Studies on the Jacobi sum and its application to the Leopoldt conjecture

数论的发展与数学的许多领域密切相关，最近它被应用于物理和工程。在本研究中，我们从综合的角度进行研究。本项目的研究人员参加了相关会议，与相关研究人员进行了讨论，收集了大量相关文献，并利用计算机对问题进行了分析。在我们的研究过程中，我们认识到高斯和与雅可比和研究的重要性，即我们坚信利奥波德猜想与高斯和之间的关系。利奥波德猜想说，有限代数数域的单位，在有理整数环上是相乘独立的，在p进整数环上也是相乘独立的。这是一个非常重要的猜想，也是一个非常困难的开放问题。这个猜想等价于L函数在1处不消失。另一方面，本研究的主要研究者在一定条件下，利用高斯和给出了L函数在1处不消失的代数证明。这意味着利奥波德猜想与雅可比和之间有着密切而深刻的关系。他也从不同的角度对这一猜想得到了部分肯定的回答。考虑到与上同调理论、积分表示、代数数域伽罗瓦群的结构、分支理论、p进L函数理论、希尔伯特范数剩余符号的显式公式及其非标记化的关系，对其进行研究显得十分重要。

Number theory has been developed relating closely to many areas in mathematics, and recently it is applied to physics and engineering. In the present research, we researched from the integrated standpoint. Investigators in this project attended related conferences, discussed the problem with related researchers, collected many related references, and analyzed the problem using computers. In the process of our research we realized the importance of studies on Gauss sums and Jacobi sums, namely we firmly believed the relation between the Leopoldt conjecture and Gauss sums. The Leopoldt conjecture says that units, which are multiplicatively independent over the ring of rational integers, of a finite algebraic number field are also multiplicatively independent over the ring of p-adic integers. It is very important conjecture and is still a very difficult open problem.This conjecture is equivalent to the nonvanishing of L function at 1. On the other hand, head investigator in the present research gave an algebraic proof of nonvanishing of L function at 1 using Gauss sums under certain conditions. This implies close and deep relation between the Leopoldt conjecture and Jacobi sums. He also obtained partial affirmative answer to the conjecture from the different standpoint. It seems to be important to pursue the research considering the relation to cohomology theory, integral representation, structure of Galois groups of algrbraic number fields, ramification theory, theory of p-adic L functions, explicit formula for Hilbert norm residue symbols, and its non-abelization.