Arithmetic Algebraic Geometry

Katz的DMS-0701395奖摘要 首席研究员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，指数和有限领域，其相关的L-功能，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。研究的特定主题包括单值群的计算、随着乘法特征标的变化而特征标和的等分布，以及一些与Z[1/n]上的固定簇和变化的素数p相关的“水平”等分布问题，对此没有理论框架，甚至是猜测的，还不存在。 该项目的广泛影响有三个方面。虽然现在评估这个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，在许多领域都有令人惊叹的实际应用（例如，电信，密码学和计算机安全，仅举几例）的大量代数几何在有限域上，其中一些可以追溯到世纪，所有这些似乎相当《双城之战》在当时正在做。 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。从最新的角度来看，该项目将促进我们对有限域情况和数域情况之间的类比的理解，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.