Automorphic Forms, Arthur Packets, and Algebraic Cycles

数论中的一个经典问题是求多项式方程组的有理数或整数解。在20世纪，数学家们意识到这个问题可以用称为代数圈的几何对象重新表述。这一认识产生了一个巨大而美丽的数学框架，现在包括一些最重要的数学未解决的问题。另一方面，二次互反律的发现（由于高斯）及其推广最终导致了朗兰兹纲领的形成，这是一个将数的对称性与分析和群论联系起来的独立的网络。这个项目中要研究的问题位于这两个不同的网络结构的界面上，因此涉及到大量算术丰富性的对象。该项目的具体目标是利用对某些高度对称函数的研究来揭示多项式方程的几何和算术信息。虽然最终目标是揭示有关多项式的复杂信息，但许多要研究的对象都有直接的实际应用。例如，椭圆曲线，这是两个变量的三次方程，在这项研究中发挥了突出的作用，也是一个重要的作用，在当代的应用，如密码学和数字签名，在商业中有广泛的用途。该项目的更广泛影响之一将是开发一门关于比特币等加密货币数学的课程，通过当前广泛流行的令人兴奋的应用推广数学。该计划亦会为研究生提供研究训练活动。以专业术语来说，研究的重点是利用自守表示的精细结构，包括局部和整体亚瑟包理论，研究代数圈的问题。具体研究的问题包括：（i）构造表示Langlands函性实例的Hodge圈，特别是酉群的Hodge圈;（ii）Oda关于Hilbert模形式的Hodge结构的因式分解的猜想;（iii）上同调平凡圈的Abel-Jacobi象与p-adic L-函数之间的关系;（iv）四元数模形式的整周期关系;（v）非调和A-包理论在局部对称空间上推广Kudla-Millson理论中的应用。该奖项反映了NSF的法定使命，并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。

A classical question in number theory is to find solutions in rational numbers or integers to systems of polynomial equations. In the twentieth century, mathematicians realized that this question can be reformulated using geometric objects called algebraic cycles. This realization gave rise to a vast and beautiful conjectural framework, which now includes some of the most important unresolved conjectures in mathematics. On the other hand, the discovery of the law of quadratic reciprocity (due to Gauss) and its generalizations lead ultimately to the formulation of the Langlands program, which is a separate web of conjectures relating the symmetries of numbers to analysis and group theory. The questions to be studied in this project lie at the interface of these two different webs of conjectures, and thus involve objects of enormous arithmetic richness. The specific goal of the project is to use the study of certain highly symmetric functions to reveal information about the geometry and arithmetic of polynomial equations. While the final goal is to reveal sophisticated information about polynomials, many of the objects to be studied have immediate practical applications. For example, elliptic curves, which are cubic equations in two variables, play a prominent role in this research and also an important role in contemporary applications such as cryptography and digital signatures, which have extensive use in commerce. One of the broader impacts of the project will be the development of a course on the mathematics of cryptocurrencies such as bitcoin, popularizing mathematics through an exciting application of broad current popular interest. The project will also provide research training activities for graduate studentsIn technical terms, the main thrust of the research is to use the fine structure of automorphic representations, including the theory of local and global Arthur packets, to study problems on algebraic cycles. Specific problems to be studied include: (i) constructing Hodge cycles that represent instances of Langlands functoriality, especially for unitary groups; (ii) Oda's conjecture on the factorization of Hodge structures of Hilbert modular forms; (iii) relations between Abel-Jacobi images of cohomologically trivial cycles and p-adic L-functions; (iv) integral period relations for quaternionic modular forms, and (v) applications of the theory of non-tempered A-packets to generalizations of Kudla-Millson theory on locally symmetric spaces. The investigator will continue to mentor graduate student research on topics related to the themes in the project.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.