2002 Fiscal Year Final Research Report Summary
Finiteness and the Counting Problem for Galais representations
| Project/Area Number |
13640001
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | KYUSHU UNIVERSITY |
|---|
Principal Investigator |
TAGUCHI Yuichiro Kyushu Univ, Mathematics, Professor, 大学院・数理学研究院, 助教授 (90231399)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshitaka Hokkaido Univ., Mathematics, Assoc.Prof., 大学院・理学研究科, 助教授 (60173720)
KANEKO Masanobu Kyushu Univ, Mathematics, Assoc.Prof., 大学院・数理学研究院, 教授 (70202017)
KOIKE Masao Kyushu Univ, Mathematics, Professor, 大学院・数理学研究院, 教授 (20022733)
KOHNO Noriko (HIRATA Noriko) Nihon Univ, Mathematics, Prof., 理工学部, 教授 (90215195)
SATOH Takekazu Saitama Univ, Mathematic, Assoc.Prof., 理学部, 助教授 (70215797)
|
|---|
| Project Period (FY) |
2001 – 2002
|
| Keywords | Galais representation / Serre's conjecture / Fontaine-Mazur's conjecture |
|---|
| Research Abstract |
(1) We proved the finiteness and non-existence of mod p Galois representations for primes p with 2 【less than or equal】 p 【less than or equal】 31 when the "reduced Serre weight" and the Artin conductor outside p are small. These results seem to be best possible as results obtained by our methods.
(2) We proved the potentially abelian case of the Finiteness conjecture of Fontaine-Mazur. Namely, there exist only finitely many isomorphism classes of semisimple potentially abelian geometric p-adic representations of a number field of finite degree over the rationals with a given dimension, bounded inertial level, and given Hodge-Tate type.
(3) We proved the induction formula for mod l Galois representations.
(4) We improved the p-adic method by T.Satoh of the fast computation of the number of rational points of an elliptic curve over a finite field F_<pN>. Our algorithm works in O(N^<2μ+0.5>) time and O(N^2) memory.
|
