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

我与Ki-Seng Tan（台湾）和Lee Joongul（韩国）共同研究了由有限域上的数域和一元函数域的l函数的特殊值得到的关于Stickelberger元的Gross猜想的Tate的改进的推广，并提出了一个新的猜想。结果，我成功地证明了在一定条件下初等阿贝尔扩展的广义猜想。此外，我还在数域的情况下证明了Tate的原始细化。本文还研究了椭圆曲线和阿贝尔变的一些算术性质。特别地，我证明了椭圆曲线的Tate-Shafarevich群的3部分在有理数域上的无界性。虽然卡塞尔已经知道了这一点，但我还是用一组新的椭圆曲线证明了这一点。进一步证明了Catalan曲线的Hodge猜想，这是我在研究代数曲线雅可比变换的算术性质方面的成果之一。作为一个相关的课题，期望在费马曲线的zeta全等函数上得到应用，我研究了Jacobi和的纯度问题，并负解了Evans提出的一个问题。Sato研究了p进整数上对称矩阵的局部密度，并将Schulze-Pillot的结果推广到更高层次的情况。藤井研究了黎曼ζ函数在临界线上的参数行为。大杉研究了环的戈伦斯坦性。

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.