Researcher Database
Last Updated :2024/07/25
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Profile and Settings
Profile and Settings
Name (Japanese)
KeijiName (Kana)
MatsumotoName
201501044792707311
Alternate Names
Achievement
Research Interests
- 超幾何関数 テータ関数 ねじれコホモロジー群 周期写像 保型形式 配置空間 モジュライ空間 算術幾何平均 超幾何微分方程式 交点数 ねじれホモロジー群 周期行列 平均反復 ねじれコホモロジー 合流 漸近展開 双曲幾何 局所系係数(コ)ホモロジー群 双曲空間 モノドロミー群 双曲幾何学 ねじれホモロジー 周期関係式 パッフ形式 基本群 交点形式 データ関数 twisted (co) homology 超幾何 周期積分
Research Areas
- Natural sciences / Algebra
- Natural sciences / Basic analysis
- Natural sciences / Geometry
- Natural sciences / Mathematical analysis
Research Experience
- 2008/11 - Today Hokkaido University Faculty of Science Department of Mathematics
- 2007/04 - 2008/10 Hokkaido University Faculty of Science Department of Mathematics
- 2006/04 - 2007/03 Hokkaido University Faculty of Science Department of Mathematics
- 1999/08 - 2006/03 Hokkaido University Graduate School of Science Department of Mathematics
Published Papers
- Keiji MatsumotoKYUSHU JOURNAL OF MATHEMATICS 71 (2) 329 - 348 1340-6116 2017/09We give the monodromy representations of local systems of twisted homology groups associated with Lauricella's system F-D(a, b, c) of hypergeometric differential equations under mild conditions on parameters. Our representation is effective even in some cases when the system F-D (a, b, c) is reducible. We show that invariant subspaces under our monodromy representations are given by the kernel or image of a natural map from a finite twisted homology group to locally finite one.
- Hideyuki Majima, Keiji Matsumoto, Nobuki TakayamaTohoku Mathematical Journal 52 (4) 489 - 513 0040-8735 2000 [Refereed][Not invited]
We present a theory of intersection on the complex projective line for homology and cohomology groups defined by connections which are regular or not. We apply this theory to confluent hypergeometric functions, and obtain, as an analogue of period relations, quadratic relations satisfied by confluent hypergeometric functions. © 2000 Applied Probability Trust. - KACHI Naoki, MATSUMOTO Keiji, MIHARA MasateruKyushu J. Math. Graduate School of Mathematics, Kyushu University 53 (1) 163 - 188 1340-6116 1999
- Keiji Matsumoto, Takeshi SasakiKyushu Journal of Mathematics 50 (1) 93 - 131 1340-6116 1996 [Refereed][Not invited]
- Matsumoto Keiji, Sasaki Takeshi, Yoshida MasaakiMemoirs of the Faculty of Science, Kyusyu University. Series A, Mathematics 九州大学 47 (2) 283 - 381 0373-6385 1993/09 [Refereed][Not invited]
- Matsumoto Keiji, M YOSHIDACOMPOSITIO MATHEMATICA 86 (3) 265 - 280 0010-437X 1993/05 [Refereed][Not invited]
- Cho Koji, Matsumoto Keiji, Yoshida MasaakiMemoirs of the Faculty of Science, Kyusyu University. Series A, Mathematics 九州大学 47 (1) 119 - 146 0373-6385 1993/03 [Refereed][Not invited]
Let X = X (3, 6) be the configuration space of six lines in general position in the projective plane; the space can be thought of the quotient space
X = GL (3) ? M*(3, 6; C)/(C*)6,
where M*(3, 6; C) is the set of 3 × 6 complex matrices such that any 3 by 3 minor does not vanish. In the previous paper [MSY], we constructed a projective compactification X of X and showed that it is the Satake compactification of the quotient of the 4-dimensional Hermitian symmetric domain H (2, 2) = {z∈M (2, 2; C)|(z − z*)/2i > 0} by an arithmetic group, say Λ acting on H (2, 2).
In this paper, we make a detailed study of combinatorial properties of the smooth affine variety X, the singular projective variety X and the parabolic parts of the arithmetic group Λ, and make, as an application of these, a toroidal compactification of X, which gives a non-singular model of X. We also treat several other arithmetic subgroups commensurable with Λ. As an appendix, we study the variety X as a toric variety. - Monodromy of the hypergeometric differential equation of type (3,6). II. The unitary reflection group of order 29⋅37⋅5⋅7.Matsumoto, Keiji, Sasaki, Takeshi, Takayama, Nobuki, Yoshida, MasaakiAnn. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (4) 617 - 631 1993 [Refereed][Not invited]
- Keiji Matsumoto, Takeshi Sasaki, Nobuki Takayama, Masaaki YoshidaDuke Mathematical Journal 71 (2) 403 - 426 0012-7094 1993 [Refereed][Not invited]
- Matsumoto, Keiji, Sasaki, Takeshi, Yoshida, MasaakiInternational J. of Math. 3 (1) 1 - 164 1992 [Refereed][Not invited]
- Matsumoto Keiji, M YOSHIDAPROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 67 (4) 125 - 127 0386-2194 1991/04 [Refereed][Not invited]
MISC
- Matsumoto Keiji RIMS Kokyuroku 1518- 76 -86 2006/10 [Not refereed][Not invited]
- 松本 圭司, 寺杣 友秀 数理解析研究所講究録 1296- 80 -91 2002/12 [Not refereed][Not invited]
- 松本 圭司 数理解析研究所講究録 1212- 50 -64 2001/06 [Not refereed][Not invited]
- 松本 圭司 数理解析研究所講究録 909- 208 -211 1995/05
- 松本 圭司 数理解析研究所講究録 756- 85 -90 1991/06
- 松本 圭司 数理解析研究所講究録 729- 50 -61 1990/10
- MATSUMOTO Keiji, SASAKI Takeshi, YOSHIDA Masaaki RIMS Kokyuroku 681- 103 -109 1989/02
Books etc
- Koornwinder, T. H., Stokman, Jasper V. (ContributorChapter 3)
Cambridge University Press 2020 (ISBN: 9781107003736) xii, 427 p.
Research Projects
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 2019/04 -2023/03Author : 松本 圭司パラメーターが (1/2,1/2,1) の超幾何微分方程式に関する Schwarz 写像の逆写像は、SL(2,Z) のレベル２の主合同部分群Γ(2) の作用に関して不変な複素上半平面上の関数となり、それはλ関数と呼ばれている。その関数は、 ２つのテータ定数の４乗の比で表示される。パラメーターが (1/2,1/2,1) の超幾何級数の変数にλ関数を代入した関数は、あるテータ定数の２乗と一致するという Jacobi の公式が古典的に知られている。 パラメーターが (1/4,3/4,1) の超幾何微分方程式に関する Schwarz 写像とその逆写像を考察することにより、Jacobi の公式の類似公式を得ることができた。その公式において、上半空間の変数を２倍にした場合に、関数たちの変化を追跡することで、超幾何関数がみたす変数変換公式を与えた。そしてこの公式から２つの拡張された平均を定義して、これらの平均を繰り返し行うことで得られる極限の表示公式を与えた。それらの結果は以下のページで公開されている。 http://arxiv.org/abs/2202.11856 パラメーターが (1/12,5/12,1) と (1/6,1/2,1) の超幾何微分方程式に関する Schwarz 写像とその逆写像を考察することで、SL(2,Z) と SL(2,Z)とΓ(2)のある中間群の作用に関して不変な複素上半平面上の関数を構成し、Jacobi の公式の類似公式を得ることができた。これらの公式を組み合わせることで、超幾何関数がみたす関数等式を与えた。それらの結果は以下のページで公開されている。 http://arxiv.org/abs/2203.07617
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2015/04 -2020/03Author : Terasoma TomohideWe study special functions and period integrals arising from special varieties, such as hypergeometric functions, multiple polylogarithm functions, multiple zeta values, etc. from a geometric point of view. We give explicit presentation of geometric objects such as inverse period functions, and unexpected relation between them. We try to find geometric origin lying behind observed phenomena. Our strategy is to apply modern strong algebra geometric technic, namely powerful tool of algebraic cycles and motives. We also try to explain phenomena of relation between depth filtrations and moduli space of elliptic curves. Up to now, naive way of constructing Hodge realization of mixed Tate motives is still unclear. We also try to clarify conjectured construction by Bloch-Kriz. Moreover recently, we found a method to construct new algebraic cycles on abelian varieties, which seems to be useful to prove the algebraicity of Weil Hodge cycles.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2016/04 -2019/03Author : Matsumoto Keiji, TERASOMA Tomohide, KANEKO Jyoichi, OHARA KatsuyoshiWe introduce a hypergeometric system of differential equations in two variables with rank 9. We give integral representations of solutions forming a basis of a local solution space to this system, and study its monodromy representation, which describes global behavior of solutions to this system. We study a period map for a 2-dimensional family of K3-surfaces by using Abel-Jacobi map for a family of algebraic curves of genus 2. We express its inverse in terms of theta functions. We give a new approach to study Lauricella's hypergeometric system F_D by introducing relative twisted (co)homology groups and intersection forms on them.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2013/04 -2017/03Author : Takayama Nobuki, HARAOKA Yoshishige, MATSUMOTO KeijiWe give an algorithm to evaluate numerically big A-hypergeometric polynomials. An efficient implementation to evaluate numerically the matrix hypergeometric function 1F1 is given. We study divergent series solutions of A-hypergeometric equations and those of Heun type ordinary differential equations in terms of the Borel summability. These theoretical results help to analyze these functions globally and we hope that they give a new method for a numerical analysis of these functions.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2013/04 -2016/03Author : Matsumoto Keiji, OHARA Katsuyoshi, TERASOMA Tomohide, YOSHIDA MasaakiPeriod integrals of algebraic varieties can be regarded as hypergeometric functions, and they satisfy a system of linear differential equations so that a vector space of its local solutions (a local solution space) is finite dimensional. For some of such systems, I characterize the monodromy representation which describe a global property of a map defined by a basis of its local solution space, and the connection matrix of the first order differential equation with a vector-valued unknown function equivalent to the original system. In this study, I make clear their structures by using the intersection forms defined between twisted (co)homology groups which are induced from period integrals.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2011/04 -2015/03Author : TERASOMA Tomohide, HANAMURA MasakiWe construct a complex using semi-algebraic set and prove generalized Cauchy formula to construct a Hodge realization functor of mixed Tate motives. We introduce a motivic filtration which gives a depth filtration. We construct surfaces which have big images of cycle maps from higher Chow groups to cohomologies. We study Schwarz maps for reducible hypergeometric systems of two variable with a special parameter. We describe the inverse period map using theta function. We give a description of the image of Abel-Jacobi map corresponding to a family of genus two curves.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2010 -2012Author : MATSUMOTO Keiji, TERASOMA Tomohide, SHIGA Hironori, YOSHIDA Masaaki, OHARA KatsuyoshiWe construct several formulas for hypergeometric functions and systems of differential equations of multi-variables. (1) By using the intersection forms defined between twisted (co)homology groups, we give monodromy representations and connection matrices of Pfaffians for some of them. (2) For hypergeometric functions with special parameters related to periods of algebraic varieties, we give some identities between automorphic forms and hypergeometric functions. (3) Some limits defined by the iteration of several means of several terms are expressed by hypergeometric functions of multi-variables.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2008 -2011Author : SHIMADA Ichiro, KIMURA Shun-ichi, ISHII Akira, TAKAHASHI Nobuyoshi, TAKAHASHI Hiroki, SUMIHIRO Hideyasu, HIRANOUCHI Toshiro, MATSUMOTO Makoto, ITO Hiroyuki, MUTSUMI Saito, OKA Mutsuo, KONDO Shigeyuki, MATSUMOTO Keiji, TERAO Hiroaki, ISHIKAWA GooBy writing various computer programs for the computational research of lattices and applying them to lattices of algebraic cycles on K3 surfaces(or related algebraic varieties), we obtained many geometric consequences. In particular, we classified the Zariski pairs of simple plane curves of degree 6 by introducing a notion of Z-splitting curves, and described their adjacency relations. We also presented an algorithm to determine the primitivity of a lattice of algebraic curves in the lattice of topological cycles for a given complex algebraic surface.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2007 -2010Author : YOSHIDA MasaakiWe succeeded to find a good discretization of the hyperbolic Schwarz map for the Airy equation. This is the starting point of the study of singularities of discrete surfaces. For the hypergeometric differential equation of type (3,6), we found a relation between the two monodromy groups - arithmetic group acting of the domain of type IV, and the maximal non-real finite complex reflection group. We described chambers cut out by six planes in general position in the 3-space. Veronese arrangements of hyperplanes in real projective spaces re studied. A set of generators of the monodromy group of the Appell-Lauricella's hypergeometric equation of type FA is obtained.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2007 -2010Author : YAMAGUCHI Keizo, IZUMIYA Shuichi, ONO Kaoru, ISHIKAWA Goo, MATSUMOTO Keiji, HURUHATA Hitoshi, YATSUI Tomoaki, NAKAI Isao, OZAWA Tetasuya, SASAKI TakeshiWe determined the class of differential equations of finite type, which admits extraordinarily rich infinitesimal symmetries, among the classes of differential equations of finite type obtained from the Se-ashi's principle. Moreover we write up explicitly the model equations of this class of finite type equations. Furthermore we investigated the fundamental Reduction procedure in the fields of Contact Geometry of second Order and formulated two fundamental Reduction Theorems.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2005 -2008Author : MATSUMOTO Keiji, ONO Kaoru, NAKAMURA Iku, SHIMADA Ichiro, IWASAKI Katsunori, TERASOMA Tomohide, YOSHIDA Masaaki, ONO Kaoru, NAKAMURA Iku, IWASAKI Katsunori, SHIMADA Ichiro, TERASOMA Tomohide, YOSHIDA Masaakiテータ関数や超幾何関数のみたす関数等式を多数与えた。テータ関数のみたす関数等式によりWhitehead link と Borromean ringsの補空間に入る双曲構造を解明した。また、超幾何関数のみたす関数等式より、いくつかの多項版の算術幾何平均を定め、それらの値の表示公式を与えた。
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2004 -2007Author : TERASOMA Tomohide, OGISO Keiji, YOSHIKAWA Ken-ichi, HOSONO Shinobu, MATSUMOTO Keiji(1) We define a homomorphism from the polylog complex defined by Goncharov to the extension group of mixed Tate motives defined by Bloch and Kriz. The existence of this map is conjectured by Beilinson-Deligne. Beilinson and Deligne assume some conjectures which are equivalent to the KP1 conjecture and Beilinson-Soule conjecture. We constructed the homomorphism without assuming these conjecture using bar constructions and recovery principle for them. (2) We show the differential graded category equivalence between the category of comodules of differential graded Hopf algebra and that of differential graded complex over differential graded category associated a differential graded algebra Using this equivalence, we showed that the Hopf coalgebra constructed form the Deligne algebra classifies the category of variation of mixed Tate Hodge structures. (3) We give a description of the pro-p completion of algebraic varieties of positive characteristic, which classifies the Tannakian category of Fp local systems in terms of Bar constructions. To get a product structure on the corresponding Hopf algebra, we introduce a homotopy shuffle product. Via this shuffle product, we get a notion of group like elements which give the description of pro-p completion. (4) We define arithmetic-geometric mean for hyperelliptic cureves of higher genus, which is a generalization of Gauss arithmetic geometric mean- Moreover, we showed that they coincides with certain determinant of periods of hyperellipitic curves using Thomae's formlula. We showed that this is also equal to a period of certain Calabi-Yau varieties. (5) We construct certain algebraic correspondence between Jacobian varieties and Calabi-Yau varieties obtained by a double covering of three dimensional projective space branched at the projective dual of Caylay Octad of genus three cureves. More over the third cohomology of the Calabi-Yau varieties are not exterior product in general by looking infinitesimal variation of Hodge structures.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2004 -2007Author : NAKAMURA Iku, YOSHIOKA Kota, KONDO Shigeyuki, WENG Lin, KATO Fumiharu, MATSUMOTO KeijiNakamura proved a new theorem on McKay correspondence of a simple singularity C2/G, G being a finite group of SL(2,C). The theorem shows certain natural modules V and Vdagger have simple structures. Among others, the structure of Vdagger explains completely the known bijetive correspondence of the extended Dynkin diagram and all the irreducible representations of G.. The paper is now in print. He also proved an important vanishing theorem for degenerate quasi-abelian varieties. Weng is constructing a new important theory of geometric class field theory, which is modeled after Mumford's stability of GIT, Seshadri-Narashiman's theory of unitary vector bundles. In this program he defined nonabelian L-functions, and in some cases he proved a theorem analogous to Riemann hypothesis. Kato studied with Fujiwara the fundamental theory of p-adic geometry and rigid geometry. He proved also that Mumford fake projective plane and the other known fake projective planes are among Shimura varieties. Kondo constructed a uniformization by a 5-dimensional complex ball of the moduli of ordered 8 points of the projective line, by using Borcherds modular forms. Matsumoto gave a very precise description of a link and its complement in S3 by using real theta functions. Yoshioka proved a formula of counting the number of instantons on certain complex surfaces with Nakajima.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2003 -2006Author : TAKAYAMA Nobuki, NORO Masayuki, FUKUYAMA Katsushi, MASUDA Tetsu, OAKU Toshinori, SAITO MutsumiWe have obtained the following results 1. We constructed vol(A)-linearly independent convergent series solutions for A-hypergeometric differential-difference equations. 2. We prove that any local Grober fan is a polyhedral fan. As an application of this fact, we give an algorithm of computing local BS polynomials, that of computing local tropical varieties, and discuss a relation of slopes and local Grobner fan. 3. We gave a tangent cone algorithm to study D-modules locally. We have more results. As to these, refer to Japanese version of this research report and papers.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2002 -2005Author : YOSHIDA Masaaki, SASAKI Takeshi, IWASAKI Katsunori, MIMACHI Katsuhisa, MATSUMO Keiji, CHO KojiI got the following results concerning the hypergeometric functions. 1)Studied the (co)homology groups attached to Selberg-tpe integrals, evaluated the intersection numbers, and discovered a combinatorial properties of the Selberg functions. 2)Presented co-variant function theory. Found the kappa function, and a 3-parameter families of hypergeometric polynomials, which are very different from the classical ones. 3)Found a new infinite-product formula for the elliptic modular function Lambda. 4)studied combinatorial-topologically the shape of the Schwarz triangles when the inner angles are general. 5)Studied the Whitehead-link-complement group, constructed automorphic functions for this group, and embedded the quotient space to a Euclidean space. 6)Studied the behavior of the solutions of the hypergeometric equation when the exponent-diffences are pure-imaginary, and studied the relation between the space of parameters and the Teichmuler space of genus 2 curves. 7)Invented the theory of hyperbolic Schwarz map. The target of the Schwarz map has been the sphere. Our hypergeometric one has the 3-dimensional hyperbolic space as its target. Group theoretically it is more natural 8)Studied the surfaces on which 3-dimensional Lie group acts, especially ones on which SL(2,R) acts.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2001 -2003Author : MATSUMOTO Keiji, SHIMADA Ichiro, SAITO Mutsumi, MAEDA YoshitakaThe head of investigator MATSUMOTO Keiji constructed period maps and automorphic forms derived from the inverses of period maps for certain families of algebraic varieties by using Prym varieties of algebraic curves. In fact, it was shown that the period map for the family of smooth cubic surfaces could be expressed in terms of periods of the Prym varieties for curves of genus 10. Automorphic forms on the 4-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. For the family of the 4-fold coverings of the complex projective line branching at eight points, the period map from this family to the 5-dimensional complex ball was constructed by using the Prym varieties of these curves. Automorphic forms on the 5-dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. SAITO Mutsumi showed that the ring of differential operators on affine tone varieties and the algebra of symmetries of the system of A-hypergeometric differential equations were anti-isomorphic, and classified systems of A-hypergeometric differential equations combinatonally under these symmetries. He studied the condition that the graded ring gr(D(R_A)) was finitely generated, and gave the composition factors of the ring R_A of functions on any tone variety as a D(R_A)-module. SHIMADA Ichiro showed that if the singularity of each singular fiber was not bad for an algebraic fiber space, the boundary homomorphism from the second homotopy group of the base space to the fundamental group of any general fiber could be constructed. He showed that the fundamental group of the complement of a resultant hypersurface was commutative. He also showed that any supersingular K3 surface could be expressed as a branched double cover of the projective plane.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2000 -2003Author : NAKAMURA Iku, KATSURA Toshiyuki, SHINODA Ken-ichi, SUWA Tatsuo, NAKAJIMA Hiraku, SAITO MasahikoCertain compactification of moduli space of abelian varieties was studied as well as moduli spaces of G-orbits for a finite subgroup G of SL(2,C) and SL(3,C). The main issues we have in mind are as follows (a) Study of a resolution of singularity of the quotient C^3/G as a moduli space (b) study of Kempf stability and compactification of moduli spaces (c) A canonical ompactification SQ_
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 2001 -2002Author : 寺杣 友秀, 小木曽 啓示, 島田 伊知朗, 斎藤 毅, 松本 圭司, 吉川 謙一1 テータ関数を使ったモジュライ空間の射影空間への埋め込みの構成。ドリーニュ=モストウにより分岐条件が与えられたときの射影直線の分岐被覆のモジュライが対称領域の開集合となるための一つの十分条件があたえられた。そのひとつの場合である8点で分岐する場合の周期領域のテータ埋め込みを構成した。これは4重被覆で与えられるが、中間の超楕円曲線の二重被覆のプリム多様体の特別な2分点におけるテータ指標の多項式で与えられる。8次の対称群が2元体上6次元の二次形式を保つ群として実現されるが、この群の作用を用いて、プリム多様体の商で主偏極をもつアーベル多様体の集合を対称的に扱うことが重要である。さらにこの作用をもって105個のテータ指標の多項式が構成され、これを使ったモジュラー埋め込みが複比を用いた多項式による埋め込みとなっていることが示される。 2 アーベル多様体の代数的サイクルの変形理論による構成。円分体を虚数乗法として持つアーベル多様体がそのコホモロジーに関するある表現論的条件をみたすとき、因子類群では生成されないホッジサイクルがヴェイユにより構成された。このサイクルが代数的サイクルで生成されるかどうかは懸案である。このアーベル多様体がある曲線のヤコビアンの商として得られている場合を考えてみる。このヤコビアンの変形が与えられたとき、この曲線の被覆のある変形でそのヤコビアンが与えられたヤコビアンの変形を商としてもつとき、ヴェイユのホッジサイクルは代数的であることをしめした。いまのところ、一般的な状況でこのような曲線の被覆は知られていないが、曲線が楕円曲線であるときは必ず存在することがわかっている。 3 極大退化曲線族に関する極限ホッジ構造とその算術的写像類群の周期への応用。ハイン氏との共同研究。極大曲線族の極限ホッジ構造をドリンフェルトアソシエーターにより表現することによりその周期が多重ゼータ値により記述される。マンフォードとショットキーによる構成法の比較により示した。さらにこれを写像類群及びその算術部分の研究に応用した。このとき相対完備化がモチーフ的であることを証明した。累次高次順像をつかうドリーニュ=ゴンチャロフの方法を相対完備化の時に用いた。
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2000 -2001Author : SAITO Mutsumi, SHIBUKAWA Youichi, MATSUMOTO Keiji, YAMASHITA Hiroshi, WACHI Akihito, YAMADA Hiro-fumiWith support of many examples with a computer, and by communication with world-wide experts in several fields, we obtained the following results. Mutsumi Saito generalized the classification theorem of A-hypergeometric systems to the cases when A is inhomogeneous and/or when we work in the analytic category. He also gave a dimension formula for the log-free series solutions when A is homogeneous, and a rank formula and the proof of the equivalence of Cohen-Macaulayness with the condition that the ranks are the same at all parameters, when A is homogeneous, and the convex hull of A is a simplex. Hiroshi Yamashita obtained some results useful to know when an isotropy representation is irreducible. Furthermore he systematically constructed nonzero quotient representations of isotropy representations attached to discrete series. Keiji Matsumoto clarified a pairing between twisted cohomology groups associated with generalized Airy functions. Writting a base of twisted cohomology groups by Young diagrams, he showed that for the base, the pairing can be explicitly written by skew-Schur polynomials. Youichi Shibukawa solved the classification problem for R operators. For the simplest affine Lie algebra A_1^<(1)> , using two of its realizations, Hiro-Fumi Yamada discovered the weight vectors are written by a modular version of Schur functions and Schur's Q-functions respectively. Akihito Wachi has studied the structure of generalized Verma modules, in particular, their irreducibility, emphasizing their relations with invariant functions.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 1999 -2001Author : YOSHIDA Masaaki, MIMACHI Katsuhisa, IWASAKI Katsunori, SASAKI Takeshi, HANAMURA Masaki, MATSUMOTO KeijiWe got the following results concerning hypergeometric functions. 0) Studied systems of linear partial differential equations modeled after grassmannians. 1) Investigated the Hodge structure of twisted cohomology groups and Got many integral formulae involving got twisted Riemann inequalities absolute values in the integrands. 2) Got the uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces. Proved that it is the restriction of the higher dimensional hypergeometric differential equation onto a d. 3) Developed the intersection theory for twisted cycles : got determinant formulae for not necessarily genetic hyperplane arrangements. Got partial results in the case that some quadratic hypersurfaces get into the arrangements. 4) Found a hyperlybolic structure on the real locus of the moduli space of marked cubic surfaces. Found that the corresponding group is the hyperbolic Coxeter group ; Constructed automorphic forms by the help of a modular embedding. 5) Made a geometric study of the hypergeometric function with Found that the monodromy groups turns out to be scottky imaginary exponents. Groups of genes 2. Constructed a modular ttu with rasp. To the monodromy group.
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1999 -2000Author : 松本 圭司超幾何微分方程式の線形独立な解を並べることで、超幾何微分方程式の定義域から射影空間への多価写像が得られる。この多価写像の像が対称空間でモノドロミー群がこの対称空間に作用する数論的な群になり、かつ逆写像が一価となるものがいくつか知られている。これらの例たちはいずれも合流がおきていない確定型の特異性をもつもののみである。合流型の超幾何微分方程式に関してでは起こり得ないこの特別な状況をよりよく理解するために、その逆写像の具体的な記述を代数幾何学や保形関数論でも特に重要であると考えられるものに対して寺杣氏(東大・数理)と共同で以下のように行った。 3次元射影空間の3次曲面Xで分岐するcyclicな3重被覆Yに対して、そのIntermediate Jacobian J(Y)は、主偏極をもつアーベル多様体である。J(X)はあるリーマン面Cとその上のinvolutionσに関するPrym varietyとして実現できる。Prym varietyの周期を記述する微分方程式は4変数階数5のF_Dと呼ばれる超幾何微分方程式であり、線形独立な解を並べることで3次曲面のモジュライ空間から4次元複素超球への多価写像が得られる。そのモノドロミー群はZ[ω]係数のユニタリ群のレベル(1-ω)の主合同群となる。テータ関数を用いて表示されるIntermediate Jacobian J(Y)上の有理型関数をリーマン面CからJ(Y)へのある正則写像を用いてC上に引き戻すことができる。こうして得られるC上の有理型関数たちを用いて3次曲面Xからリーマン面Cを構成した際の情報を記述することができる。実際E_6型のワイル群が作用する80個のIntermediate Jacobian J(Y)の3等分点を利用して、逆写像を具体的に与えることができる。
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 1998 -1999Author : FURUSAWA Masaaki, KOMORI Youhei, IMAYOSHI Yoichi, KAMAE Tetsuro, MATSUMOTO Keiji, MOCHIZUKI TakuroWe proved the fundamental lemma for the unit element in the Hecke algebra for two relative trace formulas for GSp(4). Our ultimate goal is to prove Bocherer's conjecture on the central critical values of the quadratic twists of the spinor L-functions associated to holomorphic Siegel eigen cusp forms of degree two. The announcements of the fundamental lemma have been published in C. R. Acad. Sci. Paris and the details of the proof will appear elsewhere. In the course of the proof of the fundamental lemma, we evaluated certain matrix argument Kloosterman sums explicitly in terms of the classical GL(2) Kloosterman sums. We remark that our Kloosterman sum is a special case of the generalized Kloosterman sum which appears in the Fourier coefficients of the Poincare series for the Siegel modular group. Our result on the Kloosterman sum may be of some independent interest, since it is rare that such generalized Kloosterman is evaluated explicitly. Our second conjectural trace formula is related to the quadratic base charge for GSp (4). Our result suggests that the Jacquet-Ye criterion for the quadratic base change for GL(2) generalizes to GSp(4). This clearly deserves some further investigation. Finally our result implies that it is important to study the whole L-packet when we study the special values of automorphic L-functions. It seems very interesting to clarify the relationship between the period part of the special value expected by our result and Deligne's conjecture.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 1996 -1998Author : YOSHIDA Masaaki, MATSUMOTO Keiji, WATANABE Fumihiko, HANAMURA Masaki, KANEKO Masanobu, KATO FumiharuHypergeometric integrals found by Euler was re-formulated in terms of a modern language by many authors : the dualﾟCpairing of twisted homologies and twisted cohomologies. Expected intersection theories were established by M.Kita and Yoshida for homologies, and by K.Cho and Matsumoto for cohomologies. Further developments are in progress, especially those for confluent case by Matsumoto. These can be considered to be twisted versions of Riemann's equality for period integrals. Twisted versions of Riemann's inequality were found, via twisted Hodge theory, by Hanamura and Yoshida. Modular interpretations of configuration spaces. Let X(k, n) be the configuration space of n-point-sets in the k-1-dimensional projective space. Several configuration spaces can be presented as quotient spaces of symmetric spaces under discontinuous groups ; the original one is X(2, 4) * H/GAMMA(2), where H is the upper half space and GAMMA(2) is an elliptic modular group. Yoshida found, with Matsumoto and T.Sasaki, a modular interpretation of the space X(3, 6) through hepergeometric function of type (3, 6)), which can be summerized as X(3,6) {z * M2(C) I (z -z*)/2i> O}/GAMMA, where GAMMA is an arithmetic subgroup acting on the hermitian symmetric domain of type IV.Yoshida wrote two books about this interpretation. Kaneko found, with D.Zagier, automorphic forms which connect hypergeometric functions and supersingular elliptic curves. Kaneko found a new arithmetic formulae for the Fourier coefficients of j(gamma). F.Watanabe established a new very transparent way to find Okamoto transformations for Painlv_ functions by using the Takano's construction of the phase spaces. F.Kato is ambitiously trying to find examples of algebraic varieties which are p-adically uniformized by Drinfeld symmetric spaces and the uniformizing differential equations ; he already found, with M.Ishida, new fake projective planes, and studied their uniformizations complex anlytically as well as p-adically.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 1996 -1996Author : SUMIHIRO Hideyasu, YOSHIOKA Kouta, FURUSHIMA Mikio, MATSUMOTO Keishi, SUGANO Takashi, TANISAKI ToshiyukiIn this project, we studied vector bundles on manifolds from the following various points of view : 1)algebraic geometric method, 2)algebraic analytic method, 3)number theoretic method. In 1), we obtained (1) a necessary and sufficient condition for a rank 2 bundle on projective space P^n (n*4) to split into line bundles which gives us a new approach to important conjectures concerning splitting of vector bundles on P^n, (2) we classified Non-projective Moishezon compactifications (X,Y) of affine C^3 space by clarifing numerically effectiveness of the boundary divisor Y and moreover, showed an equivalence between the Stein ess of a C^1- fiber space over a non-Stein manifold S and the triviality of certain rank 2 bundle on S., (3) we proved under some conditions on the first Chern class, the rationality of the moduli of bundles on surfaces with elliptic curves as fibers and have determined the Picard group and Albanese map of the moduli of bundles on elliptic surfaces. In 2), we studied (1) heighest weight modules of semi-simple Lie algebras, especially those corresponding to compact Hermitian symmetric spaces, (2) we have introduced the definition of Radon transformation on Flag manifolds of general type and founed usefulness of bundles to study the Radon transformation, (3) we got a duality concerning generalized hypergeometric functions by using the intersection theory on twisted cohomology group and the exterior products. In 3), (1) we gave an expression by integrals in terms of their Fourier coefficients of the L-functions which are liftings of cusp forms with haif integer weights to the modular forms on orthogonal groups and proved the meromorphic continuation and the functional equation under some technical conditions which can be viewed as a generalization of Kohnen-Skoruppa's result on quadratic Siegel cusp forms, (2) we showed an categorical equivalence between the category of p-adic Galoi representations over local fields with positive characteristic and the category of etale differential modules with Frobenius map and in particular, the ones whose representation of inertial group factors through finite representations correspond to overconvergent modules, (3) investigated sheaficatin of p-adic Hodge theory and their relativeness, which are analogue to Riemann-Hilbert correspondence in the case of complex manifolds.
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1996 -1996Author : 松本 圭司, 都築 暢夫, 木幡 篤孝, 菅野 孝史, 隅広 秀康一般化された超幾何関数は、変数が射影空間上の点の配置であって多くのパラメーターをもった関数である。この関数には変数を点の配置の双対にとりかえ、すべてのパラメーターの符号を変えるという作用が定まる。この作用に対する双対性の公式が、ねじれ(コ)ホモロジー群の交点理論、外積構造およびGauss-Manin systemの対称性から組合わせ論的にとてもきれいな形で得られ、公式内にあらわれるガンマ関数で表される因数の幾何学的な意味も解明された。 ねじれコホモロジー群における交点数の初等的計算手段が確立された。この理論を用いることにより、高階の局所系に付随するするねじれ(コ)ホモロジー群に関する交点理論が高山、小原氏(神戸大 理)との共同研究により完成した。この理論の他の応用として合流型超幾何関数に関するねじれ(コ)ホモロジー群における交点理論の研究が木村氏(熊本大 理)、原岡氏(熊本大 教養)との共同で展開されている。漸近展開可能な関数の解析がこの研究で重要であることが判明したので、この方面の専門家である真島氏(お茶の水大 理)との共同研究が始められた。 代数多様体のベクトル束上の接続と超幾何関数に関する研究が研究分担者隅広氏と開始された。 超幾何関数から自然に構成できる周期写像の逆写像として現われる対称空間上の保型形式に関して研究分担者菅野氏がcuspのまわりにおける展開定理を得た。この保型形式に関する無限積展開やL-関数に関しても研究が進んでいる。 有限体やp進体上の超幾何関数の研究の基礎となるこれらの体係数のねじれ(コ)ホモロジー群の同型定理や交点理論の確立に向けて研究分担者都築氏との共同研究が始まった。 ねじれ(コ)ホモロジー群の交点行列、Lie代数およびルート系に関する研究が研究代表者と研究分担者木幡氏で開始された。
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1995 -1995Author : 松本 圭司Gaussの超幾何微分方程式はもとよりその他の多くの一般化された超幾何微分方程式は解の積分表示が知られている。この積分表示をねじれコホモロジー群とねじれホモロジー群のpairingとみなし、これらの群の幾何学的構造から超幾何微分方程式を考察することがこの研究の主題であった。古典的に知られているようにRiemann面の周期積分の研究では(コ)ホモロジー群の交点と積分とを結び付けるRiemannの周期関係式が重要であるが、射影直線上にn個の一位の極をもつconnection formからきまるねじれコホモロジー群に対する交点理論とこの場合のRiemannの周期関係式にあたる公式が研究代表者と九大数理趙康治氏により得られた。さらに上記の結果とねじれ(コ)ホモロジー群がもつ外積構造からk重とl重の二つの積分表示をもつ(k,k+l)型超幾何微分方程式の双対性に関する理論が完成された。 また、高位の極をもつconnection formからきまるねじれ(コ)ホモロジー群に対する上記の理論を構築することにより、合流型と呼ばれる超幾何微分方程式についての研究も進められている。現在、交点理論や周期関係式等があきらかにされつつある。 さらに射影空間内のいくつかの超平面とひとつの二次超曲面の配置が定義域となる超幾何微分方程式を導入し、その対称性、隣接関係式、古典的によく知られた二変数超幾何微分方程式Appell's F_1,F_4へ退化、あるK3曲面の族に関する周期写像との関係等に関する結果が研究代表者と神戸大理佐々木武氏により得られた。
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1994 -1994Author : 松本 圭司, 吉田 正章ねじれホモロジー群の交点理論は、九大 数理 吉田正章氏(分担者)と金沢大 教養 喜多通武氏により射影直線上での場合だけでなく一般の射影空間上に対して完成された。このことから超幾何微分方程式のモノドロミ-群もつ不変なエルミート形式がねじれサイクルの交点数を計算することにより具体的にしかも容易に求まるようになった。しかも与えられた超局面たちの配置がかなり複雑なものに対しても計算できるようになっているので、交点行列の行列式を計算することで構成したねじれサイクルたちが一次独立であることを調べることができるようになった。ねじれコホモロジー群の交点理論は、研究代表者と九大 数理 趙 康治氏により射影直線上での場合において完成された。さらに一般の射影空間上に対してもほぼ完成している。ねじれコホモロジー群の交点、ねじれホモロジー群の交点、積分という三つのpairingを結び付けることができ、Riemannの周期関係式のねじれ版を得ることができた。応用として多変数超幾何級数Lauricella's F_Dがみたす二次関係式をこの幾何学的な関係式より導くことができた。さらにこれらの理論の外積を考えることにより、k×l行列が変数となる一般化された(k,l)型超幾何級数に関する二次関係式が得られているだけでなく、超幾何周期行列の双対性に関する理論が喜多通武氏との共同研究によりできつつある。一方、吉田正章氏(分担者)と趙 康治氏により射影直線上のねじれ(コ)ホモロジー群と分岐被覆でできるRiemann面との(コ)ホモロジー群の比較定理も完成されている。また、合流型超幾何関数やq-analogへの応用も試みられている。
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1993 -1993Author : 吉田 正章, 山口 孝男, 坂内 英一, 三町 勝久, 趙 康治, 松本 圭司1)射影平面上の6直線で分岐する二重coverとしてのK3曲面は4次元族をなす。この族の周期写像は(3,6;1/2)型超幾何微分方程式の射影的解となり,その局所的,大域的性質を調べた。 2)上記周期写像の逆写像をI型領域上のテーター凾数を用いて書いた。 3)(k,n)型超幾何微分方程式のモノドロミイ群の生成元を求めるアルゴリスムを与え,(3,6)型の場合に計算を実行した。 4)〓(線型関数の複素ベキ)×有理関数の積分はtwisted cohomologyとtwisted homologyのpairingと解釈できる。これらのtwisted homologyの交叉理論とtwisted cohomologyの交叉理論(一次元の場合)を完成した。 5)Sn-対称性をもつ退化した超平面配置に属する超幾何型積分のみたす微分方程式系を求めて,その性質を研究した。 6)アソシエーションスキームの分類問題と代数的組合せ論の方法を用いてのスピンモデルなどの数理物理学的対象の研究. 7)Alexaudrov空間の等長変換群がLie群になる事を証明し,それを用いて断面曲率と直径が各々下と上に一様に有界なリーマン多様体の基本群の有限性に関する結果を得た。
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1992 -1992Author : 吉田 正章, 山崎 正, 松本 圭司, 趙 康治X(k,n)でk-1次元射影空間上の一般の位置にあるn点の配置空間とするこの郁間は不変式論,代数幾何,及び超幾何凾数論的に大切である。 (1)X(2,5)及びX(2,6)のMumfordコンパクト化を具体的に埋め根みを与えることによって得た。 (2)X(k,n)のMumfordコンパクト化のPoincare多様式を(k,n)が小さい時に計算した。 (3)X(2,8)を5次元球体をPicardモジュラー群で割ったものとして表現し、組合を的な性質を調べた。 (4)X(3,6)と射影平面上6体の線上で2重に分枝するK3曲面のもジュライ,周積分としての超幾何凾数,保型形式等の関係をくわしく調べた、またX(3,6)の組合せ的,代数キカ的な性質を調べた。 X(k,n)を自然な定義域にもつ超幾何凾数をE(k,n:α)と書く。 (5)E(k,n:α)のモノドロミイ群を求めるアルゴリスムを得た。とくにE(3,6:α)のときは生成元を具体的に求めた。 (6)E(k,n:α)のEyler積分表示はtwistedコホモロジイとホモロジイの双対pairingと見なすことができるが、twistedホモロジイの交叉数理論を構築した。 (7)E(3,6:α)の局所微合幾何学的研究.等角構造にarlachした微分方程式として。
- 日本学術振興会：科学研究費助成事業Date (from‐to) : 1991 -1991Author : 山崎 正, 山田 美枝子, 趙 康治, 白谷 克巳, 松本 圭司, 吉田 正章1.山崎はSiegel上半空間の微分作用素の応用を調べた。具体的に言うと、対称行列の空間上の調和関数を考え、その変数にSiegel上半空間の微分作用素を代入して得られる作用側は保型形式をある種の保型性を持つ関数に写す。特にGL(n)の対称テンソル表現に対応する調和関数を考え、それをblock対角成分に制限すると、そのそれぞれの対角成分上の(ベクトル値)保型形式のテンソル積になる。これに対しGarrettの用いた方法を適用することによりKlingen型の(ベクトル値)Eisenstein級数のFourier展開の具体的表示が求められた。この方法は当然他の場合にも応用出来るはずである。又、ここで用いた微分作用素と志村五郎氏が研究している“数論的"微分作用素との関連を明らかにすることも今後の問題である。これらの研究の過程で表現論との関係の重要性が明白になった。今後は局所体上のReductiveな群およびその巾零群による拡大に対するHecke理論やその表現を研究したい。 2.吉田ー松本はModuli問題に関連する保型関数を調べた。例えば2次元射影空間上の6本の直線上で分岐する2重被覆から生じるK3曲面の族の周期写像の逆写像がI型領域上のtheta関数を用いて具体的に与えられることを示した。 3.趙はコンパクトKahler多様体上の安定ベクトル束のversal族上にWeilーPetersson型の距離を構成し、それはKahler距離であることを示し更に特に曲線の場合にその正則断面曲率が非負になることを示した。