Towards a P-Adic Analytic Local Langlands Correspondence

研究者和他的同事们研究了p-adic域上拓扑向量空间中p-adic群的表示理论，目的是通过“p-adic解析局部朗兰兹对应”将这种表示与算术联系起来。虽然仍处于早期阶段，但调查员、他的合作者彼得·施耐德（Peter Schneider）以及其他人（包括罗伯特·科尔曼和巴里·马祖尔）的工作，以及最近的马修·埃默顿（Matthew Emerton）和克里斯托夫·布鲁尔（Christophe Breuil）的工作，都提供了存在这种对应关系的证据。 研究者希望这种方法能在p-adic L-函数、p-adic伽罗瓦表示和p-adic自守形式之间提供一种概念上的联系，就像经典的朗兰兹对应对复L-函数、自守形式和ell-adic表示所做的那样。当前数论研究的一个原则是朗兰兹纲领，它提出了与代数群相关的某些复解析函数（称为“自守形式”）和多项式方程类的整数解的数量之间的深层关系，通过称为“zeta函数”的函数。怀尔斯对费马定理的证明，通过建立朗兰兹定理的一个非常特殊的例子，惊人地证明了这一思想的力量。 这个领域的第二个重要原则是这样一种思想，即充分理解数论不仅需要研究真实的和复数上方程的几何行为，而且还需要研究不太广为人知的p进数领域。事实上，数论中许多重要的经典对象，如在朗兰兹计划中很重要的模形式和L-函数，都有p-adic版本。 这些p-adic版本捕获了从经典情况中无法获得的重要信息。 本项目旨在通过将朗兰兹纲领的哲学扩展到p-adic分析的设置来扩展我们对自守形式和zeta函数的p-adic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.