Project/Area Number  09640018 
Research Category 
GrantinAid for Scientific Research (C)

Section  一般 
Research Field 
Algebra

Research Institution  Iwate University 
Principal Investigator 
KOJIMA Hisashi Iwate University, Derartment Faculty of Education, Professor, 教育学部, 教授 (90146118)

CoInvestigator(Kenkyūbuntansha) 
吉岡 正典 電気通信大学, 電気通信学部, 助手 (90272749)
中村 健一 電気通信大学, 電気通信学部, 助手 (40293120)
小藤 俊幸 電気通信大学, 電気通信学部, 助教授 (30234793)
伊東 裕也 電気通信大学, 電気通信学部, 助教授 (30211056)
TAYOSHI Takao University of electrocommunications, Faculty of electrocommnucation, propessor, 電気通信学部, 教授 (60017382)
KOMIYAMA Haruo Iwate University, Faculty of Education, lecture, 教育学部, 講師 (90042762)
NUMATA Minoru Iwate University, Faculty of Education, professor, 教育学部, 教授 (50028255)
MIYAI Akio Iwate University, Faculty of Education, assistant, 教育学部, 助手 (70003960)
中嶋 文雄 岩手大学, 教育学部, 教授 (20004484)
OSHIKIRI Genichi Iwate University, Faculty of Education, professor, 教育学部, 教授 (70133931)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 1998 : ¥1,300,000 (Direct Cost : ¥1,300,000)
Fiscal Year 1997 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  modular forms of half integral weight / modular forms / zeta functions / critical values of zeta functions / Fourier coefficients of modular forms / Riemannian fo / compact leaf / 半整数の重さのモジュラ形式 / モジュラ形式 / ゼータ関数 / ゼータ関数の特殊値 / モジュラ形式のフーリエ係数 / リーマン葉層 / コンパクト葉 / モジュラー形式 / 半整数の重さのモジュラー形式 / 保型形式 
Research Abstract 
(1) Under assumptions of the multiplicity one theorem of Hecke operators, H.Kojima deduced an explicit relation between the square of Fourier coefficients a(4n) at a. fundamental discriminant 4n of modular forms f(x)=SIGMAepsilon(1)^kn=0,1(4), n>0 a(n)e[nzl belonging to the Kohnen's space of half integral weight (2k+1)/ and of arbitrary odd level N with primitive character x and the critical value of the zeta function of the modular form F which is the image of f under the Shimura correspondence PSI.Our methods of the proof are the same as those of Shimura. We treated the excluding case in the Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight and our results gave a generalization and development of Shimura' results. Moreover, using this method, we derived an analogous results in the case of Maass wave forms of half integral weight belonging to Kohnen's spaces. (2) Oshikiri proves that if the codimension of a bundlelike foliation F of a Riemannian manifold (M, g) with positive sectional curvature is even, then F hasa compact leaf, and that if the codimension of F is odd, then F has a leaf whose closure is a codimension (q 1) closed submanifold of M. (3) As ingredients for the order problem of the Riemann zetafunction, Miyai investigated alternative explicit formulas for various arithmetic exponential sums relating to the problem. On constructing the cosinus form for the Atkinson phase function f(T, n), he gave an explicit formula for zeta(1/+iT)^2. (4) Tayoshi considered the equation of the vibration of a elastic string in 3dimensional space. Supposing Hooke's law and certain Lagrangian density, on the basis of analytic mechanics, he derived a system of nonlinear partial differential equations, which, in our expectation, describes the vibration of the string. Moreover, he obtained some stationary solutions.
