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

本文的主要研究成果分为三类：(1)插值多项式成对求逆的权值的界，成对求逆不仅是椭圆曲线密码学的一个重要问题，而且对于任何基于Diffie-Hellman问题难度的密码体制都是一个重要的问题。在研究中，我们得到了插值多项式的一个次数界。此外，插值多项式的权重是大特征素域上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Eisenstein级数的特定值的公式.(2)Galois表示和同余公式：证明了具有小导体N的有理数域的二维mod2 Galois表示的不存在性.作为应用，证明了2N级模型空间上的Hecke作用在特征2中幂零.作为应用，证明了某些模形式的傅立叶系数和某些组合量的同余.(3)推广的Mahler测度和Zeta函数的特殊值：将经典的Mahler测度推广到多元积分和.对特殊函数的显式计算给出了Riemann Zeta函数在正奇数时的特异值的一个公式。

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.