Research on arithmetic and geometry for algebraic varieties in positive characteristic.

本研究的目的是：1)深入研究正特征代数簇，特别是椭圆(拟椭圆)曲面、K3曲面和Calabi-Yau三重曲面。2)为正特征奇点理论提供一个新的视角。在这三年的研究中，我们得到了以下与上述目的相关的进展：1)我们给出了特征为2和3.2的不可提升的Calabi-Yau三重曲面的新的例子。3)通过对二维有理双点的模的深入研究，发现了三维奇点的新的病理现象，这必将推动三维正特征正则奇点的研究，这些结果对正特征代数簇的研究具有重要意义。得到了特征2中E8型奇点的变形与特征2中的Mordell-Weil格之间的显式关系。作为应用，我们给出了特征2中超奇异K3曲面模空间的普适族的显式构造。最后，我们还研究了Mordell-Weil格理论与对椭圆曲线密码体制的指数演算攻击之间的关系，但没有得到有效的攻击策略。

The purposes of this research are1) deep study of the algebraic varieties in positive characteristic, especially, elliptic (qasi-elliptic) surfaces, K3 surfaces and Calabi-Yau threefolds2) to give a new insight for the theory of singularities in positive characteristic.For these 3 years research term, we got the following advances related with the above purposes.1) We gave new examples for non-liftable Calabi-Yau threefolds in characteristic both 2 and 3.2) In the process on the construction above, we investigated the quasi-elliptic surfaces deeply, and we got various new 3-dimensional raional singularities which have crepant resolutions by giving the explicit resolution processes.3) By deep study on the moduli of 2-dimensional rational double points, we fond NEW pathological phenomena in 3-dimensional singularities, which must give a progress on the study of 3-dimensional canonical singularities in positive characteristic.Each results play an important role on the research of algebraic varieties in positive characteristic. We are going to pursue further study on these subjects.We also got the explicit relation between a deformation of a singularities of type E_8 and the Mordell-Weil lattices in characteristic 2. As an application of this, we gave the explicit construction of the universal family of moduli space of supersingular K3 surfaces in characteristic 2.Finally, we also studied the relationship between the theory of Mordell-Weil lattices and the index calculus attack to the elliptic curve cryptosystems, but we could not get the useful strategy for the attack.