The study of arithmetic and analytic property of automorphic forms and zeta function associated with them and numerical analysis

自守形式及其相关zeta函数的算术和解析性质研究及数值分析

基本信息

批准号：
09640018
负责人：
KOJIMA Hisashi
金额：
$ 1.86万
依托单位：
Iwate University (1998)The University of Electro-Communications (1997)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

(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 bundle-like 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 zeta-function, 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 3-dimensional 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.

（1）在Hecke运算符的多个假设的假设下，H.Kojima推断了傅立叶系数A（4n）的平方之间的明确关系。模块化形式F（x）= Sigmaepsilon（-1）^KN = 0,1（4），N> 0 a（n）E [NZL属于Kohnen的一半积分重量（2K+1）/和具有原始特征X的关键形式的关键形式的Zimular Zim of Zim of Zim forts f s sh Zimular for的图像ZIM f的基本效果X的基本差异X = 0,1（4），在原始形式x和MOMIDAL SH的关键形式下是ZIM的Zim forts f zimultar f f的基本X和ZIM的关键值，这是ZETA的关键值，这是Zim of Zim of Zim of Zim for的图像， PSI。我们的证明方法与Shimura的方法相同。我们处理了Shimura论文中有关Hilbert模块化形式的傅立叶系数的一半积分重量的傅立叶系数，我们的结果给出了Shimura结果的概括和发展。此外，使用这种方法，我们得出了类似的结果，在属于Kohnen空间的Maass Wave的一半积分重量的情况下。（2）Oshikiri证明，如果捆绑弯曲（M，G）的类似束的叶状（m，g）的束形叶子的叶状f的编成相吻合，则具有正面曲率，然后是fl hasa codimen off assa fof fof fof fof fof fof fof hasa fof flef fof fof hasa flef fof hasa fof fof fof hasa fof fof hasa fof fof fof hasa fof fof hasa fof fof fof。 MI（3）作为Riemann Zeta功能的顺序问题的成分的封闭式（Q-1），Miyai研究了与该问题有关的各种算术指数总和。在为阿特金森相函数f（t，n）构造cosinus形式时，他给出了| zeta（1/+it）|^2的明确公式。他假设Hooke定律和某些拉格朗日密度在分析力学的基础上，他得出了一个非线性偏微分方程的系统，在我们的期望下，该方程描述了字符串的振动。此外，他获得了一些固定的解决方案。