Studies in arithmetic algebraic geometry

项目技术描述：该项目涉及a)有限域上代数变量的研究，这些变量上的指数和，它们相关的l函数，这些对象如何在族中变化，以及决定控制这种变化的相关单群。B)各种“水平”问题，我们从整数的情况开始，对各种素数p进行模化，并询问上述a)部分问题的答案如何随p.C变化。c)关于给定特征p中的椭圆曲线族（除了模族）的朗-特罗特类型的各种问题。解决a)类型问题的一些主要技术工具是群论，傅里叶变换和反常束理论。对于B)和C)类型的问题，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有三个方面。虽然现在评估这一特定项目的社会影响还为时过早，但在过去的二十年里，有限领域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）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。该项目建议扩展我们对有限领域中已经提出的非常有趣的数学问题的理解，这些问题的答案很可能在未来产生更广泛的社会影响。第二，该项目提出了对新的、非常有趣的数学问题的研究，对这些问题的考虑甚至还没有一个理论框架；在这些问题上的任何进展都可能为目前缺失的理论框架指明方向。第三，该项目将加深我们对有限域情况和数字域情况之间类比的理解，这些类比已经在塑造我们对数字域情况的思考中发挥了重要作用。

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.