Research on algorithm to compute special values for non holomorphic Eisenstein series
Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants|
|Research Institution||Tokyo Institute of Technology(2004-2005)|
SATOH Takakazu Tokyo Tech., Graduate School of Mathematics, Assoc.Prof., 大学院理工学研究科, 助教授 (70215797)
YANAI Hisae Saitama Univ., Dept. of Mathematics, Lecturer, 理学部, 講師 (10008865)
GON Yasuro Kyushuu Univ., Dept. of Mathematics, Assoc.Prof., 大学院数理学研究院, 助教授 (30302508)
TAGUCHI Yuuichiro Kyushuu Univ., Dept. of Mathematics, Assoc.Prof., 大学院数理学研究院, 助教授 (90231399)
|Project Period (FY)
2003 – 2005
Completed(Fiscal Year 2005)
|Budget Amount *help
¥2,200,000 (Direct Cost : ¥2,200,000)
Fiscal Year 2005 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2004 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2003 : ¥800,000 (Direct Cost : ¥800,000)
|Keywords||Eisenstein series / finite fields / elliptic curves / Galois representations / Eisenstein級数 / 特殊値 / Verheul写像 / 等分多項式 / 近正則保型形式|
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.
Research Products (25results)