LEAPS-MPS: Elliptic Dedekind Sums, Eisenstein Cocycles, and p-adic L-Functions

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This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2).Number theory studies the properties of prime numbers and has important applications in physics, computer science, cyber security, and other areas. There are important classes of numbers extending the usual integers and corresponding classes of functions known as modular forms. The goal of this project is to extend known results about classical modular forms over the usual integers to more general settings. It is expected that different phenomena will appear in this context, which will yield new and interesting applications to number theory and beyond.The project will develop explicit methods and results in relation to the arithmetic of Bianchi modular forms, elliptic Dedekind sums, and L-functions over imaginary quadratic fields. Building upon previous work, the PI will continue the study of this circle of ideas and explore their connections to open problems and conjectures, namely (1) a generalization of Sczech’s Eisenstein cocycles over imaginary quadratic fields with applications to p-adic L-functions, (2) elliptic Dedekind sums attached to discrete subgroups of SL(2,C) and arithmetic applications such as to conjectures of Sharifi, and (3) Bianchi period polynomials associated to binary Hermitian forms, with applications to special values of L-functions.The broader impacts of this project are through the research training and mentorship of undergraduate students in a Primarily Undergraduate Institution — many of whom being first-generation students and members of under-represented minorities.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.