岡崎　武生OKAZAKI Takeoオカザキ　タケオ

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プロフィール情報

• 姓

  岡崎, オカザキ

• 名

  武生, タケオ

学位

• 理学博士

研究キーワード

• 保型形式, 代数多様体, 整数論

研究分野

• 自然科学一般, 代数学

学歴

• 1999年04月, 2004年03月, 大阪大学, 理学研究科, 数学
• 1994年04月, 1998年03月, 東京大学

担当経験のある科目（授業）

• 代数続論, 奈良女子大学
• 現代数学概説, 奈良女子大学
• 代数概論, 奈良女子大学
• 数物通論１, 奈良女子大学
• 数物の展開, 奈良女子大学
• 線形代数学Ⅱ（A), 奈良女子大学
• 数学の展開, 奈良女子大学
• 数物の歩き方, 奈良女子大学
• 数学通論Ⅰ, 奈良女子大学
• 複合自然構造特論I, 奈良女子大学
• 保型表現論と代数学 演習, 奈良女子大学
• 代数入門演習, 奈良女子大学
• 線形代数学III, 奈良女子大学
• 数学通論I, 奈良女子大学
• 線形代数II, 奈良女子大学
• 保型表現論と代数学演習, 奈良女子大学
• 代数入門 演習, 奈良女子大学
• 代数入門, 奈良女子大学
• 線型代数学Ⅱ, 奈良女子大学
• 保型表現論と代数学, 奈良女子大学
• 体とガロア理論, 奈良女子大学
• 微分積分学概論Ⅰ(Ａ), 奈良女子大学

Ⅱ．研究活動実績

論文

• 査読あり, 英語, Advances in Theoretical and Mathematical Physics, On some Siegel threefold related to the tangent cone of the Fermat quartic surface, Takeo Okazaki; Takuya Yamauchi, Let Z be the quotient of the Siegel modular threefold Asa(2, 4, 8) which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple FZ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential (2, 0)-form on Z. The variety Z is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of Z. One of these, denoted by W, is a kind of Igusa compactification such that the boundary ∂W is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform g of weight 3. Then we construct an automorphic representation π of GSp2(A) related to g and an explicit vector EZ sits inside π which creates a vector valued (non-cuspidal) Siegel modular form of weight (3, 1) so that FZ coincides with EZ in H2,0(∂W) under the Poincaré residue map and various identifications of cohomologies., 2017年, 21, 3, 585, 630, 研究論文（学術雑誌）, 10.4310/ATMP.2017.v21.n3.a1

  DOI URL
• 査読あり, 英語, AMERICAN JOURNAL OF MATHEMATICS, ENDOSCOPIC LIFTS TO THE SIEGEL MODULAR THREEFOLD RELATED TO KLEIN'S CUBIC THREEFOLD, Takeo Okazaki; Takuya Yamauchi, Let A(11)(lev) be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let x be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Pi(i), 0 <= i <= 4 from two elliptic modular forms f circle times x(i) of weight 2 and g circle times x(i) of weight 4 with complex multiplication by Q(root-11) such that Pi(i infinity) gives a non-holomorphic differential form on A(11)(lev) for each i, 0 <= i <= 4. Then their spinor L-functions are of form L(s - 1, f circle times x(i))L(s,g circle times x(i)) such that L(s,g circle times x(i)) does not appear in the L-function of A(11)(lev) for any i, 0 <= i <= 4. The existence of such lifts is motivated by the computation of the L-function of Klein's cubic threefold which is a birational smooth model of A(11)(lev)., 2013年02月, 135, 1, 183, 205, 研究論文（学術雑誌）
• 査読あり, 英語, J. Number Theory, 合同部分群ガンマ（2,4,8）に関するL-関数, 岡崎武生, 2012年, 132, 54-78
• 査読あり, 英語, Math. Ann, Saito-Kurokawa type lift to $S_3(\Gamma^{1,3}(2))$, 岡崎武生, 2008年, 208, 589-601
• 査読あり, 英語, JOURNAL OF NUMBER THEORY, L-functions of S-3(Gamma(4, 8)), Takeo Okazaki, We prove most of B. van Geemen and D. van Straten's conjectures on the explicit description of Andrianov L-functions of Siegel cuspforms of degree 2 of weight 3 for the group Gamma(4, 8), which are contained in [B. van Geemen, D. van Straten, The cuspform of weight 3 on Gamma(2) (2, 4, 8), Math. Comp. 61 (204) (1993) 849-872]. These L-functions are related to the Galois representations on the Siegel modular threefold Gamma(4,8) \ h(2) as determined by B. van Geemen and N. Nygaard [B. van Geemen, N.O. Nygaard, On the geometry and arithmetic of some Siegel modular threefolds, J. Number Theory 53 (1995) 45-87]. (c) 2006 Elsevier Inc. All rights reserved., 2007年07月, 125, 1, 117, 132, 研究論文（学術雑誌）, 10.1016/j.jnt.2006.11.005

  DOI URL
• 査読あり, 英語, AMERICAN JOURNAL OF MATHEMATICS, Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces, T Okazaki, This paper gives complete proofs of R. Salvati Manni and J. Top's conjectures which are contained in the paper, "Cusp forms of weight 2 for the group Gamma(4, 8), Amer J Math. 115 (1993), 455-486." One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over Q of the hyper-elliptic curve y(2) = x(5) - x equals the Andrianov L-function of Siegel cusp form Theta of degree 2 and weight 2 which is four products of the Igusa theta constants. Regarding the Theta as a function obtained by the Yoshida lifting from a pair of elliptic modular forms, we prove the conjecture., 2006年02月, 128, 1, 139, 165, 研究論文（学術雑誌）

MISC

• 査読無し, 英語, ADVANCES IN THEORETICAL AND MATHEMATICAL PHYSICS, INT PRESS BOSTON, INC, On some Siegel threefold related to the tangent cone of the Fermat quartic surface, Takeo Okazaki; Takuya Yamauchi, Let Z be the quotient of the Siegel modular threefold A(sa)(2, 4, 8) which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple F-Z of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential (2,0)-form on Z. The variety Z is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of Z. One of these, denoted by W, is a kind of Igusa compactification such that the boundary. W is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform g of weight 3. Then we construct an automorphic representation. of GSp(2)(A) related to g and an explicit vector E-Z sits inside. which creates a vector valued (non-cuspidal) Siegel modular form of weight (3, 1) so that F-Z coincides with E-Z in H-2,H-0(partial derivative W) under the Poincare residue map and various identifications of cohomologies., 2017年, 21, 3, 585, 630
• 査読無し, その他, Advances in Theoretical and Mathematical Physics, フェルマー４次曲面の接錐に関するジーゲル３次元多様体について, OKAZAKI Takeo, 2017年, 21, 3
• 査読無し, 英語, AMERICAN JOURNAL OF MATHEMATICS, JOHNS HOPKINS UNIV PRESS, ENDOSCOPIC LIFTS TO THE SIEGEL MODULAR THREEFOLD RELATED TO KLEIN'S CUBIC THREEFOLD (vol 135, pg 183, 2013), Takeo Okazaki; Takuya Yamauchi, 2015年08月, 137, 4, 1147, 1147, その他, 10.1353/ajm.2015.0030

  DOI URL
• 査読無し, 英語, American Journal of Mathematics, Endoscopic lifts to the siegel modular threefold related to Klein's cubic threefold, Takeo Okazaki; Takuya Yamauchi, Let Alev A11lev be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let χ be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Πi, 0≤i≤4 from two elliptic modular forms f ⊗χi of weight 2 and g⊗Xi of weight 4 with complex multiplication by Q(√ -11) such that Πi∞ gives a non-holomorphic differential form on Alev A11lev for each i, 0 ≤ i ≤ 4. Then their spinor L-functions are of form L(s-1,f ⊗χi)L(s,g⊗χi) such that L(s,g⊗χi) does not appear in the L-function of Alev A11lev for any i, 0 ≤ i ≤ 4. The existence of such lifts is motivated by the computation of the L-function of Klein's cubic threefold which is a birational smooth model of Alev A11lev., 2013年02月, 135, 1, 183, 205, 10.1353/ajm.2013.0002

  DOI URL
• 査読無し, 英語, AMERICAN JOURNAL OF MATHEMATICS, JOHNS HOPKINS UNIV PRESS, ENDOSCOPIC LIFTS TO THE SIEGEL MODULAR THREEFOLD RELATED TO KLEIN'S CUBIC THREEFOLD, Takeo Okazaki; Takuya Yamauchi, Let A(11)(lev) be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let x be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Pi(i), 0 <= i <= 4 from two elliptic modular forms f circle times x(i) of weight 2 and g circle times x(i) of weight 4 with complex multiplication by Q(root-11) such that Pi(i infinity) gives a non-holomorphic differential form on A(11)(lev) for each i, 0 <= i <= 4. Then their spinor L-functions are of form L(s - 1, f circle times x(i))L(s,g circle times x(i)) such that L(s,g circle times x(i)) does not appear in the L-function of A(11)(lev) for any i, 0 <= i <= 4. The existence of such lifts is motivated by the computation of the L-function of Klein's cubic threefold which is a birational smooth model of A(11)(lev)., 2013年02月, 135, 1, 183, 205, 10.1353/ajm.2013.0002

  DOI URL
• 査読無し, その他, J. Number Theory, 合同部分群ガンマ（2,4,8）に関するL-関数, 岡崎 武生, 2012年, 132, 54, 78
• 査読無し, 英語, JOURNAL OF NUMBER THEORY, ACADEMIC PRESS INC ELSEVIER SCIENCE, L-functions of S-3 (Gamma(2)(2, 4, 8)), Takeo Okazaki, van Geemen and van Straten [B. van Geemen. D. van Straten, The cuspform of weight 3 on Gamma(2) (2, 4, 8), Math. Comp. 61 (1993) 849872] showed that the space of Siegel modular cusp forms of degree 2 of weight 3 with respect to the so-called Igusa group Gamma(2)(2, 4,8) is generated by 6-tuple products of Igusa theta constants, and each of them are Hecke eigenforms. They conjectured that some of these products generate Saito-Kurokawa representations, weak endoscopic lifts, or D-critical representations. In this paper, we prove these conjectures. Additionally, we obtain holomorphic Hermitian modular eigenforms of GU(2, 2) of weight 4 from these representations. (C) 2011 Elsevier Inc. All rights reserved., 2012年01月, 132, 1, 54, 78, 10.1016/j.jnt.2011.06.015

  DOI URL
• 査読無し, 英語, MATHEMATISCHE ANNALEN, SPRINGER, A Siegel modular threefold and Saito-Kurokawa type lift to S-3(Gamma(1,3)(2)), Takeo Okazaki; Takuya Yamauchi, Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Gamma(1,3)(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Gamma(0)(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915-937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Gamma(1,3)(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3., 2008年07月, 341, 3, 589, 601, 10.1007/s00208-007-0204-1

  DOI URL
• 査読無し, 英語, MATHEMATISCHE ANNALEN, SPRINGER, A Siegel modular threefold and Saito-Kurokawa type lift to S-3(Gamma(1,3)(2)), Takeo Okazaki; Takuya Yamauchi, Hulek and others conjectured that the unique differential three-form F (up to scalar) on the Siegel threefold associated to the group Gamma(1,3)(2) comes from the Saito-Kurokawa lift of the elliptic newform h of weight 4 for Gamma(0)(6). This F have been already constructed as a Borcherds product (cf. Gritsenko and Hulek in Int Math Res Notices 17:915-937, 1999). In this paper, we prove this conjecture by using the Yoshida lift and we settle a conjecture which relates our theorem. A remarkable fact is that the Yoshida lift using the usual test function cannot give the Saito-Kurokawa type lift of weight 3 associated to the group Gamma(1,3)(2). So important task is to find special test functions for the Yoshida lift at the bad primes 2 and 3., 2008年07月, 341, 3, 589, 601, 10.1007/s00208-007-0204-1

  DOI URL
• 査読無し, その他, J. Number theory, On L-functions of $S_3(\Gamma_2(4,8))$, 岡崎 武生, 2007年, 125, 117, 132
• 査読無し, その他, J. Number theory, On L-functions of $S_3(\Gamma_2(4,8))$, OKAZAKI Takeo, 2007年, 125, 117, 132
• 査読無し, 英語, American Journal of Mathematics, Proof of R. Salvati Manni and J. Top's conjectures on siegel modular forms and abelian surfaces, Takeo Okazaki, This paper gives complete proofs of R. Salvati Manni and J. Top's conjectures which are contained in the paper, "Cusp forms of weight 2 for the group Γ(4,8), Amer. J. Math. 115 (1993), 455-486." One of the conjectures claims that the Hasse-Weil zeta function corresponding to the Jacobian variety defined over ℚ of the hyper-elliptic curve y2 = x5 - x equals the Andrianov L-function of Siegel cusp form Θ Θ of degree 2 and weight 2 which is four products of the Igusa theta constants. Regarding the Θ as a function obtained by the Yoshida lifting from a pair of elliptic modular forms, we prove the conjecture., 2006年02月, 128, 1, 139, 165, 10.1353/ajm.2006.0008

  DOI URL
• 査読無し, その他, Amer. J. Math, Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces, 岡崎 武生, 2006年, 128, 139, 165, 10.1353/ajm.2006.0008

  DOI URL
• 英語, 数理解析研究所講究録, 京都大学, New Forms for $GU(2,2)$ (Modular forms and automorphic representations), 岡崎 武生, 2015年11月, 1973, 131, 140

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• 英語, 数理解析研究所講究録, 京都大学, Paramodular form on $GSp(2, \mathbb{A})$ (Automorphic representations, automorphic $L$-functions and arithmetic), 岡崎 武生, 2009年07月, 1659, 27, 36

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• 英語, 数理解析研究所講究録, 京都大学, $\theta$-correspondence for PGSp(4) and PGU(2,2) (Automorphic forms, trace formulas and zeta functions), 岡崎 武生, 2011年10月, 1767, 96, 102

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• 英語, 数理解析研究所講究録, 京都大学, Base change lift type spinor L-functrion of $GSp_2(\mathbb{Q})$ (Automorphic Representations, Automorphic Forms, L-functions, and Related Topics), 岡崎 武生, 2008年10月, 1617, 187, 192

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講演・口頭発表等

• 岡崎武生, 東京電機大学 第６回 数学講演会, いくつかの代数群における新形式理論, 2017年03月, 日本語, 国内会議
• 岡崎武生, RIMS研究集会「モジュラー形式と保型表現」, NewForm for GU(2,2), 2015年02月, 英語
• 岡崎武生, 早稲田大学整数論シンポジウム, SU(2,2)の三つ子幾何的保型形式, 2012年03月, 日本語, 坂田裕, 早稲田大学, 国内会議
• 岡崎武生, p-進表現論勉強会, GSp(4)の新形式, 2012年02月, 日本語, 原下 秀士, 国内会議

共同研究・競争的資金等の研究課題

• 保型形式と岩澤理論, 0, 0, 0, 競争的資金
• 基盤研究(C), 2015年04月01日, 2020年03月31日, 15K04783, 保型形式のNewForm理論と岩澤理論への応用, 岡崎 武生, 日本学術振興会, 科学研究費助成事業, 奈良女子大学, 4550000, 3500000, 1050000, 次数2のSiegel 保型形式のNewformとは何であるべきか?どのような性質をもっているのか?といった問題に取り組んだ.この問題は局所体上の代数群GSp(4)の既約 許容表現論の問題である. Roberts, Schmidt氏 [2007 年] によって構築されたPGSp(4)のgeneric既約許容表現(Whittaker 模型を持つ表現)のNewform理論をGSp(4)に拡張し, また, 非generic-表現である斎藤-黒川リフトにおいて, Bessel 模型のNewform理論を構築した., kaken

  対象リソースIRI
• 若手研究(B), 2007年, 2007年, 19740014, 代数多様体と保型形式の関係や構成, 岡崎 武生, 日本学術振興会, 科学研究費助成事業, 大阪大学, 1000000, 1000000, kaken

  対象リソースIRI
• 若手研究(B), 2012年04月01日, 2015年03月31日, 24740017, 保型形式、代数多様体と岩澤理論の関係, 岡崎 武生; 山内 卓也, 日本学術振興会, 科学研究費助成事業, 奈良女子大学, 2600000, 2000000, 600000, 符号(2,2)のユニタリー群上保型表現の関数等式を与え, 新形式理論を構築した. 新形式を固定する離散群として, ディーパラモジュラー群を定義した. 特に, 符号(2,2)のユニタリー群上の保型表現がジーゲル保型形式とテータ対応している場合, ディーパラモジュラー群で固定されるシャライカピリオドを持つことを示した. ジーゲル保型形式とのテータ対応と新形式理論を使うことにより, ファンーヒーメンとファン-ストラテンのジーゲル保型形式に関する予想も解決した., kaken

  対象リソースIRI