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

我研究，与谭基成（台湾）和李Joongul（韩国），泰特的细化的格罗斯猜想的Stickelberger元素从特殊值的L-函数的数域和功能领域的一个变量在有限域的推广，并制定了一个新的猜想。结果，在一定条件下成功地证明了初等交换扩张的广义猜想。此外，我证明了泰特的原始细化在数域的情况下。研究了椭圆曲线和交换簇的一些算术性质。特别是，我证明了无界的3部分的泰特-Shafarevich组的椭圆曲线在有理数域。虽然这已经知道卡塞尔，我证明了它的一个新的家庭的椭圆曲线。证明了Catalan曲线的Hodge猜想，这是我研究代数曲线的Jacobian簇的算术性质的结果之一。作为一个相关的课题，为了应用于Fermat曲线的全等zeta函数，研究了Jacobi和的纯性问题，解决了Evans提出的一个问题，Sato研究了p-adic整数上对称矩阵的局部稠密性，将Schulze-Pillot的一个结果推广到更高层次的情形。藤井研究的行为的论点的黎曼zeta函数的临界线。Ohsugi研究Toric环的Gorensteinness。

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.