Study on interrelations among quadratic forms, automorphic forms and related various zeta-functions

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I studied, with Ki-Seng Tan (Taiwan) and Lee Joongul (Korea), a generalization of Tate's refinement of the Gross conjecture on the Stickelberger element obtained from special values of L-functions of number fields and function fields of one-variable over finite fields, and formulated a new conjecture. As a result, I succeeded in proving the generalized conjecture for elementary abelian extensions under certain conditions. Moreover, I proved Tate's original refinement in the case of number fields. I also studied some arithmetic properties of elliptic curves and abelian varieties. In particular, I proved unboundedness of the 3-part of the Tate-Shafarevich group of elliptic curves over the rational number field. Although this has been already known by Cassels, I proved it for a new family of elliptic curves. Furthermore, I proved the Hodge conjecture for Catalan curves, and this is one of my results in the study of arithmetic properties of jacobian varieties of algebraic curves. As a related topic, expecting application for congruent zeta functions of Fermat curves, I studied purity problem of Jacobi sums and solved negatively a problem posed by Evans.Sato studied the local density of symmetric matrices over p-adic integers and extended a result of Schulze-Pillot to higher level cases. Fujii studied the behavior of the argument of the Riemann zeta function on the critical line. Ohsugi studied Gorensteinness of toric rings.