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

设M是正特征代数闭域κ上主极化阿贝尔曲面的模叠叠，π：X-M是普适族。对于整数h，我们设置M^&lt；(H)&gt；={X∈M|HeightΦx[大于或等于]h}。取对应于主极化阿贝尔曲面(A，D，∈)的点xσM，并假设形式布劳尔群Φ&gt；_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以及M^&lt；(H)&gt；在x处的切空间同构于{Im H^1(A，Z_h)}∩D^〓⊂H^1(A，Ω^1_A)。设X是k上n维非奇异完备代数簇，H_μR(X)是X的de Rham上同调群，则H_&lt；DR&gt；(X)有Hodge滤子H_&lt；DR&gt；(X)=F_0⊃F_1⊃…⊃F_n和Frobenius映射F作用于H_&lt；DR&gt；(X)。我们用a(X)=max{i|F^*H_&lt；dr&gt；(X)⊂F_i}定义a-数。我们可以证明，对于阿贝尔变种X，这个数与F.Oort定义的a数重合。如果Hodge to de Rham谱序列在E_1级退化，则F^*导出一个映射H^n(X，Ox)=F_0/F_1-H_&lt；DR&gt；(X)。因此，我们有a(X)=max{i|F^*H^n(X，Ox)⊂F_i}，我们可以计算不同品种的这个数字。明确了Artin-Mazur形式群的a-数与高度h之间的关系。最后，对于n[大于或等于]3维的Calabi-Yau簇X，我们证明了在假设H^0(X，Ω^I_X)=0(i=1，2)下，_pk-H^1(⊂^1_X)的自然同态NS(X)/Pns(X)是客观的。至于密码学，T.Okamoto et al.给出了公钥密码体制RSA-OAEP的安全性的精确证明。

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.