Functoriality in the Mod-p Langlands Program

数论是数学的一个分支，主要研究整数的性质和多项式方程的整数解，是最古老的数学学科之一。 表示论是另一个同样有影响力的数学分支，它量化了几何对象（如正方形或氢原子）的对称性，并在物理学中有重要用途。 虽然这两个领域看似无关，但朗兰兹纲领（Langlands Program）将这两个领域紧密联系在一起，朗兰兹纲领是一套庞大的理论，允许数论和表示论之间的结果和定理转移。 理解这些知识是至关重要的，因为一个学科的工具可以被引入到另一个学科来解决以前难以解决的问题（费马大定理的证明就是一个很好的例子）。 这将朗兰兹纲领推向了当前研究的前沿。 本项目的目的是建立一个局部形式的具有mod p系数的Langlands纲领的实例，以便将表示论中的信息转化为算术数据，本项目的背景是mod p向量空间上的p-adic约化群（如GL_2（Q_p））的表示论。 这样的表示是非常复杂的，主要目标之一是使用导出范畴，以便更精确地将这样的表示与微分分次Hecke代数上的模联系起来。 这将允许使用新的工具来理解不同群体的朗兰兹对应之间的关系。 除此之外，PI和他的合作者计划使用已知的自守基变化的实例和自守形式的整体理论来开发p-adic酉群的mod p Langlands对应。 这将丰富已知的mod p朗兰兹对应的实例，表明他们是兼容的函式constructions.This奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Number theory is the branch of mathematics that deals with properties of whole numbers and whole number solutions to polynomial equations, and stands as one of the oldest mathematical disciplines. Representation theory, another equally influential branch of mathematics, quantifies symmetries of geometric objects (such as a square or a hydrogen atom), and has important uses in physics. Though seemingly unrelated, these two areas are intimately linked by the Langlands Program, a vast set of conjectures that allows for the transfer of results and theorems between number theory and representation theory. It is of paramount importance to understand these conjectures, since tools from one discipline can be imported to tackle previously intractable problems in another (the proof of Fermat's Last Theorem being a prime example). This has pushed the Langlands Program to the forefront of current research. The present project seeks to establish instances of a local version of the Langlands Program with mod p coefficients, so that information from representation theory can be transferred into arithmetic data.The setting of the current project lies within the representation theory of p-adic reductive groups (such as GL_2(Q_p)) on mod p vector spaces. Such representations are exceedingly intricate, and one of the main goals is to use derived categories in order to more precisely relate such representations to modules over differential graded Hecke algebras. This will allow for the use of new tools to understand the relationships between Langlands correspondences for varying groups. In addition to this, the PI and his collaborators plan to use known instances of automorphic base change and the global theory of automorphic forms to develop a mod p Langlands correspondence for p-adic unitary groups. This would enrich the known instances of mod p Langlands correspondences by showing that they are compatible with functorial constructions.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.