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.

令M为正面特征的代数闭合场κ上主要两极分化的Abelian表面的模量堆栈，让令π：x -m为普遍家族。对于整数H，我们设置了m^<（h）> = {x∈M|高度φx[大于或相等] h}。采用点x∈M，该点与主要极化的阿贝尔表面（a，d，σ）相关，并假定正式brauer groupφ> _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x时m^<（h）>的切线空间对{im h^1（a，z_h）}∩d^d^〓h^1（a，ω^1_a）。现在，让x成为k的k上的非发挥完整的代数变化，让h_μr（x）为X的de rham共同体学组。然后，H_ <dr>（x）具有Hodge fltration h_ <dr>（x）= f_0 = f_0 f_0 f_1 f_1 f_1 f_1 f_1 f_ f_n和frobenius mmapping m_________________ <dr.我们通过a（x）= max {i |定义了a-number f^*h_ <dr>（x）⊂f_i}。我们可以证明，对于Abelian品种X，此数字与F. oort定义的A-number相吻合。如果hodge到de rham光谱序列将在e_1级别退化，则f^*诱导映射h^n（x，ox）= f_0/f_1/f_1-h_ <dr>（x）。因此，我们有一个（x）= max {i | f^*h^n（x，ox）⊂f_i}，我们可以为各种变体计算此数字。我们还明确了A-number与Artin-Mazur正式小组的高度H之间的关系。最后，对于尺寸n的calabi-yau品种x [大于或相等] 3，我们表明自然同构ns（x）/pns（x）of_p k-h^1（ω^1_x）⊂h^2_ h^2_ <dr>（x）在假设H^0（x，x，x，x，ω^i___ x）下是在假设下。至于密码学，T。Okamoto等。关于公共钥匙密码系统的安全性提供了确切的证据，该系统称为RSA-OAEP。