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Studies on agebraic geometry in positive characteristic, coding theory and cryptography

正特征、编码理论和密码学中的年龄数几何研究

基本信息

批准号：
12554001
负责人：
KATSURA Toshiyuki
金额：
$ 6.02万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

项目摘要

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.

设 M 为正特性代数闭域 κ 上主极化阿贝尔曲面的模堆，并设 π: X ―― M 为普适族。对于整数 h，我们设置 M^<(h)> = {X ∈ M |高度Φx【大于等于】h}。取对应于主极化阿贝尔面 (A,D,σ) 的点 x ∈ M，并假设形式布劳尔群 Φ>_A 的高度 h 是有限的。然后，我们可以证明 Im H^1(A, Z_h) = 7 - h 并且 M^<(h)> 在 x 处的切空间同构于 {Im H^1(A,Z_h)}∩D^〓 ⊂ H^1(A,Ω^1_A)。现在，令 X 为 n 维 k 上的非奇异完全代数簇，并令 H_μR(X) 为 X 的 de Rham 上同调群。则 H_<dR>(X) 具有霍奇过滤 H_<dR>(X) = F_0 ⊃ F_1 ⊃ … ⊃ F_n 且 Frobenius 映射 F 作用于 H_<dR>(X)。我们通过 a(X) = max{i | 定义 a 数F^*H_<dR>(X) ⊂ F_i}。我们可以证明，对于阿贝尔簇 X，这个数与 F. Oort 定义的 a 数一致。如果 Hodge 到 de Rham 谱序列在 E_1 级退化，则 F^* 会产生映射 H^n(X,Ox) = F_0/F_1 ―― H_<dR>(X)。因此，我们有 a(X) = max{i | F^*H^n(X,Ox) ⊂ F_i} 我们可以计算各种变量的这个数字。我们还明确了Artin-Mazur形式群的a数与高度h之间的关系。最后，对于维度 n 【大于或等于】3 的 Calabi-Yau 变体 X，我们证明自然同态 NS(X)/pNS(X) OF_p k ── H^1(Ω^1_X) ⊂ H^2_<dR>(X) 在假设 H^0(X,Ω^i_X) = 0 (i = 1, 2) 下是内射的。至于密码学，T. Okamoto 等人。对公钥密码体制RSA-OAEP的安全性给出了精确的证明。