| 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)
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| 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.
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