Functoriality in the Mod-p Langlands Program

数论是数学的一个分支，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。表象理论是另一个同样有影响力的数学分支，它量化几何对象(如正方形或氢原子)的对称性，在物理学中有重要的用途。尽管这两个领域看似不相关，但朗兰兹计划将这两个领域紧密联系在一起，朗兰兹计划是一组巨大的猜想，允许在数论和表象理论之间转移结果和定理。理解这些猜想是至关重要的，因为可以从一个学科引入工具来解决另一个学科以前难以解决的问题(费马大定理的证明就是一个最好的例子)。这将朗兰兹计划推向了当前研究的前沿。本项目旨在建立具有mod p系数的局部形式的朗兰兹规划的实例，以便将表示理论中的信息转化为算术数据.本项目的背景在于mod p向量空间上的p-进还原群(如GL_2(Q_P))的表示理论.这种表示非常复杂，主要目的之一是使用派生范畴，以便更精确地将这种表示与微分分次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通信之间的关系。除此之外，PI和他的合作者计划利用已知的自同构基变化的实例和自同构形的整体理论来发展p-ady酉群的mod p-朗兰兹对应。这将丰富已知的现代朗兰兹通信实例，表明它们与功能结构兼容。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.