| Project/Area Number |
12554001
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (B)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 展開研究 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
KATSURA Toshiyuki The University of Tokyo, Graduate School of Mathmatical Sciences, Professor, 大学院・数理科学研究科, 教授 (40108444)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
TERASOMA Tomohide The University of Tokyo, Graduate School of Mathmatical Sciences, Associate. Professor, 大学院・数理科学研究科, 助教授 (50192654)
OKAMOTO Kazuo The University of Tokyo, Graduate School of Mathmatical Sciences, Professor., 大学院・数理科学研究科, 教授 (40011720)
OKAMOTO Tatsuaki NTT, Institute on Information Sharing Laboratory, Chief Researcher., 情報流通プラットフォーム研究所, 主席研究員
TAKAYAMA Nobuki Kobe University, Faculty of Sciences, Professor, 理学部, 教授 (30188099)
KATO Akishi The University of Tokyo, Graduate School of Mathmatical Sciences, Associate Professor., 大学院・数理科学研究科, 助教授 (10211848)
|
|---|
| Project Period (FY) |
2000 – 2002
|
| Project Status |
Completed (Fiscal Year 2002)
|
| Budget Amount *help |
¥9,400,000 (Direct Cost: ¥9,400,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2001: ¥2,800,000 (Direct Cost: ¥2,800,000)
Fiscal Year 2000: ¥3,700,000 (Direct Cost: ¥3,700,000)
|
|---|
| Keywords | Positive characteristic / Artin-Mazur formal group / Cartier operator / Moduli space / Chow group / Calabi-Yau variety / Abelian surface / Cryptography / ネロン・セヴェリ群 / イリュージー層 / ハイト / 符号 / ド・ラムコホモロジー群 / フロベニウス写像 / ホッジフィルトレーション / a-数 / 形式的ブラウワー群 / モジュウイ空間 / 公開鍵暗号 / 量子計算機 / NP問題 |
|---|
| Research Abstract |
Let M be the moduli stack of principally polarized abelian surfaces over an algebraically closed field κ of positive characteristic, and let π: X ―― M be the universal family. For an integer h, we set M^<(h)> = {X ∈ M | height Φx 【greater than or equal】 h}. Take the point x ∈ M which correspoads to a principally polarized abelian surface (A,D,σ), and assume that the height h of the formal Brauer group Φ>_A is finite. Then, we could prove that Im H^1(A, Z_h) = 7 - h and that the tangent space of M^<(h)> at x is isomorphic to {Im H^1(A,Z_h)}∩D^〓 ⊂ H^1(A,Ω^1_A). Now, let X be an nonsingular complete algebraic variety over k of dimension n, and let H_μR(X) be the de Rham cohomology group of X. Then, H_<dR>(X) has the Hodge fltration H_<dR>(X) = F_0 ⊃ F_1 ⊃ … ⊃ F_n and the Frobenius mapping F acts on H_<dR>(X). We define the a-number by a(X) = max{i | F^*H_<dR>(X) ⊂ F_i}. We can show that for an abelian variety X this number coincides with the a-number defined by F. Oort. If the Hodge to de Rham spectral sequence degenerates at E_1-level, then F^* induces a mapping H^n(X,Ox) = F_0/F_1 ―― H_<dR>(X). Therefore, we have a(X) = max{i | F^*H^n(X,Ox) ⊂ F_i} and we can compute this number for various varieties. We also make clear the relation between the a-number and the height h of the Artin-Mazur formal group. Finally, for a Calabi-Yau variety X of dimension n 【greater than or equal】 3, we showed that the natural homomorphism NS(X)/pNS(X) OF_p k ―― H^1(Ω^1_X) ⊂ H^2_<dR>(X) is iujective under the assumption H^0(X,Ω^i_X) = 0 (i = 1, 2). As for the cryptography, T. Okamoto et al. gave a precise proof on the security of the public-key cryptosystem which is called RSA-OAEP.
|
