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

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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.