P-adic Analysis in Algebraic Geometry over finite fields

这一建议的很大一部分考虑了L函数，它是由指数和族产生的几何伽罗瓦表示所附的函数。以前的研究主要集中在L表示理论的技巧上，因为它们特别适合于利用欧拉特征和零极点的权界进行次数计算。PI在Dwork，Adolphson，Sperber，Robba，wan等人的启发下，提出了对这类L函数的p-进研究。这将是一项一般性的研究，包括许多常见的原型，如广义的艾利家族、超级克鲁斯特曼家族和德沃克斯家族。该提案的其他方面集中在继续对除数的Zeta函数的p-addy研究，以及对镜像流形的算术异同的研究的继续。伽罗瓦表示编码了数论中许多深刻而重要的问题。事实上，正是通过他们的研究，著名的费马最后定理最终被Wiles等人确定为肯定的。在1995年。最近，暗示费马的Serre猜想和Sato-Tate猜想也归结为关于伽罗瓦表示的陈述。伽罗瓦表示经常被与其相关联的L函数所研究。这只是一个简单的函数，它对伽罗瓦表示的大部分(如果不是基本上全部)进行编码。即使是关于这些L函数的最基本的问题，在很大程度上仍然是未知的。例如，黎曼假设是一个猜想，描述了L函数的零点在特定伽罗瓦表示上的精确位置。在这份提案中，PI计划使用p-进分析中的工具和扩展工具来研究几何情况下产生的伽罗瓦表示。

A large portion of this proposal considers L-functions attached to geometric Galois representations arising from families of exponential sums. Previous studies have focused on techniques from l-adic representation theory since they are especially amenable to degree computations via Euler characteristic and bounds for the weights of the zeros and poles. The PI proposes a p-adic study of such L-functions using techniques inspired by Dwork, Adolphson, Sperber, Robba, Wan, et al.. This would be a general study including many common archetypes, such as the generalized Airy family, the hyper-Kloosterman family, and the Dwork family. Other aspects of the proposal focus upon a continuation of the p-adic study of the zeta function of divisors, and acontinuation of the study into the arithmetic similarities and differences of mirror manifolds. Galois representations encode many deep and significant problems within number theory. In fact, it was through their study that the celebrated Fermat's Last theorem was finally settled in the affirmative by Wiles et al. in 1995. More recently, Serre's conjecture, which implies Fermat, and the Sato-Tate conjecture, also came down to statements about Galois representations. Galois representations are frequently studied by their associated L-function. This is simply a function which encodes much, if not essentially all, of the Galois representation. Even the most elementary questions about these L-functions are still largely unknown. For instance, the Riemann hypothesis is a conjecture describing the precise location of the zeros of the L-function attached to a specific Galois representation. In this proposal, the PI plans to study Galois representations which arise from geometric situations using, and extending, tools from p-adic analysis.