数论中的p-adic方法

It is well known that Kloosterman sum and exponential sum are basic topics in number theory. By using Dwork's p-adic theory and method, particularly by using Dwork trace formula, we investigate the p-adic Newton polygon of L-function of dimension 3 twisted Kloosterman sum. We explore the asymptotic behavior of p-adic Newton polygon of L-function of twisted exponential sum of one-variable and two-variable polynomials of small degree. We study the p-adic Newton polygon of L-function of exponential sum of binomial x^d+ax^{d-1}. By using p-adic method, we study the periodicity of the arithmetic function associated with the least common multiple of consecutive quadratic progression terms and consecutive cubic progression terms and determine the smallest periods. By p-adic and analytic methods, we investigate the asymptotic behavior of the least common multiple of consecutive quadratic progression terms and consecutive cubic progression terms. This project will be important in science, and will make active and important contributions to number theory and algebraic geometry

利用Dwork的p-adic理论和方法,特别地利用Dwork迹公式,研究三维扭曲Kloosterman和的L函数的p-adic牛顿多边形，研究一元和二元多项式的扭曲指数和的L函数的p-adic牛顿多边形及其渐近行为,特别地,研究低次数多项式的扭曲指数和Ｌ函数的p-adic牛顿多边形的渐近行为和形如x^d+ax^{d-1}的二项式的指数和Ｌ函数的p-adic牛顿多边形。利用p-adic方法和解析方法,研究二次及高次级数连续项的最小公倍数的渐近公式和与二次及高次级数连续项的最小公倍数相关联的周期算术函数的周期性及最小正周期。本项目具有较重要的科学理论意义和价值，而且也将会在数论和代数几何中产生积极影响。

利用p-adic方法和解析方法,给出了二次及高次级数连续项最小公倍数的渐近公式，给出了与二次级数连续项最小公倍数相关联的算术函数是最终周期的刻画，并确定了它的最小正周期. 研究了有限域上的扭曲指数和L函数的p-adic牛顿多边形, 取得了阶段性进展. 还利用p-adic方法证明关于第二类Stirling数的2-adic赋值的两个Lengyel猜想成立. 研究了算术级数和高次多项式序列的倒数的初等对称函数的整性问题，所得结果加强细化了Erdos和Niven所得结果. 利用线性传递子,构造了有限域上几类新的置换多项式,得到了有限域上逆Dickson多项式的几个置换性质. 本项目取得的成果具有较重要的科学理论意义和价值，而且也将会在数论和p-adic分析中产生积极影响.