Studies in arithmetic algebraic geometry

该项目的技术说明：该项目涉及A）研究有限域上的代数簇，这些簇上的指数和，它们的相关L函数，这些对象如何在族中变化，以及决定这种变化的相关单值群。B）各种“水平”问题，我们从整数的情况开始，对各种素数p取模约化，并询问上面A）部分问题的答案如何随p而变化。C）关于给定特征p中椭圆曲线的族而非模族的Lang-Trotter类型的各种问题。攻击A）类型问题的一些主要技术工具是群论，傅立叶变换，以及反常层理论对于类型B）和C）的问题，工具很少，甚至设计和实施数值实验来猜测什么应该是真的也可能是不平凡的。项目的更广泛的意义和重要性：项目的更广泛的意义和重要性是三方面的。虽然现在评估这个特定项目的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。该项目旨在扩展我们对有限域上已经提出的非常有趣的数学问题的理解，这些问题的答案在未来可能会产生更广泛的社会影响。第二，该项目提出了新的，非常有趣的，数学问题的调查，为考虑还没有甚至存在一个理论框架;在这些问题上的任何进展可能指向目前缺乏的理论框架的方式。 第三，这个项目将加深我们对有限域和数域之间的类比的理解，这些类比在塑造我们对数域的思考方面已经发挥了重要作用。

Technical description of the project: The project is concerned withA) the study of algebraic varieties over finite fields, exponential sums on such varieties, their associated L-functions, how these objects vary in families, and the determination of the associated monodromy groups which govern this variation. B)Various "horizontal" questions where we start with a situation over the integers, reduce modulo various primes p, and ask how the answers to the questions of section A) above vary with p.C) Various questions of Lang-Trotter type about families, other than modular families, of elliptic curves in a given characteristic p.Some of the main technical tools for attacking questions of type A) are grouptheory, Fourier Transform, and the theory of perverse sheaves. For questionsof types B) and C), there are very few tools, and even designing and implementing numerical experiments to guess what should be true can be nontrivial.Broader significance and importance of the project: The broader significance and importance of the project is three-fold. While it is too soon to appraise the 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. The project proposes to extend our understanding of already posed, extremely interesting mathematical questions over finite fields, questions the answers to which may well in the future have a broader societal impact. Second, the project proposes the investigation of new, extremely interesting, mathematical questions, for whose consideration there does not yet exist even a theoretical framework; any progress on such questions may point the way to the presently missing theoretical framework. Third, the project will deepen our understanding of the analogies between the finite eld case and the number field case, analogies which have already played a important role in shaping our very thinking about the number field case.