Insights from Number Theory: From Arithmetic Geometry to Quantum Topology

Venturing beyond the traditional confines of number theory, this dissertation explores various intersections of the theory of modular forms with other disciplines. In particular, this body of work concerns interconnected discoveries spanning the subjects of the representation theory of the symmetric group, the arithmetic-geometric mean and its finite-field analogue, and newly defined invariants of 3-manifolds.

We first derive asymptotic formulas for two families of zeros within the character tables of the symmetric groups, focusing on those indexed by p-core partitions for primes p>3. These results answer a question of McKay and shed light on the properties of large p-core partitions. These results originally appeared in a joint paper with Ono.

We also revisit the classical arithmetic-geometric mean through a modern lens by investigating a finite field analogue first defined and studied by Griffin, Ono, Saikia, and Tsai. In drawing parallels with the work of Gauss on the classical arithmetic-geometric mean, we uncover the underlying structure of "jellyfish swarms" -- directed graphs that organize the finite field arithmetic-geometric mean. These swarms serve as a novel framework for organizing Legendre elliptic curves over finite fields. This exploration yields new identities for Gauss' class numbers and offers insights into the interplay between jellyfish sizes and the orders of certain elements in related class groups appearing as endomorphism rings of elliptic curves. These results originally appeared in an expository paper with Ono.

Lastly, we extend our research to the realm of quantum modular forms, which surprisingly arise in the study of 3-manifold invariants. Inspired by the pioneering work of Lawrence and Zagier, we give infinite families of quantum modular invariants whose radial limits toward roots of unity may be thought of as a deformation of the Witten-Reshetikhin-Turaev invariants. We use a recently developed theory of Akhmechet, Johnson, and Krushkal (AJK) which extends lattice cohomology and BPS q-series of 3-manifolds. As part of this work, we provide the first calculation of the AJK series invariant for an infinite family of 3-manifolds. These results originally appeared in a joint paper with Liles.