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)

  Yasuda

• Name (Kana)

  Seidai

• Name

  202001019806009457

Achievement

Research Areas

• Natural sciences / Algebra / arithmetic geometry
• Natural sciences / Algebra / number theory

Research Experience

• 2020/10 - Today Hokkaido University Faculty of Science Department of Mathematics
• 2012/04 - 2020/09 Osaka University Graduate School of Science Department of Mathematics
• 2007/04 - 2012/03 Kyoto University Research Institute for Mathematical Sciences Applied Mathematics Research Section
• 2002/05 - 2007/03 Kyoto University Research Institute for Mathematical Sciences Applied Mathematics Research Section

Education

• 1998/04 - 2001/03  The University of Tokyo  Graduate School of Mathematical Sciences
• 1996/04 - 1998/03  The University of Tokyo  Graduate School of Mathematical Sciences
• 1994/04 - 1996/03  The University of Tokyo  Faculty of Science  Department of Mathematics

Published Papers

• Local newforms for the general linear groups over a non-archimedean local field

  Hiraku Atobe, Satoshi Kondo, Seidai Yasuda

  Forum of Mathematics, Pi 10 2022 

  Abstract In [14], Jacquet–Piatetskii-Shapiro–Shalika defined a family of compact open subgroups of p-adic general linear groups indexed by nonnegative integers and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of nonnegative integers. For the proof, we introduce the Rankin–Selberg integrals for Speh representations.
• Regularity of quotients of Drinfeld modular schemes

  Satoshi Kondo, Seidai Yasuda

  Pacific Journal of Mathematics 304 (2) 481 - 503 0030-8730 2020/02/12 [Refereed]
   

  Let A be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal I⊂A, Drinfeld defined the notion of structure of level I on a Drinfeld module. We extend this to that of level N, where N is a finitely generated torsion A-module. The case where N=(I−1/A)d, where d is the rank of the Drinfeld module, coincides with the structure of level I. The moduli functor is representable by a regular affine scheme. The automorphism group AutA(N) acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of Γ0 and of Γ1. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups.
• Belyi’s theorem in characteristic two

  Yusuke Sugiyama, Seidai Yasuda

  Compositio Mathematica 156 (2) 325 - 339 0010-437X 2020/02 [Refereed]
   

  We prove an analogue of Belyi’s theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called pseudo-tameness for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a ‘pseudo-tame’ rational function by showing the vanishing of an obstruction class. Finally, we construct a tamely ramified rational function from the ‘pseudo-tame’ rational function.
• Category of mixed plectic Hodge structures

  Kenichi Bannai, Kei Hagihara, Shinichi Kobayashi, Kazuki Yamada, Shuji Yamamoto, Seidai Yasuda

  Asian Journal of Mathematics 24 (1) 31 - 76 1093-6106 2020 [Refereed]
  
• First and second K-groups of an elliptic curve over a global field of positive characteristic

  Satoshi Kondo, Seidai Yasuda

  Annales de l’institut Fourier 68 (5) 2005 - 2067 0373-0956 2018/11 [Refereed]
   

  In this paper, we show that the maximal divisible subgroup of groups K_1 and K_2 of an elliptic curve E over a function field is uniquely divisible. Further those K-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of E, which is an elliptic surface over a finite field.
• Sites whose topoi are the smooth representations of locally prodiscrete monoids

  Satoshi Kondo, Seidai Yasuda

  Journal of Algebra 502 382 - 496 0021-8693 2018/05 [Refereed]
   

  We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined). Examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups. This is a generalization of the fact that the topos associated with the étale site of a scheme is equivalent to the category of sets with continuous action by the étale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.
• The radius of convergence of the p-adic sigma function

  Kenichi Bannai, Shinichi Kobayashi, Seidai Yasuda

  Mathematische Zeitschrift 286 (1-2) 751 - 781 0025-5874 2017/06 [Refereed]
   

  The purpose of this article is to investigate the radius of convergence of the p-adic sigma function of elliptic curves, especially when p is a prime of supersingular reduction. As an application, we prove certain p-divisibility of critical values of Hecke L-functions of imaginary quadratic fields at inert primes.
• Finite real multiple zeta values generate the whole space Z

  Seidai Yasuda

  INTERNATIONAL JOURNAL OF NUMBER THEORY 12 (3) 787 - 812 1793-0421 2016/05 [Refereed][Not invited]
   

  We prove that the Q-vector space generated by the multiple zeta values is generated by the finite real multiple zeta values introduced by Kaneko and Zagier.
• On some applications of integral p-adic Hodge theory to Galois representations

  Go Yamashita, Seidai Yasuda

  JOURNAL OF NUMBER THEORY 147 721 - 748 0022-314X 2015/02 [Refereed][Not invited]
   

  We explicitly construct an analytic family of n-dimensional crystalline representations by using integral p-adic Hodge theory. This is a generalization of results by Berger, Li, and Zhu and by Dousmanis. We show, by using Kisin's method, that the part of a universal deformation ring related to the above constructions is connected. From this we obtain an explicitly described subclass of potentially diagonalizable representations in the sense of Barnet-Lamb, Gee, Geraghty and Taylor. This yields automorphy lifting theorem and potential automorphy theorem, in which the condition at p is weakened. (C) 2014 The Authors. Published by Elsevier Inc.
• On two higher Chow groups of schemes over a finite field

  Satoshi Kondo, Seidai Yasuda

  DOCUMENTA MATHEMATICA 20 737 - 752 1431-0643 2015 [Refereed][Not invited]
   

  Given a separated scheme X of finite type over a finite field, its higher Chow groups CH-1(X, 1) and CH-2(X, 3) are computed explicitly.
• The Hoffman basis of the space of multiple zeta values

  Yasuda Seidai

  RIMS Kokyuroku Bessatsu 京都大学 51 (51) 375 - 433 1881-6193 2014/10 [Refereed][Not invited]
   

  Recently Brown [Br1] gave a proof of a conjecture by Hoffman [Hof2] that the Q-vector space generated by the multiple zeta values is generated by the set, called the Hoffman basis, of multiple zeta values of a certain special type. In this article we give a survey of this topic including an outline of the proof by Brown [Br1]. At the end of the article we also give some applications of the result and mention some open problems.
• The Riemann-Roch theorem without denominators in motivic homotopy theory

  Satoshi Kondo, Seidai Yasuda

  JOURNAL OF PURE AND APPLIED ALGEBRA 218 (8) 1478 - 1495 0022-4049 2014/08 [Refereed][Not invited]
   

  The Riemann-Roch theorem without denominators for the Chern class maps on higher algebraic K-groups with values in motivic cohomology groups in the context of motivic homotopy theory is proved. (C) 2013 Elsevier B.V. All rights reserved.
• The l-parity conjecture for abelian varieties over function fields of characteristic p>0

  Fabien Trihan, Seidai Yasuda

  COMPOSITIO MATHEMATICA 150 (4) 507 - 522 0010-437X 2014/04 [Refereed][Not invited]
   

  Let A/K be an abelian variety over a function field of characteristic p>0 and let l be a prime number (l = p allowed). We prove the following: the parity of the corank r(l) of the l-discrete Selmer group of A/K coincides with the parity of the order at s = 1 of the Hasse-Weil L-function of A/K. We also prove the analogous parity result for pure l-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin-Tate conjecture.
• Explicit t-expansions for the elliptic curve y^2=4(x^3 + Ax + B)

  Seidai Yasuda

  PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 89 (9) 123 - 127 0386-2194 2013/11 [Refereed][Not invited]
   

  For an elliptic curve E : y^2 = 4(x^3 + Ax + B) over a field of characteristic not equal 2, we explicitly compute the pullback to the formal completion of E at the origin of some important objects on E including the functions x, y and the invariant differential w = dx/y in terms of the formal parameter t = -2x/y.
• Zeta elements in the K-theory of Drinfeld modular varieties

  Satoshi Kondo, Seidai Yasuda

  MATHEMATISCHE ANNALEN 354 (2) 529 - 587 0025-5831 2012/10 [Refereed][Not invited]
   

  Beilinson (Contemp Math 55:1-34, 1986) constructs special elements in the second K-group of an elliptic modular curve, and shows that the image under the regulator map is related to the special values of the L-functions of elliptic modular forms. In this paper, we give an analogue of this result in the context of Drinfeld modular varieties.
• Local L and epsilon factors in Hecke eigenvalues

  Satoshi Kondo, Seidai Yasuda

  JOURNAL OF NUMBER THEORY 132 (9) 1910 - 1948 0022-314X 2012/09 [Refereed][Not invited]
   

  Formulas (Theorems 3.5 and 4.1) which express the local L-factor and the local epsilon factor of an irreducible admissible representation of GL(d) over a non-archimedean local field in terms of the eigenvalues of some explicitly given Hecke operators are derived. (C) 2012 Elsevier Inc. All rights reserved.
• On the second rational K-group of an elliptic curve over global fields of positive characteristic

  Satoshi Kondo, Seidai Yasuda

  PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 102 1053 - 1098 0024-6115 2011/06 [Refereed][Not invited]
   

  Let E be an elliptic curve over a global field of positive characteristic. Let r be the order of zero at s = 0 of the Hasse-Weil L-function with bad factors removed. The Parshin conjecture on the vanishing of higher rational K-theory of projective smooth schemes over finite fields implies dim_Q K_2(E) \otimes_Z Q = r. It is shown that dim_Q K_2(E) \otimes_Z Q >= r.
• Product structures in motivic cohomology and higher Chow groups

  Satoshi Kondo, Seidai Yasuda

  JOURNAL OF PURE AND APPLIED ALGEBRA 215 (4) 511 - 522 0022-4049 2011/04 [Refereed][Not invited]
   

  It is shown that the product structures of motivic cohomology groups and of higher Chow groups are compatible under the comparison isomorphism of Voevodsky (2002) [11] This extends the result of Weibel (1999) [14] where he used the comparison isomorphism which assumed that the base field admits resolution of singularities The mod n motivic cohomology groups and product structures in motivic homotopy theory are defined and it is shown that the product structures are compatible under the comparison isomorphisms.
• Non-negativity of the Fourier coefficients of eta products associated to regular systems of weights

  Seidai Yasuda

  PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES 46 (3) 549 - 563 0034-5318 2010/09 [Refereed][Not invited]
   

  We prove Saito's conjecture [9, Conjecture 13 5] about the non-negativity of the Fowler coefficients of the eta products associated to regular systems of weights
• Haramonic and equianharmonic equaitons in the Grothendieck-Teuchmuller group. III

  Hiroaki Nakamura, Hiroshi Tsunogai, Seidai Yasuda

  JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU 9 (2) 431 - 448 1474-7480 2010/04 [Refereed][Not invited]
   

  We study behaviours of the `equianharmonic' parameter of the Grothendieck-Teichmuller group introduced by Lochak and Schneps. Using geometric construction of a certain one-parameter family of quartics, we realize the Galois action on the fundamental group of a punctured Mordell elliptic curve in the standard Galois action on a specific subgroup of the braid group B^_4. A consequence is to represent a matrix specialization of the `equianharmonic' parameter in terms of special values of the adelic beta function introduced and studied by Anderson and Ihara.
• On Haagerup's list of potential principal graphs of subfactors

  Marta Asaeda, Seidai Yasuda

  COMMUNICATIONS IN MATHEMATICAL PHYSICS 286 (3) 1141 - 1157 0010-3616 2009/03 [Refereed][Not invited]
   

  We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.
• The product formula for local constants in torsion rings

  Seidai Yasuda

  JOURNAL OF MATHEMATICAL SCIENCES-THE UNIVERSITY OF TOKYO 16 (2) 199 - 230 1340-5705 2009 [Refereed][Not invited]
   

  Let p be a rational prime and K a local field of residue characteristic p. In this paper, we prove the product formula for local epsilon_0-constants defined in [Y1].
• Local constants in torsion rings

  Seidai Yasuda

  JOURNAL OF MATHEMATICAL SCIENCES-THE UNIVERSITY OF TOKYO 16 (2) 125 - 197 1340-5705 2009 [Refereed][Not invited]
   

  Let p be a rational prime and K a local field of residue characteristic p. In this paper, generalizing the theory, of Deligne [De1], we construct a theory of local epsilon_0-constants for representations, over a complete local ring with an algebraically closed residue field of characteristic not equal p, of the Weil group W_K of K.
• Laurent Lafforgue 氏の業績 : 関数体上のGL_rに対する Langlands 対応の確立

  安田 正大

  数学 日本数学会 60 (4) 415 - 424 0039-470X 2008/10 [Not refereed][Not invited]
   

  2002年の国際数学者会議において Fields 賞を受賞した，Laurent Lafforgue 氏の業績を紹介することが本稿の目的である.
• Local ε0-characters in torsion rings

  Seidai Yasuda

  Journal de Theorie des Nombres de Bordeaux 19 (3) 763 - 797 1246-7405 2007 [Refereed][Not invited]
   

  Let p be a rational prime and K a complete discrete valuation field with residue field k of positive characteristic p. When k is finite, generalizing the theory of Deligne [1], we construct in [10] and [11] a theory of local ε0-constants for representations, over a complete local ring with an algebraically closed residue field of characteristic ≠ p, of the Weil group W_K of K. In this paper, we generalize the results in [10] and [11] to the case where k is an arbitrary perfect field.
• K-THEORETIC ELEMENTS ON DRINFELD MODULAR VARIETIES AND SPECIAL L-VALUES(Algebraic Number Theory and Related Topics)

  KONDO SATOSHI, YASUDA SEIDAI

  RIMS Kokyuroku 京都大学数理解析研究所 1521 (1521) 66 - 69 1880-2818 2006/10 [Not refereed][Not invited]
  
• 志村曲線のCMサイクルとSafarevich-Tate群 (代数的整数論とその周辺)

  安田 正大

  数理解析研究所講究録 京都大学数理解析研究所 1097 (1097) 139 - 143 1880-2818 1999/04 [Not refereed][Not invited]
  

MISC

• The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits building of PGL of a non-archimedean local field in positive characteristic and modular symbols

  Satoshi Kondo, Seidai Yasuda  2014/06/27  [Not refereed][Not invited]
   

  We study the homology and the Borel-Moore homology with coefficients in
  $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits
  building of $\mathrm{PGL}$ of a nonarchimedean local field of positive
  characteristic by an arithmetic subgroup (a special case of the general
  definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)).
  We define an analogue of modular symbols in this context and show that the
  image of the canonical map from homology to Borel-Moore homology is contained
  in the sub $\mathbb{Q}$-vector space generated by the modular symbols.
  By definition, the limit of the Borel-Moore homology as the arithmetic group
  becomes small is isomorphic to the space of $\mathbb{Q}$-valued automorphic
  forms that satisfy certain conditions at a distinguished (fixed) place (namely
  that it is fixed by the Iwahori subgroup and the center at the place). We show
  that the limit of the homology with $\mathbb{C}$-coefficients is identified
  with the subspace consisting of cusp forms. We also describe an irreducible
  subquotient of the limit of Borel-Moore homology as an induced representation
  in a precise manner and give a multiplicity one type result.
• Hoffman予想について (特集 ｢想定外｣の数学)

  安田 正大  数学セミナー  51-  (1)  8  -12  2012/01  [Not refereed][Not invited]
  

Presentations

• Local newforms for the general linear groups  [Invited]

  Seidai Yasuda

  Automorphic form, automorphic L-functions and related topics  2022/01
• Vincent Lafforgue による関数体の Langlands 対応の構成  [Invited]

  安田 正大

  代数的整数論とその周辺  2019/12
• Integral structures of two dimensional crystalline representations  [Invited]

  Seidai Yasuda

  p-adic methods in arithmetic geometry at Sendai, 2019  2019/11
• Cotangent complex and Postnikov towers  [Invited]

  Seidai Yasuda

  Yatsugatake Workshop, 2019  2019/09
• Depth graded structures  [Invited]

  Seidai Yasuda

  Multiple zeta values and related topics  2019/06
• 重さ (p^2+1)/2 以下の 2 次元クリスタリン表現の整構造.  [Invited]

  安田 正大

  早稲田大学整数論セミナー  2019/04
• Modular complexes and dimensions of derived double shuffle modules  [Invited]

  Seidai Yasuda

  第17回北陸数論研究集会  2018/12
• A product of Eisenstein series and special L-values over the rational function field  [Invited]

  Seidai Yasuda

  NTCS Seminar on Number Theory  2018/12
• Derived double shuffle Lie algebra and the Steinberg modules  [Invited]

  Seidai Yasuda

  NTCS Seminar on Number Theory  2018/12
• Foliations I, II  [Invited]

  Seidai Yasuda

  八ヶ岳ワークショップ, 2018 The conservative conjecture  2018/09
• 「p進多重ゼータ値」から「有限多重ゼータ値」へ  [Invited]

  安田 正大

  第２６回整数論サマースクール「多重ゼータ値」  2018/09
• 結合子と結合子関係式  [Invited]

  安田 正大

  第２６回整数論サマースクール勉強会「モチヴィック多重ゼータ値」  2017/08
• Drinfeld modular 多様体上の zeta 元について  [Invited]

  安田 正大

  九大数理談話会  2017/06
• Linearized and derived double shuffle Lie algebras  [Invited]

  Seidai Yasuda

  Workshop: Johnson homomorphisms and related topics  2017/05
• pseudo-tame rational functions on curves in characteristic two  [Invited]

  Seidai Yasuda

  Weekly Seminar of the Laboratory of Algebraic Geometry and its Applications  2017/03
• Ihara bracket for gorup schemes  [Invited]

  Seidai Yasuda

  Low dimensional topology and number theory IX  2017/03
• Belyi's theorem in charcteristic two  [Invited]

  Seidai Yasuda

  p-adic methods in arithmetic geometry at Sendai, 2016  2016/10
• Etale theta functions, mono-theta enviroments, and [IUTeichI] \S1-\S3, I, II  [Invited]

  Seidai Yasuda

  Inter-iniversal Teichmuller Theory Summit 2016  2016/07
• Topic on multiple zeta values  [Invited]

  Seidai Yasuda

  NCTS Number Theory Seminar  2015/12
• モチフィック多重ゼータ値と有限多重ゼータ値  [Invited]

  安田 正大

  日本数学会２０１５年度秋季総合分科会  2015/09
• Motivic and finite multiple zeta values  [Invited]

  Seidai Yasuda

  Bousfield localizations form a set: a workshop in memory of Tetsusuke Ohkawa  2015/08
• 階数２のWach加群の族の構成  [Invited]

  安田 正大

  九大代数学セミナー  2015/06
• Grids and the associated monoids  [Invited]

  Seidai Yasuda

  上智大学数学談話会  2015/05
• Integrality of p-adic multiple zeta values and application to finite multiple zeta values  [Invited]

  Seidai Yasuda

  東京北京パリ数論幾何セミナー  2015/04
• Outlines: Kisin's proof of Breuil-Mezard conjecture  [Invited]

  Seidai Yasuda

  Winter school of p-adic Hodge theory  2015/01
• Integrality and a conjectural relation between $p$-adic multizeta values and truncated multiple harmonic sums  [Invited]

  Seidai Yasuda

  Novel visage of arithmetic and derived geometry  2014/10
• p-adic multiple zeta values and truncated multiple harmonic sums  [Invited]

  Seidai Yasuda

  Workshop on Multiple Zeta Values  2014/08
• eriods of Mixed Tate Motives and Multiple Zeta Values  [Invited]

  Seidai Yasuda

  2014 NCTS Lecture Series on Number Theory  2014/05
• 有限実多重ゼータ値と $p$ 進多重ゼータ値  [Invited]

  安田 正大

  第 20 回関西多重ゼータ研究会＆第 7 回多重ゼータ研究集会  2014/02
• GL_d の smooth 表現の Galois 圏論的解釈と保型 Euler 系  [Invited]

  安田 正大

  九州代数的整数論 2013  2013/02
• 多重ゼータ値についての最近の進展とドゥリーニュ・伊原予想  [Invited]

  安田 正大

  代数的整数論とその周辺  2012/12
• Galois represetations attached to Siegel modular forms I、ＩＩ  [Invited]

  Seidai Yasuda

  The 15th Hakuba Autumn Workshop on Number Theory  2012/11
• p-adic representations and p-adic Hodge theory  [Invited]

  Seidai Yasuda

  L-functions and Arithmetic  2012/10
• Brown 氏の研究における余積構造の利用法: Hoffman 基底と深さ filtration  [Invited]

  安田 正大

  関西多重ゼータ研究集会（第１１回）  2012/09
• 整係数 2 次元 p 進表現の構成  [Invited]

  安田 正大

  第 57 回代数学シンポジウム  2012/08
• Bernstein center I, II  [Invited]

  Seidai Yasuda

  勉強会 「p 進代数群の表現論」  2012/02
• Some hypergeometric polynomials and reductions of crystalline representations with moderate Hodge-Tate weights  [Not invited]

  Seidai Yasuda

  Workshop on p-adic arithmetic geometry and motives  2012/01
• Stability of a higher Chow group of an elliptic curve  [Invited]

  Seidai Yasuda

  Workshop on arithmetic geometry 2011  2011/10
• Euler 系とその応用について  [Invited]

  安田 正大

  北大数論幾何学セミナー  2011/01
• Construction of extensions by $K_2$  [Invited]

  Seidai Yasuda

  第１３回白馬整数論オータムワークショップ「被覆群上の保型表現・保型形式」  2010/11
• 安定跡公式と志村多様体  [Invited]

  安田 正大

  第18回整数論サマースクール「アーサー・セルバーグ跡公式入門」  2010/09
• $GL(n)$のガロア表現と局所及び大域ラングランズ対応 (Introduction to Clozel, Harris-Taylor and Taylor-Yoshida)  [Invited]

  安田 正大

  $GSp(4)$ の数論を中心とした基礎的ワークショップ  2010/08
• Iwasawa theory and higher Fitting ideals  [Invited]

  Seidai Yasuda

  Workshop on Iwasawa Theory over Function Fields of Characteristic $p>0$  2010/04
• モジュラ曲線の直積のK_3元について  [Invited]

  安田 正大

  東北大学・代数セミナー  2010/01

Association Memberships

• THE MATHEMATICAL SOCIETY OF JAPAN   

Research Projects

• New developments in the anticyclotomic Iwasawa theory and special value formulas on L-functions

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

  Date (from‐to) : 2022/04 -2027/03 

  Author : 小林 真一, 太田 和惟, 大坪 紀之, 千田 雅隆, 中村 健太郎, 安田 正大
• 数論的対象の背後にある幾何学の発見・構築を通じたL関数・ガロア表現の研究

  日本学術振興会：科学研究費助成事業 基盤研究(B)

  Date (from‐to) : 2021/04 -2026/03 

  Author : 安田 正大, 古庄 英和, 山下 剛
• Strategic research to construct motivic units using new symmetry

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

  Date (from‐to) : 2018/06 -2023/03 

  Author : 坂内 健一, 志甫 淳, 寺杣 友秀, 勝良 健史, 小林 真一, 安田 正大, 山本 修司

   

  昨年までの研究経過を踏まえて、総実代数体に付随する代数トーラスに対してプレクティックDeligne-Beilinsonコホモロジーを定義して、その中にポリログを定義する研究に着手した。ポリログのde Rham実現を具体的に記述しようと試みた過程で、総実代数体のHecke L関数の負の整数点の値の母関数について、新谷卓郎が研究した非標準な母関数について、この母関数を、代数トーラスの同変コホモロジー類として解釈すると、極めて自然で標準的な類を構成できることを発見した。このコホモロジー類を「新谷生成類」と呼ぶことにした。通常、高次のコホモロジー類を点に制限すると消えてしまうが、同変コホモロジー類を考えることで「点での値」をうまく定義できることが新しい発見である。当初は、プレクティックポリログのホッジ実現を完全に書ききるまで、整数論的に面白い成果は得られないと想定していたが、早い段階で、整数論の基本的な結果に対して新しい知見を得たことは、とても嬉しく感じている。上記の結果を受けて、新谷生成類の考え方をベースに、総実代数体に付随するp進ポリログ関数の定義をした。これもやはり、総実代数体の代数トーラスの同変コホモロジー類として定義した。また、この関数の等分点での制限が、p進Hecke L関数の特殊値と一致することを証明した。この成果は、有理数体の場合のColemanの古典的な結果を総実代数体の場合に一般化するものであり、今後、今回の代数トーラスやp進ポリログ関数が数論幾何的予想に対して有用であることを強く示唆する結果である。
• モチヴィックガロア群と多重ゼータ値から広がる数学ー整数論からの解放ー

  日本学術振興会：科学研究費助成事業 基盤研究(B)

  Date (from‐to) : 2018/04 -2023/03 

  Author : 古庄 英和, 田坂 浩二, 大野 泰生, 安田 正大

   

  7月にカナダのCRM研究所のプログラム「Expansions, Lie Algebras, and Invariants」に参加し、Enriquez氏と共同研究を続けRacinetが2002年に提出したdouble shuffle群のBetti側に対応する群の正体を明らかにした。1年前に発見した調和余積のBetti対応物を用いてde Rham側と同様な簡明な表示を与えることができた。この結果をpreprintにまとめ発表した。今までの「Double shufle関係式のBetti側の理論」に関する一連の共著論文（3本）をようやく書き終えたことになるが、読み直してみると複雑に入り組んでいた議論のいくつかが簡略化できそうなことに気づいたので、引き続き議論の整備をし改訂を行っていくことにした。 11月には「多重ゼータ値の諸相」の国際集会を数理解析研究所で主催した。多重ゼータ値を研究する研究者を各方面から招聘した。Enriquez氏もこの集会に招聘し共同研究のサーベイ発表してもらった。 研究分担者の田坂氏は楕円モジュラー形式の新形式を二重Eisenstein級数の基底で表示する明示公式を得た。大野氏はArakawa-Kaneko多重ゼータ関数の特殊値に関する和公式を構成した。安田氏はグラフの圏を適当に局所化することによって得られる対称性の観点から複シャッフル空間に関する Goncharovの仕事の再解釈を行なった。
• Adelic new methods on arithemetic geometry and their applications to p-adic Hodge theory and multiple L-functions

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

  Date (from‐to) : 2015/04 -2020/03 

  Author : Yasuda Seidai

   

  The research representative and Go Yamashita have constructed families of Wach modules of rank two and applied them to the study of crystalline deformation rings of dimension two. He and Satoshi Kondo have constructed lifts of the zeta elements in motivic cohomologies of Drinfeld modular varieties to their integral models satisfying norm relations, and have constructed a theory of topoi related to monoids. He and Yusuke Sugiyama have introduced a new notion of pseudo-tameness and, by using them, have proved that any algebraic curve over an algebraically closed field has a tame morphism to the projective line. He has introduced the derived double shuffle spaces and has applied them to show a double shuffle analogue of Broadhurst-Kreimer conjecture in depth four. He has found that a suitable quotient of a Hilber modular surface related to the L-function of a certain curve of genus is a Kummer surface.
• Generalization of Iwasawa theory through Galois doformation and search for new phenomena

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

  Date (from‐to) : 2014/04 -2020/03 

  Author : Ochiai Tadashi

   

  With this grant, I executed the following projects (I) Euler system thheory over deformation rings with singularity, (II) Iwasawa theory for GSp(4), (III) Iwasawa theory for Coleman families, (IV) Iwasawa theory for CM fields and CM modular forms, (V) functional equation of Selmer group in noncommutative Iwasawa theory, (VI) Euler system theory for higher rank Galois representation. Also, I organized an international workshop for the generalization of p-adic L-function.
• Strategic Reseach using Eisensterin classes to prove Conjectures in Arithmetic Geometry

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

  Date (from‐to) : 2014/04 -2019/03 

  Author : Bannai Kenichi, TAKAI Yuuki, OTA Kazuto, ONO Masataka, KIRAL Erin Mehmet

   

  Our original goal was to study the polylogarithm in the case of totally real fields. Our original goal was to study the polylogarithm via the Eisenstein class, but in course of our research, we realized the importance of a certain algebraic torus associated to a totally real field, and using the ideas from plectic structures proposed by Nekovar and Scholl, we succeeded in proving that the Shintani generating function which generates special values of Shintani zeta functions, defines a canonical class on the algebraic torus.
• A research on a new symmetry on motive with real multiplication

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

  Date (from‐to) : 2016/04 -2018/03 

  Author : kobayashi shinichi, OTA Kazuto, HAGIHAEA Kei, YAMADA Kazuki

   

  Plectic conjecture by J. Nekovar and A. Scholl is considered to give a strong impact on the study for motive with real multiplication if the program is completed. However, the program has just started. We studied the Hodge realization of the plectic cohomology. We gave an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also constructed an explicit complex calculating the extension groups in this category. This result is important to consider applications to concrete problems.
• Ramification theory of automorphic representations and arithmetic of special L-values

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

  Date (from‐to) : 2015/04 -2018/03 

  Author : Yoshi-Hiro Ishikawa

   

  Number theory investigation usually involves quite vast area of deep mathematics,like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L‐/ε‐factors. Our approach is resorting to integral presentations of L‐function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are only in the case of Real/unramified U(3).
• Study of the moduli of Galois representations of number fields and function fields

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

  Date (from‐to) : 2013/04 -2016/03 

  Author : Taguchi Yuichiro, HATTORI Shin, KURIHARA Masato, SAITO Takeshi, TAMAGAWA Akio, YASUDA Seidai, HIRANOUCHI Toshiro

   

  We have constructed a moduli scheme of Galois representations and studied its properties, and obtained some basic results. We have also obtained several related results, such as: (1) a vanishing theorem of the Galois-fixed subspace of a Galois representation of a rather general type of complete discrete valuation field (a generalization of a theorem of Imai) and its application to Iwasawa theory, (2) a result on the congruence of Galois representations and its application to non-existence theorems a la Rasmussen-Tamagawa, (3) proof of the fact that the Hecke field of a geometric Galois represntation is often (say, with density 1 primes, in certain cases) generated by the trace of the Frobenius for a single finite prime, (4) an upper bound of the number of the connected components of the Zariski closure of the image of a Galois representation.
• Ramified components of automorphic representations: local theory and its application to special L-values

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

  Date (from‐to) : 2012/04 -2015/03 

  Author : ISHIKAWA YOSHI-HIRO, TSUZUKI Masao, YASUDA Seidai, TAKANO Keiji, MIYAUCHI Michitaka

   

  Number theory investigation usually involves quite vast area of deep mathematics, like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L-/ε-factors. Our approach is resorting to integralpresentations of L-function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are in the case of Real/unramified U(3). As an application to the global problem, we got algebraicity result for all the critical values of twisted L-function of generic cuspidal representaions on U(3).
• Study of various aspects of Galois representations

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

  Date (from‐to) : 2010 -2012 

  Author : TAGUCHI Yuichiro, SAITO Takeshi, HIRANOUCHI Toshiro, YASUDA Seidai, HATTORI Shin, MIEDA Yoichi

   

  We obtained several useful results on Galois representations. In particular, for a geometric Galois representation of a complete discrete valuation fields with imperfect residue field, we proved, under suitable conditions, that its fixed subspace over a ``large'' Kummer extension of the base field is trivial, and applied this to Iwasawa Theory. Moreover, we studied the congruence of Galois representations and obtained some results on the generalization of the Rasmussen-Tamagawa conjecture.
• Study onε-factor of automorphic representations and conductor of remified components

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

  Date (from‐to) : 2009 -2011 

  Author : ISHIKAWA Yoshihiro, MORIYAMA Tomonori, YASUDA Seidai, MIYAUCHI Michitaka, TAKANO Keiji

   

  Number theory investigation usually involves quite vast area of deep mathematics, like as the Fermat Last Theolem does. The Langlands Program, which led to the settlement of FLT, has been the central strategy of arithmetic since 70s. We follow the LP to study the ramification theory of the group U(3) representations in view point of L-/ε-factors. Our approach is resorting to integral presentations of L-function of automorphic forms, whose ramified factors give us arithmetic info. The point is to find nice Whittaker functions appearing in the ramified factor. We can successfully detect where/which the nice ones are in the case of Real/unramified U(3).
• On the special values of the product of L-functions and the periods of automorphic forms defined over function fields

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

  Date (from‐to) : 2009 -2011 

  Author : KONDO Satoshi, YASUDA Seidai

   

  We verified that when an automorphic form over a function field is integrated over the maximal torus at a fixed prime, the value may be expressed in terms of special values of L-functions, when the function field is that of a projective line over a finite field. We were not able to verify this over the function field of a more general curve.
• Towards ramification theory of automorphic representations : Ramified representations and their L-factors

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

  Date (from‐to) : 2007 -2008 

  Author : ISHIKAWA Yoshihiro, MORIYAMA Tomonori, YASUDA Seidai, YOSHINO Yuji, TAKANO Keiji, WAKATSUKI Satoshi

   

  フェルマ予想(FLT)の様な数論の問題は, 非常に広範で深い理論を駆使して研究される。FLTの証明をも含み, 70年代より数論研究の支柱たり続けているLanglandsプログラムに沿って, 比較的小さい群U(3), GSp(4)の場合に, その分岐表現と付随するL-関数を研究した。方針は, L-関数を上の群を対称性にもつ保型形式という"関数"の積分変換で表示し, その積分の分岐因子を(一般化)ホイタッカー関数を通じて明示的に研究する。表現の分岐が激しくない簡易な場合に, L-因子を計算した。分岐が激しい場合にも, 部分群からのアプローチが有効で有ることが判った。
• 数論多様体の分岐とL-関数に関する研究

  日本学術振興会：科学研究費助成事業 若手研究(B)

  Date (from‐to) : 2003 -2005 

  Author : 安田 正大

   

  1.近藤智氏との共同研究を行い、以下の成果を得た。 前年度までの共同研究に登場した、正標数の大域体F,Fの素点∞、および正標数d【greater than or equal】1に対する、F上の階数dの、適当なレベル構造つきのDrinfeldモジュラー多様体のd-次Milnor K-群の元の改良および一般化を行い、Drinfeldモジュラー多様体の無限素点でのreductionと関係するBruhat-Tits buildingの数論的商に関する、GL_dの一般のモジュラーシンボル(の関数体類似)と関係づけることができた。 まだ完成していないが、これらのモジュラーシンボルが、上記の数論的商のとあるホモロジー群を生成することが証明できる見通しが立っており、それが実元すると、Drinfeldモジュラー多様体のMilnor K-群に十分多くの元が構成できたことになる。 またd=2の場合に、上記のように構成した元を用いて、関数体上の楕円曲線のK_<2->群に十分多くの元を作る事への応用を行った(プレプリント執筆中)。この方面へ応用するというアイディアは近藤氏による。当該研究者の貢献はl-進層の消滅サイクルの理論を援用して、曲線のモデルの考察を最小限にとどめる技法を開発したところにある。 2.体上の楕円曲線EのK_1群とK_2群を、Gersten複体の部分複体を用いて記述する予想を与え、E上のベクトル束の分類およびFourier-向井変換を用いて、それを証明するための計算の主要な部分を実行した。
• 数論的多様体の分岐とL-関数

  日本学術振興会：科学研究費助成事業 特別研究員奨励費

  Date (from‐to) : 2002 -2004 

  Author : 安田 正大

   

  ArtinモチーフのTate twistに対するBloch加藤予想と関数等式とのcompatibilityについて研究した結果,それが(B^<ψ=p^γ>_∩B^+_)/Z_pt^γの構造を調べることに帰着された.Artinモチーフに対するBloch加藤予想と関数等式とのcompatibilityに関する下記の結果を,ChinbergのΩ(N/K,2)不変量に関する予想と関係づけられることがわかった.また(B^<ψ=p^γ>_∩B^+_)/Z_pt^γの構造をと,導手の理論との密接な結びつきが明らかになってきた. 一咋年に自分が得た,局所Weil群の表現に対するε_0-因子の構成に関する結果が改良された,当時の結果では,係数環が剰余体が代数閉体の局所環であって,p-乗写像が全射となるものに対してしか,ε_0-定数が構成されていなかった.が、加法指標の値域を係数環と分離することにより,pが加逆となる,一般の可換noether環を係数環とする表現に対しても,同様にε_0-因子の理論が作れることがわかった. 加藤和也氏により構成されているp-進ε-元の(ψ,Γ)-加群の視点からの見直しを行った結果,rank 1の表現に対する加藤氏のp-進ε-元は,一見Coleman巾級数を用いた,技巧的な方法を用いて構成されているように見えるが,(ψ,Γ)-加群の立場から見ると,p-進ε-元は,固定した1のp-巾根のsystem ε=(ζ_)から作られる元[ε]∈Aに1∈Q_pを送ることにより得られるアーベル群の準同型Q_p→A^×を,通常の加法指標の類似と思い,Tateによるε-因子の構成と同様の構成を実施して構成したものである,という自然な見方ができることがわかった. 係数をp-加逆な局所環に一般化したところでの,Langlands対応の問題は,定式化をすることがまず困難であるという問題があることがわかった.不分岐なところで考えると,表現そのものではなく,表現行列の固有多項式しか問題にしていない感が強い.Tameの部分に何らかの対応らしいものを見出すことが勝負だと思われる.ε-因子はそもそも表現行列の固有多項式にしか依存しないことも判明した. 対応の確立のためには,tameな場合が本質的であると思われるが,それには,Bushnell, Kutzkoのtypeの理論を用いた,ε-因子の構成の理論(Bushnell, Henniart)と,自分のε_0-元の構成との関連をもっと追う必要があろう.