CAREER: Quantifying congruences between modular forms

Michigan State University, East Lansing MI

Investigators

Abstract

Number theory is the study of the most basic mathematical objects, whole numbers. Because whole numbers are so fundamental, number theory has connections with all major areas of mathematics. For instance, consider the problem of finding the whole-number solutions to a given equation. One can consider the shape given by the graph of that equation, or the set of symmetries that the equation has, or the function whose coefficients come from counting the number of solutions over a variety of number systems. The geometric properties of this shape, the algebraic properties of these symmetries, and the analytic properties of this function are all intimately related to the behavior of the equation’s whole-number solutions. Number theorists use techniques from each of these mathematical areas, but also, in the process, uncover surprising connections between the areas whereby discoveries in one area can lead to growth in another. One part of number theory where the connections between geometry, algebra, and analysis are particularly strong is in the field of modular forms. The proposed research focuses on an important and well-known type of relation between different modular forms called congruence and aims to compute the number of forms that are congruent to a given modular form and uncover the number-theoretic significance of this computation. Many of the conjectures that drive this project were found experimentally, through computer calculations. The main educational objective is to contribute to the training of the next generation of theoretical mathematicians in computational and experimental methods. To achieve this, the Principal Investigator (PI) will design software modules for a variety of undergraduate algebra and number theory courses that provide hands-on experience with computation. In addition, the PI will supervise undergraduates in computational research experiments designed to numerically verify conjectures made in the project and to explore new directions. Congruences between modular forms provide a link between two very different types of objects in number theory: algebraic objects, like Galois representations, and analytic objects, like L-functions. This link has been used as a tool for proving some of the most celebrated results in modern number theory, such as the Main Conjecture of Iwasawa theory. The proposed research pushes the study of congruences in a new, quantitative direction by counting the number of congruences, not just determining when a congruence exists. The central hypothesis is that this quantitative structure of congruences contains finer information about the algebraic and analytic quantities involved than the Main Conjecture and its generalizations (such as the Bloch—Kato conjecture) can provide. 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.