CAREER: Slopes of p-adic Modular Forms

职业：p-adic 模形式的斜率

• 批准号: 1752703
• 负责人: Liang Xiao
• 金额: $ 40万
• 依托单位: University of Connecticut
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1752703&HistoricalAwards=false
• 关键词: CAREER; Slopes; adic; Modular; Forms

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open conjectures in the area, including Gouvea's conjecture on slope distribution, the Gouvea-Mazur conjecture, and an unpublished conjecture of Breuil-Buzzard-Emerton, as well as a partial result on the irreducibility of the Coleman-Mazur eigencurve.This award reflects NSF's statutory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.

在数字理论中，自然要问一个固定的质数P，两个整数的差异是否可以由P（或P的功率）排除，在这种情况下，我们说两个整数是一致的Modulo P（或其功率）。 P-ADIC数字在20世纪引入，以捕获整数之间的一致性关系：如果p的高力量可以将它们的差异分开，则将两个整数视为“近距离”。这样，我们可以在整数上执行微积分，但具有不同的距离定义。事实证明，在计算问题的实际应用以及理论上的应用（例如Fermat的最后一个定理证明）中，这一概念已被证明是一个有力的工具。 在这个项目中，PI将使用P-ADIC演算研究数字理论中的问题。 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特征形式的Q扩展系数的第三个系数，目的是获得有关近期Bergdall和Pollack最近猜想的新见解，这将通过将其与P-Adic Local Langlands计划联系起来，从而导致证明。 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通过使用基金会的知识分子和更广泛影响的评论标准来通过评估来支持。