有限域上代数簇的有理点与指数和及其应用

Algebraic varieties over finite fields are an important research object in number theory, especially in Diophantine geometry. Exponential sums have a close relationship with the computation of rational points on such algebraic varieties. This project plans to study the rational points and exponential sums on algebraic varieties over finite fields as well as their applications. More precisely, we will mutually combine the methods of estimates of exponential sums, Newton polygon and p-adic analysis, etc., to study mainly the problems related to L-functions (including zeta functions), arithmetic properties of some special algebraic varieties, and value sets involving Weil sums. These new problems have arisen during the previous research and the applicant has made the preliminary work and published some paper. This project is a continuation and deepening of the last one, which will not only improve the known results and (partially) solve several conjectures, but also have potential practical applications to cryptography and coding theory.

有限域上的代数簇是数论尤其是丢番图几何中重要的研究对象.指数和与此类代数簇有理点的计算有着密切联系.本项目拟研究定义在有限域上的代数簇的有理点与指数和及其应用；具体地讲，我们将综合应用指数和估计、牛顿多边形及p-adic分析等方法，重点研究与L-函数（含zeta函数）、某些特殊代数簇的算术性质以及包括Weil和在内的值集等有关的问题.这些问题是在前期的研究中发现的新问题，项目申请者已做了一些初步的工作，且均有成果发表.本项目的研究是前期项目的继续和深化，不仅将改进已知的结论和（部分）解决若干猜想，而且在密码学和编码学中也具有潜在的应用价值.

利用p-adic Gamma函数和Gross-Koblitz公式研究了有限域上一类扭曲指数和的L-函数并给出了其具体表达式，计算了此类L-函数的零点和极点的倒数的斜率并确定了相对应的p-adic牛顿多边形. 利用p-adic超几何函数研究了Dwork超曲面的有理点问题，修正和改进了Barman与Goodson等人的相关结论. 发现了推广的Markoff-Hurwitz 类型方程的解数公式并部分了解决了Carlitz提出的问题. 改进了Deligne-Weil定理在特殊情形下关于指数和的估计. 利用Teichmüller提升改进了Chevalley-Warning-Ax-Katz型估计，并对Clark的问题给出了否定的回答. 利用Dwork迹公式和Wan分解理论研究了有限域上一类指数和的L-函数并计算了与之对应的Hasse多项式.