## OKAZAKI TakeoFaculty Division of Natural Sciences Research Group of Mathematics Associate Professor |

Last Updated :2021/06/02

- automorphic forms, algebraic variety, number theory

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

On some Siegel threefold related to the tangent cone of the Fermat quartic surface

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., International Press of Boston, Inc., 2017, Advances in Theoretical and Mathematical Physics, 21 (3), 585 - 630, doiScientific journal

ENDOSCOPIC LIFTS TO THE SIEGEL MODULAR THREEFOLD RELATED TO KLEIN'S CUBIC THREEFOLD

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)., JOHNS HOPKINS UNIV PRESS, Feb. 2013, AMERICAN JOURNAL OF MATHEMATICS, 135 (1), 183 - 205, web_of_scienceScientific journal

$L$-functions of $S_3(\G(2,4,8))$

2012, J. Number Theory, 132, 54-78Saito-Kurokawa type lift to $S_3(\Gamma^{1,3}(2))$

2008, Math. Ann, 208, 589-601L-functions of S-3(Gamma(4, 8))

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., ACADEMIC PRESS INC ELSEVIER SCIENCE, Jul. 2007, JOURNAL OF NUMBER THEORY, 125 (1), 117 - 132, doi;web_of_scienceScientific journal

Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and Abelian surfaces

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., JOHNS HOPKINS UNIV PRESS, Feb. 2006, AMERICAN JOURNAL OF MATHEMATICS, 128 (1), 139 - 165, web_of_scienceScientific journal

On some Siegel threefold related to the tangent cone of the Fermat quartic surface

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., INT PRESS BOSTON, INC, 2017, ADVANCES IN THEORETICAL AND MATHEMATICAL PHYSICS, 21 (3), 585 - 630, web_of_science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

2017, Advances in Theoretical and Mathematical Physics, 21 (3)Endoscopic lifts to the siegel modular threefold related to Klein's cubic threefold

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, American Journal of Mathematics, 135 (1), 183 - 205, doiENDOSCOPIC LIFTS TO THE SIEGEL MODULAR THREEFOLD RELATED TO KLEIN'S CUBIC THREEFOLD

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)., JOHNS HOPKINS UNIV PRESS, Feb. 2013, AMERICAN JOURNAL OF MATHEMATICS, 135 (1), 183 - 205, doi;web_of_science合同部分群ガンマ（2,4,8）に関するL-関数

2012, J. Number Theory, 132, 54 - 78L-functions of S-3 (Gamma(2)(2, 4, 8))

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., ACADEMIC PRESS INC ELSEVIER SCIENCE, Jan. 2012, JOURNAL OF NUMBER THEORY, 132 (1), 54 - 78, doi;web_of_scienceA Siegel modular threefold and Saito-Kurokawa type lift to S-3(Gamma(1,3)(2))

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., SPRINGER, Jul. 2008, MATHEMATISCHE ANNALEN, 341 (3), 589 - 601, doi;web_of_scienceA Siegel modular threefold and Saito-Kurokawa type lift to S-3(Gamma(1,3)(2))

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., SPRINGER, Jul. 2008, MATHEMATISCHE ANNALEN, 341 (3), 589 - 601, doi;web_of_scienceOn L-functions of $S_3(\Gamma_2(4,8))$

2007, J. Number theory, 125, 117 - 132On L-functions of $S_3(\Gamma_2(4,8))$

2007, J. Number theory, 125, 117 - 132Proof of R. Salvati Manni and J. Top's conjectures on siegel modular forms and abelian surfaces

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, American Journal of Mathematics, 128 (1), 139 - 165, doi

New Forms for some algebraic groups

東京電機大学 第６回 数学講演会, Mar. 2017NewForm for GU(2,2)

RIMS workshop ``Modular forms and Automorphic Representations’’, Feb. 2015Triple Geometric Automorphic forms on SU(2,2)

早稲田大学整数論シンポジウム, Mar. 2012, 坂田裕, 早稲田大学Newforms for GSp(4)

p-進表現論勉強会, Feb. 2012, 原下 秀士