Towards a P-Adic Analytic Local Langlands Correspondence

研究者和他的同事研究了p进域上拓扑向量空间中p进群的表示理论，目的是通过“p进解析局部朗兰兹对应”将这种表示与算术联系起来。尽管还处于早期阶段，研究者和他的合作者彼得·施耐德，以及包括罗伯特·科尔曼和巴里·马祖尔在内的其他人，以及最近的马修·埃默顿和克里斯托夫·布勒伊，已经提供了这样一种通信存在的证据。研究者希望这种方法将提供p进l函数、p进伽罗瓦表示和p进自同构形式之间的概念联系，就像经典朗兰兹对应对复l函数、自同构形式和ell-adic表示所做的一样。当前数论研究的一个原则是朗兰兹程序，它提出了与代数群（称为“自同构形式”）相关的某些复解析函数与多项式方程类的整数解的数量之间的深刻关系，通过称为“zeta函数”的函数作为媒介。怀尔斯对费马定理的证明，通过建立朗兰兹猜想的一个非常特殊的例子，有力地证明了这一思想的力量。该领域的第二个重要原则是，要充分理解数论，不仅需要研究实数和复数方程的几何行为，还需要研究较少为人所知的p进数领域。事实上，数论中许多重要的经典对象，如模形式和l函数，在朗兰兹计划中很重要，都有p进的版本。这些p进的版本捕获了从经典情况中无法获得的重要信息。本项目提出通过将朗兰兹纲领的哲学扩展到p进分析的设置来扩展我们对自同构形式和ζ函数的p进版本的理解。

The investigator and his colleagues study the representation theory of p-adic groups in topological vector spaces over p-adic fields, with the goal of relating such representations to arithmetic by means of a "p-adic analytic local Langlands correspondence." Though still in its early stages, work of the investigator, his collaborator Peter Schneider, and others including Robert Coleman and Barry Mazur, and more recently Matthew Emerton and Christophe Breuil, have provided evidence that such a correspondence exists. The investigator has the hope that this approach will provide a conceptual link between p-adic L-functions, p-adic Galois representations, and p-adic automorphic forms, in the same way that the classical Langlands correspondence does for complex L-functions, automorphic forms, and ell-adic representations. One principal of current research in number theory is the Langlands program, which proposes a deep relationship between certain complex analytic functions related to algebraic groups called "automorphic forms" and the number of integer solutions to classes of polynomial equations, through the medium of functions called "zeta functions." The power of this idea was strikingly demonstrated by Wiles' proof of Fermat's theorem, which proceeded by establishing one very special case of the Langlands conjectures. A second important principal in this field is the idea that a full understanding of number theory requires the study, not only of the geometric behavior of equations over the real and complex numbers, but also over the less widely known fields of p-adic numbers. Indeed, many of the important classical objects of number theory, such as modular forms andL-functions, that are important in the Langlands project, have p-adic versions. These p-adic versions capture important information not accessible from the classical situation. This project proposes to extend our understanding of the p-adic versions of automorphic forms and zeta functions by extending the philosophy of the Langlands program to the setting of p-adic analysis.