2005 Fiscal Year Final Research Report Summary
Research on algorithm to compute special values for non holomorphic Eisenstein series
| Project/Area Number |
15540008
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Tokyo Institute of Technology (2004-2005)
Saitama University (2003) |
|---|
Principal Investigator |
SATOH Takakazu Tokyo Tech., Graduate School of Mathematics, Assoc.Prof., 大学院理工学研究科, 助教授 (70215797)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
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
|
| Keywords | Eisenstein series / finite fields / elliptic curves / Galois representations |
|---|
| Research Abstract |
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
(24 results)
