A local and global study of arithmetic varieties determined by arithmetic differential equations

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We constructed an arithmetic family of Calabi-Yau varieties on the projective line whose period integral is a generalized hypergeometric function, and determined the relative cohomology of Betti, de Rham, etale and crystalline realizations. This family is a higher dimensional version of Legenedre's family of elliptic curves. The family of Calabi-Yau is obtained by a desingularization which is given explicitly, and has a semistable degeneration at 0. The cohomologies can be calculated by applying weight spectral sequences. Moreover, we have a result on modularity for rational fibers.