Arithmetic Algebraic Geometry

主要研究者建议继续在算术代数几何方面的研究，特别是有限域上变的l-进上同调，有限域上的指数和及其相关的l-函数，单群的确定，以及这种确定在前面问题中的应用。一些主要的工具是群论，傅立叶变换，和倒行逆施理论。特定的研究主题包括单群的计算，随着乘法字符变化的字符和的均匀分布，以及一些“水平”均匀分布问题，这些问题附加于Z[1/n]和变化素数p上的固定变量，这些问题没有理论框架，甚至推测，尚存在。这个项目的广泛影响是三重的。虽然现在评价这一特定项目的广泛社会影响还为时过早，但在过去的二十年里，在有限领域中，大量代数几何在许多领域（例如，电信、密码学和计算机安全，仅举几例）中得到了惊人的实际应用，其中一些可以追溯到19世纪，所有这些在当时都是相当神秘的。在更直接的范围内，该项目将导致与博士后研究员、研究生和高级本科生在理论合作和进行计算机实验方面的大量互动。从最狭隘的角度来看，该项目将促进我们对有限域情况和数域情况之间类比的理解，这些类比已经在塑造我们对数论的思考中发挥了重要作用。

Abstract for the award DMS-0701395 of Katz The principal investigator proposes to continue work in arithmetic algebraic geometry, especially the l-adic cohomology of varieties over finite fields, exponential sums over finite fields, their associated L-functions, the determination of monodromy groups, and the application of that determination to the earlier questions. Some of the main tools are group theory, Fourier Transform, and the theory of perverse sheaves. Particular topics of investigation include the calculation of monodromy groups, the equidistribution of character sums as the multiplicative character varies, and some 'horizontal' equidistribution questions attached to fixed varieties over Z[1/n] and varying primes p for which no theoretical framework, even conjectural, yet exists. The broader impact of this project is three-fold. While it is too soon to appraise the wide societal impact of this particular project, the last two decades have seen stunning practical application in many fields (e.g., telecommunications, cryptology, and computer security, to name just a few) of a great deal of algebraic geometry over finite fields, some of which goes back to the nineteenth century, and all of which seemed quite arcane at the time it was being done. On a more immediate scale, the project will lead to a great deal of interaction with postdoctoral fellows, graduate students, and advanced undergraduates, both in theoretical collaborations and in the carrying out of computer experiments. From the narrowest point of view, the project will advance our understanding of the analogies between the finite field case and the number field case, analogies which have already played an important role in shaping our very thinking about number theory.