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

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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.