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Research on algorithm to compute special values for non holomorphic Eisenstein series

非全纯艾森斯坦级数特殊值计算算法研究

基本信息

批准号：
15540008
负责人：
SATOH Takakazu
金额：
$ 1.41万
依托单位：
Tokyo Institute of Technology (2004-2005)Saitama University (2003)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

项目摘要

Major results of the research are classified into three classes.(1)Bounds of weights of interpolating polynomials for pairing inversion.Pairing inversion is an important problem not only for elliptic curve cryptography but also for any cryptosystem based on difficulty of the Diffie-Hellman problem on a subgroup of the multiplicative group of finite fields which is an image of the pairing. In the research, we obtained a bound of degree of interpolating polynomials. Moreover, the weight of the interpolating polynomial is the maximum possible value for 58% of elliptic curves over a prime field of large characteristic. The proof is based on the formula which expresses coefficients of polynomial interpolation as a special values of certain holomorphic and non-holomorphic Eisenstein series.(2)Galois representations and congruence formulas :Non-existence is proved for 2-dimensional mod 2 Galois representations of the rational number field with small conductor N. As an application, it is shown that the Hecke action on the space of modular forms of level 2N is nilpotent in characteristic 2. As an application, congruences for Fourier coefficients of some modular forms and some combinatorial quantities are proved.(3)Generalized Mahler measure and special values of zeta functions :Generalization of the classical Mahler measure to multivariate integrand is given. Explicit computation for particular functions gives a formula for special values of the Riemann zeta function at positive odd integers.

主要研究成果分为三类。(1)配对逆插值多项式的权重界限。配对反转不仅对于椭圆曲线密码学来说是一个重要问题，而且对于任何基于有限域乘法群子群上的迪菲-赫尔曼问题困难的密码系统来说都是一个重要问题，该子群是配对的图像。在研究中，我们得到了插值多项式的次数界。此外，插值多项式的权重是大特征素数域上 58% 的椭圆曲线的最大可能值。证明基于将多项式插值系数表示为某些全纯和非全纯爱森斯坦级数的特殊值的公式。(2)伽罗瓦表示和同余公式：证明了具有小导体N的有理数域的二维模2伽罗瓦表示的不存在性。作为应用，证明了模空间上的Hecke作用 2N级形式在特征2上是幂零的。作为应用，证明了某些模形式和某些组合量的傅里叶系数的同余性。(3)广义马勒测度和zeta函数的特殊值：给出了经典马勒测度对多元被积函数的推广。特定函数的显式计算给出了黎曼 zeta 函数在正奇整数处的特殊值的公式。