■Faculty | Faculty Division of Natural Sciences Research Group of Mathematics | |||

■Position | Associate Professor |

Last Updated :2024/02/22

#### Name (Japanese)

Okazaki#### Name (Kana)

Takeo

- Apr. 1999, Mar. 2004, Osaka University, Graduate School, Division of Natural Science, 数学
- Apr. 1994, Mar. 1998, The University of Tokyo

- Refereed, Advances in Theoretical and Mathematical Physics, International Press of 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 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, 21, 3, 585, 630, Scientific journal, 10.4310/ATMP.2017.v21.n3.a1
- Refereed, 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)., Feb. 2013, 135, 1, 183, 205, Scientific journal
- Refereed, J. Number Theory, $L$-functions of $S_3(\G(2,4,8))$, OKAZAKI Takeo, 2012, 132, 54-78
- Refereed, Math. Ann, Saito-Kurokawa type lift to $S_3(\Gamma^{1,3}(2))$, OKAZAKI Takeo; T. Yamauchi, 2008, 208, 589-601
- Refereed, JOURNAL OF NUMBER THEORY, ACADEMIC PRESS INC ELSEVIER SCIENCE, 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., Jul. 2007, 125, 1, 117, 132, Scientific journal, 10.1016/j.jnt.2006.11.005
- Refereed, AMERICAN JOURNAL OF MATHEMATICS, JOHNS HOPKINS UNIV PRESS, 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., Feb. 2006, 128, 1, 139, 165, Scientific journal

- Not Refereed, 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, 21, 3, 585, 630
- Not Refereed, 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, 21, 3
- Not Refereed, 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, Aug. 2015, 137, 4, 1147, 1147, Others, 10.1353/ajm.2015.0030
- Not Refereed, 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., Feb. 2013, 135, 1, 183, 205, 10.1353/ajm.2013.0002
- Not Refereed, 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)., Feb. 2013, 135, 1, 183, 205, 10.1353/ajm.2013.0002
- Not Refereed, 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, 132, 54, 78
- Not Refereed, 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., Jan. 2012, 132, 1, 54, 78, 10.1016/j.jnt.2011.06.015
- Not Refereed, 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., Jul. 2008, 341, 3, 589, 601, 10.1007/s00208-007-0204-1
- Not Refereed, 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., Jul. 2008, 341, 3, 589, 601, 10.1007/s00208-007-0204-1
- Not Refereed, J. Number theory, On L-functions of $S_3(\Gamma_2(4,8))$, 岡崎 武生, 2007, 125, 117, 132
- Not Refereed, J. Number theory, On L-functions of $S_3(\Gamma_2(4,8))$, OKAZAKI Takeo, 2007, 125, 117, 132
- Not Refereed, 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., Feb. 2006, 128, 1, 139, 165, 10.1353/ajm.2006.0008
- Not Refereed, 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, 128, 139, 165, 10.1353/ajm.2006.0008

- OKAZAKI Takeo; Takeo Okazaki, 東京電機大学 第６回 数学講演会, New Forms for some algebraic groups, Mar. 2017, False
- OKAZAKI Takeo; TAKEO OKAZAKI, RIMS workshop ``Modular forms and Automorphic Representations’’, NewForm for GU(2,2), Feb. 2015
- OKAZAKI Takeo; Takeo Okazaki, 早稲田大学整数論シンポジウム, Triple Geometric Automorphic forms on SU(2,2), Mar. 2012, 坂田裕, 早稲田大学, False
- OKAZAKI Takeo, p-進表現論勉強会, Newforms for GSp(4), Feb. 2012, 原下 秀士, False

- 保型形式と岩澤理論, 0, 0, 0, Competitive research funding
- Grant-in-Aid for Scientific Research (C), 01 Apr. 2015, 31 Mar. 2020, 15K04783, Newform Theory for automorphic forms and its applications to Iwasawa Theory, Okazaki Takeo, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 4550000, 3500000, 1050000, I tried to construct a newform theory for irreducible admissible representations of GSp(4), that is Siegel modular forms of degree 2 in the classical sense. I have completed to the construction for the generic case, which is a generalization of Roberts and Schmidt for PGSp(4), and for the Saito-Kurokawa lifts, a non-generic case., kaken
- 若手研究(B), 2007, 2007, 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
- Grant-in-Aid for Young Scientists (B), 01 Apr. 2012, 31 Mar. 2015, 24740017, Automorphic forms, algebraic varieties and Iwasawa theory, OKAZAKI Takeo; YAMAUCHI Takuya, Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Nara Women's University, 2600000, 2000000, 600000, We established functional equations for automorphic representations of GU(2,2), and a New form theory corresponding to them. We call D-paramodular subgroups which fix the new forms. In particular, when the automorphic representation is distinguished, it has a D-paramodular Shalika period. By considering the theta correspondence between GSp(4) and GU(2,2), we give a proof for a conjecture of van Geemen and van Straten., kaken