Local Langlands in Families

朗兰兹程序是一套广泛的猜想，为理解l函数的神秘性质提供了一条途径，l函数编码了多项式方程解的信息。这些猜想激发了代数数论领域的广泛研究，并揭示了长期存在的开放性问题。在某些情况下，朗兰兹猜想已被证明是正确的，但仍有一些微妙之处尚不清楚。在其他情况下，它们甚至还没有被精确地表述出来。本项目拟在几何变形的背景下研究朗兰兹程序。除了数论研究目标之外，该项目还包括对教育工作者的培训：在项目的每一年，为缅因州的高中教师，特别是那些服务于农村地区的高中教师提供为期三周的暑期数论课程。作为该计划的一部分，教育工作者将为当地高中开设数论方面的课外活动或选修课程。该项目的第一个研究目标是在家族中建立所谓的局部朗兰兹猜想方面取得进展，该猜想将假定的局部朗兰兹对应升级为对应两侧积分对象的模空间之间的映射，假设有一个简短的理想列表。第二个目标是研究科中的局部朗兰兹猜想与围绕局部朗兰兹猜想的p进几何化的猜想之间的逻辑依赖性。该方法是推广逆定理和族中的Plancherel测度，以扩展先前用于一般线性群和经典群的特殊情况的论证。该项目由代数和数论项目和促进竞争研究的既定项目（EPSCoR）共同资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The Langlands program is a broad set of conjectures that provide an avenue toward understanding the mysterious properties of L-functions, which encode information about solutions to polynomial equations. These conjectures have motivated broad swaths of research in the field of algebraic number theory and have shed light on long-standing open problems. In some contexts, the Langlands conjectures have been proven true, but there remain subtleties that are poorly understood. In other contexts, they are yet to even be precisely formulated. This project proposes to study the Langlands program within the context of geometric deformation. In addition to the number theory research goals, the project also includes a training component for educators: a three-week summer number theory program during each year of the project for high school teachers throughout Maine, especially those serving rural areas. As part of the program, educators will develop extracurricular activities or elective courses in number theory for their local high school. The first research goal of the project is to make progress toward establishing the so-called local Langlands conjecture in families, which upgrades a presumed local Langlands correspondence to a map between moduli spaces of integral objects on either side of the correspondence, assuming a short list of desiderata. The second goal is to study the logical dependence between the local Langlands conjecture in families and the conjectures surrounding recent p-adic geometrizations of the local Langlands conjectures. The method is to generalize converse theorems and the Plancherel measure in families in order to extend a line of argument previously used in the special cases of general linear groups and classical groups. This project is jointly funded by the Algebra and Number Theory program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.