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

本文的研究目的是：1）深入研究正特征的代数簇，特别是椭圆代数簇（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了新的见解。与上述结果相关，我们得到了以下进展：1）给出了特征2和3中不可升的Calabi-Yau三重数的新例子; 2）在上述构造过程中，我们对拟椭圆曲面进行了深入的研究，通过给出显式的分解过程，得到了各种新的具有不同分解的三维有理奇点。3）通过对二维有理二重点模的深入研究，我们发现了三维奇点中的新的病态现象，这些结果对正特征的代数簇的研究具有重要的意义。我们还得到了E_8型奇点的形变与特征为2的Mordell-Weil格之间的显式关系。作为应用，我们给出了特征为2的超奇异K3曲面的模空间的泛族的显式构造。最后，我们还研究了Mordell-Weil格理论与椭圆曲线密码体制的指数演算攻击之间的关系，但没有得到有用的攻击策略。

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.