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

研究的主要成果分为三类。（1）对逆插值多项式的权的界对逆不仅是椭圆曲线密码学中的一个重要问题，而且对于任何基于有限域乘法群的子群上的Diffie-Hellman问题的困难性的密码系统都是一个重要问题，该子群是对逆的象。在研究中，我们得到了插值多项式次数的一个界。而且，对于大特征的素域上的椭圆曲线，58%的插值多项式的权是最大可能值。证明是基于多项式插值系数表示为某些全纯和非全纯Eisenstein级数的特殊值的公式。（2）Galois表示与同余公式：证明了具有小导体N的有理数域的二维模2 Galois表示的不存在性。作为应用，证明了2N阶模形式空间上的Hecke作用在特征2下是幂零的。作为应用，证明了某些模形式和某些组合量的Fourier系数的同余关系。（3）广义Mahler测度与zeta函数的特殊值：将经典Mahler测度推广到多元被积函数。对特殊函数的显式计算给出了黎曼zeta函数在正奇数处的特殊值的公式。

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.