Organization
School of Science Professor
Title
Professor
External link
OCHIAI TADASHI
Degree
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博士（数理科学） ( 東京大学 )
Research Interests
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Iwasawa theory
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Hida theory
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p-adic L-function
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Selmer group
Research Areas
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Natural Science / Algebra / 数論幾何学/代数的整数論
Professional Memberships
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THE MATHEMATICAL SOCIETY OF JAPAN
Papers
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Endoscopic congruences modulo adjoint $L$-values for $\mathrm{GSp}(4)$ Reviewed International journal
Tadashi Ochiai, Francesco Lemma
Kyoto Journal of Mathematics 2021
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Control theorem and functional equation of Selmer groups over p-adic Lie extensions Reviewed
Somnath Jha, Tadashi Ochiai
Selecta Mathematica 26 ( 80 ) 1 - 58 2020.11
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Specialization method in Krull dimension two and Euler system theory over normal deformation rings Reviewed
Tadashi Ochiai, Kazuma Shimomoto
Annales Mathematiques du Quebec 1 - 53 2019.4
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The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplication Reviewed
Takashi Hara, Tadashi Ochiai
Kyoto Journal of Mathematics 58 ( 1 ) 1 - 100 2018.4
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On twists of modules over noncommutative Iwasawa algebras Reviewed
Somnath Jha, Tadashi Ochiai, Gergely Zabradi
ALGEBRA & NUMBER THEORY 10 ( 3 ) 685 - 694 2016
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BERTINI THEOREM FOR NORMALITY ON LOCAL RINGS IN MIXED CHARACTERISTIC (APPLICATIONS TO CHARACTERISTIC IDEALS) Reviewed
Tadashi Ochiai, Kazuma Shimomoto
NAGOYA MATHEMATICAL JOURNAL 218 125 - 173 2015.6
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THE COLEMAN MAP FOR HIDA FAMILIES OF GSp(4) Reviewed
Francesco Lemma, Tadashi Ochiai
AMERICAN JOURNAL OF MATHEMATICS 136 ( 3 ) 729 - 760 2014.6
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Coleman map for families of ordinary Galois representations Invited
Tadashi Ochiai
Symposium on Arithmetic Geometry 2012.10
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Control theorems for Selmer groups of nearly ordinary deformations Reviewed
Olivier Fouquet, Tadashi Ochiai
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 666 163 - 187 2012.5
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SEVERAL VARIABLES p-ADIC L-FUNCTIONS FOR HIDA FAMILIES OF HILBERT MODULAR FORMS Reviewed
Tadashi Ochiai
DOCUMENTA MATHEMATICA 17 807 - 849 2012
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Jan Nekovář: Selmer Complexes, Astérisque, 310, Soc. Math. France Inst., 2006年，559ページ． Reviewed
落合 理
数学 63 ( 4 ) 453 - 458 2011
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Iwasawa Theory for nearly ordinary Hida deformations of Hilbert modular forms Invited Reviewed
Tadashi Ochiai
Proceeding of Algebraic Number Theory and Related Topics 2008 2010.4
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NOTES ON NON-COMMUTATIVE IWASAWA THEORY Reviewed
Yoshitaka Hachimori, Tadashi Ochiai
ASIAN JOURNAL OF MATHEMATICS 14 ( 1 ) 11 - 17 2010.3
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On the Selmer groups of abelian varieties over function fields of characteristic p > 0 Reviewed
Tadashi Ochiai, Fabien Trihan
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY 146 23 - 43 2009.1
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Iwasawa Theory for Hida deformations with complex multiplication Invited
Tadashi Ochiai
Oberwolfach Reports (for Algebraiche Zahlentheorie 2007) 1778 - 1781 2008.7
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The algebraic p-adic L-function and isogeny between families of Galois representations Reviewed
Tadashi Ochiai
JOURNAL OF PURE AND APPLIED ALGEBRA 212 ( 6 ) 1381 - 1393 2008.6
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Haruzo Hida: p-adic Automorphic Forms on Shimura Varieties, Springer Monogr. Math., Springer, 2004年，390ページ． Reviewed
落合 理
数学 60 ( 3 ) 324 - 329 2008
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On the two-variable Iwasawa main conjecture Reviewed
Tadashi Ochiai
COMPOSITIO MATHEMATICA 142 ( 5 ) 1157 - 1200 2006.9
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p-adic L-functions for Galois deformations and related problems on periods Invited
OCHIAI,Tadashi
Conference proceeding for autumn workshop of Number theory 2006 2006.4
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Euler system for Galois deformations Reviewed
T Ochiai
ANNALES DE L INSTITUT FOURIER 55 ( 1 ) 113 - + 2005
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A generalization of the Coleman map for HIDA deformations Reviewed
T Ochiai
AMERICAN JOURNAL OF MATHEMATICS 125 ( 4 ) 849 - 892 2003.8
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Control theorem for Greenberg's Selmer groups of Galois deformations Reviewed
T Ochiai
JOURNAL OF NUMBER THEORY 88 ( 1 ) 59 - 85 2001.5
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Control theorem for Bloch-Kato's Selmer groups of p-adic representations Reviewed
T Ochiai
JOURNAL OF NUMBER THEORY 82 ( 1 ) 69 - 90 2000.5
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l-independence of the trace of monodromy Reviewed
T Ochiai
MATHEMATISCHE ANNALEN 315 ( 2 ) 321 - 340 1999.10
Books
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現代整数論の風景
落合 理（ Role： Sole author）
日本評論社 2019.5 （ ISBN:9784535788589 ）
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岩澤理論とその展望（下）
落合理（ Role： Sole author）
岩波書店 2016.8 （ ISBN:9784000298223 ）
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Algebraic Number Theory and Re- lated Topics 2013
Tadashi Ochiai（ Role： Joint editor）
2015.9
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Algebraic Number Theory and Re- lated Topics 2012
Tadashi Ochiai（ Role： Joint editor）
2014.10
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Iwasawa theory and its perspetive I
Tadashi Ochiai（ Role： Sole author）
Iwanami Syoten 2014.9 （ ISBN:9784000298216 ）
MISC
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代数的整数論とその周辺における問題設定 Invited
落合理
数理科学 2020.10
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Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms
Tadashi Ochiai
2018.2
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円分ℤp拡大から保型形式のp進族へ (特集 現象を通して見る岩澤理論)
落合 理
数学セミナー 57 ( 1 ) 26 - 31 2018.1
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Main conjectures for higher rank nearly ordinary families -- I
Kazim Büyükboduk, Tadashi Ochiai
2017.8
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数の世界の散歩道 : 整数論に導かれて(第12回)ゼータ関数の特殊値(2)
落合 理
数学セミナー 56 ( 3 ) 80 - 86 2017.3
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数の世界の散歩道 : 整数論に導かれて(第11回)ゼータ関数の特殊値(1)
落合 理
数学セミナー 56 ( 2 ) 76 - 82 2017.2
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数の世界の散歩道 : 整数論に導かれて(第10回)ゼータの進化(2)
落合 理
数学セミナー 56 ( 1 ) 70 - 76 2017.1
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数の世界の散歩道 : 整数論に導かれて(第9回)ゼータの進化(1)
落合 理
数学セミナー 55 ( 12 ) 72 - 78 2016.12
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数の世界の散歩道 : 整数論に導かれて(第8回)整数論における局所と大域
落合 理
数学セミナー 55 ( 11 ) 75 - 81 2016.11
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数の世界の散歩道 : 整数論に導かれて(第7回)代数的整数論の源流を求めて(2)
落合 理
数学セミナー 55 ( 10 ) 66 - 72 2016.10
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数の世界の散歩道 : 整数論に導かれて(第6回)代数的整数論の源流を求めて(1)
落合 理
数学セミナー 55 ( 9 ) 58 - 64 2016.9
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数の世界の散歩道 : 整数論に導かれて(第5回)ゼータの登場(2)
落合 理
数学セミナー 55 ( 8 ) 60 - 66 2016.8
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数の世界の散歩道 : 整数論に導かれて(第4回)ゼータの登場(1)
落合 理
数学セミナー 55 ( 7 ) 60 - 66 2016.7
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Coleman Map in Coleman Families
Filippo A. E. Nuccio, Tadashi Ochiai
2016.6
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数の世界の散歩道 : 整数論に導かれて(第3回)無理数を感じたい,超越数を見極めたい
落合 理
数学セミナー 55 ( 6 ) 57 - 63 2016.6
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数の世界の散歩道 : 整数論に導かれて(第2回)広がっていく数の世界(2)
落合 理
数学セミナー 55 ( 5 ) 52 - 58 2016.5
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数の世界の散歩道 : 整数論に導かれて(第1回)広がっていく数の世界(1)
落合 理
数学セミナー 55 ( 4 ) 63 - 69 2016.4
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OCHIAI Tadashi
RIMS Kokyuroku Bessatsu 19 301 - 319 2010.6
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円分塔の高みから望む夢--岩澤健吉と岩澤理論 (特集 日本の数学者たち--和算から現代数学まで)
落合 理
現代思想 37 ( 15 ) 172 - 183 2009.12
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p-adic L-functions for Galois deformations Invited
OCHIAI, Tadashi
Trends in Mathmatics 2006.6
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現代代数学の歩み(第17回)ワイルズ
落合 理
数学セミナー 45 ( 3 ) 68 - 75 2006.3
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ガロア表現の変形に対する岩澤理論の研究について (「整数論のこの主題,自分はこう考える」若手発表会)
落合 理
数理解析研究所講究録 1256 38 - 70 2002.4
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ガロア表現の変形に対する岩澤主予想とオイラー系について (代数的整数論とその周辺)
落合 理
数理解析研究所講究録 1154 10 - 21 2000.5
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局所Weil群の作用とそのトレースについて (代数的整数論とその周辺)
落合 理
数理解析研究所講究録 1097 100 - 107 1999.4
Presentations
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p-adic Artin L-function over a CM-field Invited
Tadashi Ochiai
``Elliptic curves and the special values of L-functions" （International Centre for Theoretical Sciences - TIFR/バンガロール/インド) 2022.8
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Congruence of automorpihc representations and Special values of adjoint L-functions for GSp(4) Invited
2020.6
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Endoscopic congruences and adjoint L-values for GSp(4) Invited
Tadashi Ochiai
Analytic, geometric and p-adic aspects of automorphic forms and L-functions (RIMS/Kyoto) 2020.1
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Congruences between endoscopic representations and non-endoscopic representationsof GSp(4) modulo adjoint L-values Invited
Tadashi Ochiai
Workshop on the congruence ideals and p-adic L-functions"（Academia Sinica, Taiwan) 2019.12
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Iwasawa Main Conjecture for Galois deformation spaces and glueing of Euler systems I, II Invited
Tadashi Ochiai
CMC special weeks on number theory (Nov.11-Nov.22) （ KIAS/Korea) 2019.11
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ゼータ函数の特殊値に対するp進変形のアプローチ Invited
Tadashi Ochiai
2005.8
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Greenberg's view on generalizing Iwasawa theory via Galois deformations Invited
Tadashi Ochiai
2005.6
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p-adic L-functions for Galois deformation spaces and Iwasawa main Conjecture Invited
Tadashi Ochiai
2006.1
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p-adic L-functions for Galois deformations and related problems on periods Invited
Tadashi Ochiai
2005.9
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On the two-variable Iwasawa theory Invited
Tadashi Ochiai
2004.7
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岩澤理論の一般化と関連する諸問題について Invited
Tadashi Ochiai
2004.1
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肥田変形の岩澤理論における結果と展望 Invited
Tadashi Ochiai
2004.9
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Toward a generalization of the Iwasawa theory Invited
Tadashi Ochiai
2004.8
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2変数の岩澤主予想について Invited
Tadashi Ochiai
2003.12
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ガロア表現の変形空間における数論 Invited
Tadashi Ochiai
2003.11
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岩澤理論の一般化とガロア表現の変形空間 Invited
Tadashi Ochiai
2003.10
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岩澤理論のガロア変形的な一般化について Invited
Tadashi Ochiai
2003.7
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２変数モジュラー変形の岩澤主予想とその具体例 Invited
Tadashi Ochiai
2003.6
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non-ordinaryなガロア表現の族でのBloch-Kato指数写像の補間について Invited
落合理
数学コロキウム"（豊田中央研究所) 2013.10
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Interpolation of dual exponential maps of Bloch-Kato for nearly ordinary deformations Invited
落合理
Number Theory Forum（慶応大学) 2013.3
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Coleman map for nearly ordinary Hida deformations Invited
Tadashi Ochiai
2012.1
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Coleman map for Hida deformations of GSp(4)
Ochiai, Tadashi
2011.10
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Coleman map for families of Galois representations Invited
Tadashi Ochiai
2013.3
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Coleman map for families of ordinary Galois representations Invited International conference
落合理
Symposium on ARITHMETIC GEOMETRY (九州大学) 2012.10
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p-adic L-functions over Galois deformation spaces Invited
OCHIAI, Tadashi
2010.1
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岩澤理論におけるｐ進L函数と周期積分 Invited
落合理
2009.4
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岩澤理論に現れる可換環論の問題たちについての注意 Invited
落合理
2010.6
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Towards construction of Iwasawa theory over Galois deformation spaces Invited
Ochiai, Tadashi
2010.6
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総実体の肥田変形に関する岩澤理論 Invited
落合理
2008.12
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岩澤理論の一般化について Invited
落合理
2008.9
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Iwasawa Theory for Hilbert modular Hida deformation Invited
Tadashi Ochiai
2009.3
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La théorie d'Iwasawa pour déformations galoisiennes Invited
Tadashi Ochiai
2007.1
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Iwasawa theory for Galois deformations Invited
Tadashi Ochiai
2006.12
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Théorie d'Iwasawa pour les déformations de Hida avec multiplication complexe Invited
Tadashi Ochiai
2007.6
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La conjecture principale d'Iwasawa pour déformations de Hida et la multiplication complexe Invited
Tadashi Ochiai
2007.5
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On the two-variable Iwasawa Main Conjecture Invited
Tadashi Ochiai
2006.5
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On the Selmer groups of abelian varieties over function fields of characteristic p Invited
Tadashi Ochiai
2006.7
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La théorie d'Iwasawa à deux variables Invited
Tadashi Ochiai
2006.6
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local Bertini theorem and Euler system bound over Cohen-Macaulay deformation Invited International conference
落合理
Sendai International Conference on Arithmetic Geometry 2016 2016.1
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Generalization of Euler system theory and problems on commutative algebra Invited
落合理
Departmental Seminar 0f IIT Kanpur 2015.9
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Iwasawa theory for p-adic family of modular forms Invited
Tadashi Ochiai
2016.10
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p進モジュラー形式のColeman変形族とゼータ元 Invited
落合理
早稲田大学整数論研究集会 2016 2016.3
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Familles p-adiques des representations Galoisiennes modulaires et l'interpolation des exponentielles p-adiques de Bloch-Kato Invited
落合理
Seminaire: Groupes Reductifs et Formes Automorphes 2015.3
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Familles p-adiques des representions Galoisiennes modulaires et l'interpolation des exponentielles p-adiques de Bloch-Kato Invited
Seminaire: Groupes Reductifs et Formes Automorphes 2015.3
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岩澤理論の高次元をとりまく風景（概説講演） Invited
落合理
第60回代数学シンポジウム 2015.9
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保型L函数の特殊値と付随するp進L函数について Invited
落合理
香川セミナー 2015.5
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岩澤理論の一般化と可換環論 Invited
落合理
北海道大学談話会 2015.1
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Some basic problems on critical values and periods of automorphic L-functions in p-adic contexts Invited
落合理
研究集会"保型形式の整数論・講演会 2015.1
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Coleman map for non-ordinary family Invited
Tadashi Ochiai
Number theory seminar of University of California Los Angels 2014.11
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Remark on the cyclotomic Iwasawa Main Conjecture for ordinary Hilbert cusp forms Invited International conference
落合理
Automorphic Forms and Arithmetic (satellite to the ICM, conference number 28) 2014.8
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Coleman map for non-ordinary families of modular Galois representations Invited International conference
落合理
The sixth annual Pan Asian Number Theory (satellite to the ICM, conference number 22) 2014.8
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Local Bertini Theorem and Iwasawa Theory Invited
落合理
Number theory seminar of University of Washington 2014.10
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L-values, zeta elements and Coleman map Invited
落合理
Number theory seminar of University of California Berkeley 2014.9
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On the cyclotomic Iwasawa Main Conjecture for ordinary Hilbert cusp forms Invited
落合理
Mini-Workshop on Iwasawa theory 2014.4
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Coleman map for non-ordinary p-adic modular Galois representation Invited
Ochiai, Tadashi
2014.2
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non-ordinary familyのColeman写像 Invited
落合理
東北大学数論セミナー 2014.7
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On the Iwasawa Main Conjecture for CM Hilbert cusp forms Invited International conference
落合理
p-adic aspects of modular forms 2014.6
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Coleman map for non-ordinary family Invited
落合理
Number theory seminar of University of California Los Angels 2014.1
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GSp(4) の内視的な保型表現の mod p 合同とadjoint L 関数の特殊値 Invited
落合理
2019大分佐賀整数論研究集会 2019.10
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Congruences endoscopiques et fonction L adjointe pour GSp(4) Invited
OCHIAI Tadashi
Seminaire: Groupes Reductifs et Formes Automorphes （Universite Paris 7) 2019.3
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Congruences of modular forms and special values of L-functions Invited
OCHIAI Tadashi
2019.2
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Congruence of modular forms and Special values of adjoint L-functions Invited
Tadashi Ochiai
Number theory colloquium (Indian Statistical Institute, Kolkata) 2019.8
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Endoscopic congruences and adjoint L-values for GSp(4) Invited International conference
OCHIAI Tadashi
Recent advances in the arithmetic of Galois representations （Universita di Genova/Italy) 2019.7
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La conjecture principale d'Iwasawa pour les families p-adiques de formes modulaires Invited
落合理
Seminaire de la théorie des nombres de l'Université de Caen 2018.5
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La conjecture principale d'Iwasawa pour les families de formes modulaires p-adiques
落合理
Seminaire de la théorie des nombres de l'Université de Bordeaux 2018.5
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Overview of p-adic L-function Invited International conference
OCHIAI Tadashi
Special Values of Automorphic L-functions and Associated p-adic L-Functions (Casa Mathematica Oxaca/Mexico) 2018.10
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Endoscopic congruences modulo adjoint L-values for GSp(4) Invited International conference
Tadashi Ochiai
2018.9
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La conjecture principale d'Iwasawa pour les families p-adiques des formes paraboliques Invited
落合理
Seminaire de la théorie des nombres de l'Université de Grenoble 2018.2
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Iwasawa theory for Coleman families Invited
落合理
number theory semina at Heidelberg University 2018.2
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La conjecture principale d'Iwasawa pour les déformations galoisiennes p-adiques de rang général Invited
落合理
Seminaire de la théorie des nombres de l'Université Paris 13/ 2018.3
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高階数のp擬通常的ガロワ変形の岩澤主予想について Invited
落合理
Regulators in Niseko 2017 2017.9
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Iwasawa Main conjecture for p-adic family of elliptic modular forms Invited International conference
落合理
Iwasawa 2017 2017.7
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La conjecture principale d'Iwasawa pour les déformations galoisiennes p-adiques et le systeme d'Euler Invited
落合理
Seminaire: Groupes Reductifs et Formes Automorphes 2017.10
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Iwasawa Main Conjecture for nearly ordinary deformations of higher rank Invited International conference
落合理
workshop on p-adic L-functions and algebraic cycles 2017.9
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Special values of L-funcctions and p-adic L-function for modular forms Invited
落合理
Departmental Semina of IIT Kanpur 2016.10
-
Iwasawa theory for p-adic family of modular forms Invited
Tadashi Ochiai
2017.5
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岩澤理論の高次元化と関連する諸問題 Invited
落合理
京都大学談話会 2017.1
Research Projects
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代数的サイクル、エンドスコーピー、BSD予想
Grant number：23KF0220 2023.11 - 2026.3
日本学術振興会 科学研究費助成事業 特別研究員奨励費
落合 理, CAUCHI ANTONIO
Grant amount：\1800000 （ Direct Cost: \1800000 ）
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非可換化を軸にした数論幾何学の新しい展開
Grant number：23K17651 2023.6 - 2026.3
日本学術振興会 科学研究費助成事業 挑戦的研究(萌芽)
落合 理
Grant amount：\6240000 （ Direct Cost: \4800000 、 Indirect Cost：\1440000 ）
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変形理論、非可換化の融合による岩澤理論の新展開
Grant number：23K25763 2023.4 - 2028.3
日本学術振興会 科学研究費助成事業 基盤研究(B)
落合 理, 原 隆, 下元 数馬, 田口 雄一郎
Grant amount：\17940000 （ Direct Cost: \13800000 、 Indirect Cost：\4140000 ）
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新しいp進L関数の探求と派生する保形L関数の問題
Grant number：18K18711 2018.6 - 2023.3
日本学術振興会 科学研究費助成事業 挑戦的研究(萌芽)
落合 理
Grant amount：\6110000 （ Direct Cost: \4700000 、 Indirect Cost：\1410000 ）
今年度は特に次の計画を中心に研究活動を進めた.
(I) Eisenstein級数のp進変形族とp進L関数, (II) 新しいEuler系の構成の探求, (III) CM体 における多変数岩澤理論や非可換岩澤理論, (VI) non-ordinaryなp進変形族のp進L関数の構成とその応用
(I)に関しては, 昨年度に引き続いてHarrisやSkinnerらによる総実代数体上のユニタリ群のEisenstein級数のp進変形族を考察した. (II)に関しては, 共同研究者であるFrancesco Lemma氏とともに特にlocal system がconstantとなる一番低いcohomological weightの場合にexplicit reciprocity lawの研究を進めた. (III)に関しては, CM体の非可換岩澤理論 については共同研究者の原氏と共に, CM体の絶対ガロワ群のArtin表現に付随するp進L関数の構成と岩澤主予想の証明の細部を詰めながら論文の執筆を進めた. (VI)に関しては, 応用を考えている形でPerrin-RIouやAmice-Veluによる古典的な結果の使いにくい点が気になりadmissible measureの理論の再構成を追求した. -
Algebraic cycles and Iwasawa theory
Grant number：17H02836 2017.4 - 2022.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Grant amount：\15470000 （ Direct Cost: \11900000 、 Indirect Cost：\3570000 ）
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Generalization of Iwasawa theory for Galois deformation and related new phenomena (Fostering Joint International Research)
Grant number：16KK0100 2017 - 2018
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Fund for the Promotion of Joint International Research (Fostering Joint International Research)
Ochiai Tadashi, Lemma Francesco
Grant amount：\11700000 （ Direct Cost: \9000000 、 Indirect Cost：\2700000 ）
(1) We establish the existence of congruences between an endoscopic cuspidal automorphic representation of GSp(4) of level 1 and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at 1 of the adjoint L-function of normalized by a suitable period.
(2) We studied the construction of Euler system associated to cuspidal representations of Gap(4). We obtained partial results on norm compatibility of certain motivic elements. -
Generalization of Iwasawa theory through Galois doformation and search for new phenomena
Grant number：26287005 2014.4 - 2020.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Ochiai Tadashi
Grant amount：\16250000 （ Direct Cost: \12500000 、 Indirect Cost：\3750000 ）
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. -
Explicit study of number theory of automorphic forms of several variables related to trace formulas.
Grant number：25247001 2013.4 - 2018.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
Ibukiyama Tomoyoshi, Hayashida Shuichi, Watanabe Takao, Moriyama Tomonori, Ochiai Tadashi, Ikeda Tamotsu
Grant amount：\25090000 （ Direct Cost: \19300000 、 Indirect Cost：\5790000 ）
In study of the number theory, functions called automorphic forms are important objects which describe various arithmetic properties. We gave many results on these in this project. A theory of automorphic differential operators includes the classical theory of special functions such as Gegenbauer functions, and in our study, we gave three different complete constructions for these operators, which can be regarded as final results. We also gave explicit solutions of associated system of differential equations. We gave conjectures on comparison of
automorphic forms belonging to different domains, and structure theorems of automorphic forms by using constructive methods including automorphic differential operators. Some dimension formulas of automorphic forms and trace formulas of arithmetically important Hecke operators were given, and we clarified their relations to arithmetic invariants in some algebraic geometry over finite fields. -
Developement of Mathematics Arising from Generalization of Iwasawa Theory
Grant number：24654004 2012.4 - 2015.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research
OCHIAI tadashi
Grant amount：\3900000 （ Direct Cost: \3000000 、 Indirect Cost：\900000 ）
In this research, we studied mainly (1) behaviour of mixed characteristic complete local rings under specialization (joint with Shimomoto), (2) Iwasawa Main Conjecture for Hilbert modular forms with complex multiplication (joint with Hara), (3) Coleman map for non ordinary Galois deformation (joint with Nuccio). For (1), we finished the first paper and it was recently accepted. We are now preparing the second paper which treats arithmetic applications of the first paper. For (2), we are now in the final revision of the first paper. For (3), we are trying to fill some missing details of the paper and we hope to finish the first paper soon.
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Explicit study of arithmetic structures of automorphic forms of several variables and trace formulas
Grant number：21244001 2009.4 - 2014.3
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (A)
IBUKIYAMA Tomoyoshi, WAKATSUKI Satoshi, KATSURADA Hidenori, SATO Fumihiro, IKEDA Tamotsu, ODA Takayuki, SAITO Kyoji, NAGAOKA Shoyu, OCHIAI Tadashi, MORIYAMA Tomonori, WATANABE Takao, SUGANO Takashi, BANNAI Eiichi
Grant amount：\26520000 （ Direct Cost: \20400000 、 Indirect Cost：\6120000 ）
In study of the number theory, functions called automorphic forms and their L functions are important objects which describe various arithmetic properties. We gave many results on these in this project. Among others, we gave structure theorems and dimension formulas of some automorphic forms, and based on these, we proposed three precise new conjectures which claim that several spaces of automorphic forms whose definitions and origins are completely different have essentially the same arithmetical structures. We also gave three new lifting conjectures which claim that some parts of such spaces of automorphic forms come from easier spaces. Using the theory we exploited on special functions associated with differential operators, we newly calculated explicit values of some L functions, and gave a proof of Zagier's conjecture in 1976 and some number-theoretic applications. Together with other newly obtained results, these contribute to the development of the theory of automorphic forms.
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On various generalizations of Iwasawa theory
Grant number：20740013 2008 - 2011
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)
OCHIAI Tadashi
Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）
We obtained several results on generalized Iwasawa theory especially on the function-field analogue of Iwasawa theory studying abelian varieties over global fields of positive characteristic and the deformation-theoretic generalization of Iwasawa theory. For the algebraic side of Iwasawa theory, we proved the control theorem to study the behavior of Selmer groups on Hida deformations of Hilbert modular forms. For the analytic side of the theory, we constructed the p-adic L-functions in several variables on Hida deformations of Hilbert modular forms. We also worked on the Hida deformations of Siegel modular forms. We constructed the Coleman. Perrin-Riou map which interpolates dual exponential maps on the deformation. We also generalized our result on Euler system for deformation spaces removing the assumption that the deformation ring is regular.
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Refined Iwasawa theory and the zeta functions
Grant number：17340010 2005 - 2008
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
KURIHARA Masato, OTA Katsuhiro, TANAKA Takaaki, MIYAZAKI Takuya, SHIOKAWA Iekata, BANNAI Kenichi, MATSUNO Kazuo, HACHIMORI Yoshitaka, AOKI Miho
Grant amount：\9710000 （ Direct Cost: \8600000 、 Indirect Cost：\1110000 ）
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ガロア表現の変形空間における数論的不変量の変動の研究
Grant number：16740014 2004 - 2006
日本学術振興会 科学研究費助成事業 若手研究(B)
落合 理
Grant amount：\2900000 （ Direct Cost: \2900000 ）
本年度は変形空間の岩澤理論の研究に知見を加えるべく2変数p-進L函数と2変数セルマー群の詳細な性質を調べた.特に2変数P-進L函数に関しては北川氏やGreenberg-Stevensによるモジュラーシンボルを用いた構成と自分が以前に得たオイラー系を経由した構成との関係を適当な条件の下で明らかにすることがすることができ理解を一段階深めたと言える.2変数セルマー群に関しても関係する様々な方向への1変数特殊化での振る舞いを調べるコントロール定理を追求した.このような特殊化でのセルマー群の振る舞いを調べるコントロール定理はある程度一般的に前の仕事で調べてはいた.しかし肥田変形という特殊な状況に限る代わりに現れる全ての局所項を完全に決定して使える形にすることには労力と工夫を要した.またこの仕事によって目指すべき様々な(円分変形とは異なる)1変数変形空間に対する岩澤主予想の正しいであろう予想の定式化を得ることができた.得られた定式化においては当初の見込みとは若干異なった現象も見受けられ面白さを感じている.
そのほか正標数の函数体の上のアーベル多様体のセルマー群を調べることでCoatesによる代数体の上のアーベル多様体の非可換岩澤理論の函数体の場合での類似をどこまで検証できるかを調べる研究に着手して部分的な結果を得たいくらかの場合にはセルマー群がよい"ねじれ的性質"をもつことが確かめられた.一般に函数体においては代数体では使えない強い性質がしばしば成り立つこともあり,この方向の研究を進展させてより一般的な性質を追求することで非可換岩澤理論にも新しい手がかりを与えたい.