CAREER: Slopes of p-adic Modular Forms

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open conjectures in the area, including Gouvea's conjecture on slope distribution, the Gouvea-Mazur conjecture, and an unpublished conjecture of Breuil-Buzzard-Emerton, as well as a partial result on the irreducibility of the Coleman-Mazur eigencurve.This award reflects NSF's statutory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