CAREER: Slopes of p-adic Modular Forms

在数论中，对于一个固定的素数p，很自然地会问两个整数的差是否能被p（或p的幂）整除，在这种情况下，我们说这两个整数模p（或其幂）全等。p-adic数在世纪被引入，以捕捉整数之间的这种同余关系：如果两个整数的差可以被p的高次幂整除，则它们被认为是“靠近”的。这个概念已被证明是数论中的一个强大工具，无论是在计算问题的实际应用中，还是在理论应用中，如费马大定理的证明。 在这个项目中，PI将使用p-adic演算来研究数论中的问题。 此外，PI还将为高年级本科生和研究生举办康涅狄格州数论暑期学校，向他们介绍当代数论的主题（包括p-adic数）。更详细地说，PI将研究模形式的斜率，即模形式空间上Hecke算子在p处的特征值的p-adic赋值，或者等价地，本征形的q-展开式的p阶系数的p-adic赋值，目的是对最近的Bergdall和Pollack猜想获得新的见解，这将导致一个证明，通过将其与p-adic局部Langlands程序相关联。证明这个猜想的成功将导致该领域其他开放猜想的证明，包括Gouvea关于斜率分布的猜想，Gouvea-Mazur猜想，以及Breuil-Buzzard-Emerton未发表的猜想，以及关于科尔曼的不可约性的部分结果-该奖项反映了NSF的法定使命，并被认为是值得通过评估使用基金会的知识优点和更广泛的影响审查标准。

In number theory, it is natural to ask, for a fixed prime number p, whether the difference of two integers is divisible by p (or a power of p), in which case we say the two integers are congruent modulo p (or its power). The p-adic numbers are introduced in the 20th century to capture this congruence relation among integers: two integers are considered "close together" if their difference is divisible by a high power of p. This way, we can perform calculus on integers, but with a different definition of distance. This concept has been proved to be a powerful tool in number theory, both in practical applications to computational problems, and in theoretical applications such as the proof of Fermat's Last Theorem. In this project the PI will use p-adic calculus to study questions in number theory. In addition, the PI will organize Connecticut Summer Schools in Number Theory for advanced undergraduate students and beginning graduate students, to introduce them to topics of contemporary number theory (including the p-adic numbers).In more detail, the PI will study slopes of modular forms, that is the p-adic valuation of the eigenvalues of the Hecke operator at p on the space of modular forms, or equivalently, the p-adic valuation of the p-th coefficients of q-expansions of the eigenforms, with the goal of gaining new insight on the recent conjecture of Bergdall and Pollack, that will lead to a proof, by relating it to the p-adic local Langlands program. Success in proving this conjecture will lead to proofs of other open conjectures in the area, including Gouvea's conjecture on slope distribution, the Gouvea-Mazur conjecture, and an unpublished conjecture of Breuil-Buzzard-Emerton, as well as a partial result on the irreducibility of the Coleman-Mazur eigencurve.This award reflects NSF's statutory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.