。
Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B), 18740012

An research of algebraic K-theory of arithmetic varieti
Grants-in-Aid for Scientific Research
2005 - 2008
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。
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Kyushu University, 17340009

Research of modular and quasimodular forms arising in various areas of mathematics and their application
Grants-in-Aid for Scientific Research
2003 - 2006
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), KYUSHU UNIVERCITY, 15340014

ホッジ理論と代数的サイクルの研究
科学研究費助成事業
2002 - 2004
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。
日本学術振興会, 若手研究(B), 九州大学, 14740020

Many-sided researches on rationally connected projective varieties-Fano varieties are unirational ?
Grants-in-Aid for Scientific Research
2002 - 2004
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.
Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B), Kyushu University, 14340014

ホッジ理論と代数的サイクルの研究
科学研究費助成事業
2000 - 2001
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。
日本学術振興会, 奨励研究(A), 九州大学, 12740019