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

本研究的主要成果可分为三类。(1)配对反演插值多项式的权值界。配对反转不仅是椭圆曲线密码学的一个重要问题，而且是任何密码系统的一个重要问题，它基于有限域乘群的子群上的Diffie-Hellman问题的难度，该子群是配对的一个像。在研究中，我们得到了插值多项式的阶限。此外，插值多项式的权值是58%的椭圆曲线在大特征素域上的最大可能值。该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)伽罗瓦表示与同余公式：证明了具有小导体n的有序数场的2维模2伽罗瓦表示的不存在性。作为应用，证明了在特征2上，2N层模形式空间上的Hecke作用是幂零的。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Mahler测度和zeta函数的特殊值：将经典Mahler测度推广到多元被积函数。对特定函数的显式计算给出了正奇数处黎曼ζ函数的特殊值的公式。

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.