L-Functions and Monodromy

摘要奖DMS-0355496卡茨的主要研究者建议继续工作的算术代数几何，特别是L-进上同调的品种在有限领域，L-进理论的指数总和在有限领域，位置的“低躺零”的L-函数在有限领域，确定monodromy组，并应用该决定的早期问题。 一些主要的工具是群论，傅立叶变换和反常层理论。 调查的特定主题包括计算monodromy组，问题的L-函数的椭圆曲线，和equidistribution的字符总和作为字符的变化。该项目的广泛影响有三个方面。 虽然现在评价这个特殊项目的广泛社会影响还为时过早，但在过去的20年里，有限域上的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世纪，所有这些在当时似乎都很《双城之战》。 在更直接的规模上，该项目将导致与博士后研究员，研究生和高级本科生的大量互动，无论是在理论合作还是在计算机实验的进行中。 从最重要的角度来看，该项目将促进我们在数学的一个重要领域的知识。

Abstract for award DMS-0355496 of Katz The principal investigator proposes to continue work in arithmetic algebraic geometry, especially the l-adic cohomology of varieties over finite fields, the l-adic theory of exponential sums over finite fields, the locations of the "low-lying zeroes" of L-functions over finite fields, 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, questions about the L-functions of elliptic curves, and the equidistribution of character sums as the character varies. 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 knowledge in a vital area of mathematics.