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q-series Identities Connected to Ideals in Quadratic Fields

q-series Identities Connected to Ideals in Quadratic Fields

Wednesday, July 31, 2024 3:00 pm
STAG 163
Lucas Perryman-Deskins
Oregon State University

A certain $q$-hypergeometric series $\sigma$ was shown by Andrews, Dyson, and Hickerson to have an interpretation in terms of counting ideals in $\mathbb{Z}(\sqrt{6})$. This interpretation was used to demonstrate unique combinatorial and analytic characteristics, along with some interesting $q$-series identities. Some of these identities reflect a correspondence, predicted by class field theory, between characters on the ideals of each quadratic subfield of a biquadratic field. Cohen described that there are many quadratic fields where similar identities are theoretically discoverable, but few have been actually explored.

Cohen also constructed from $\sigma$ a related Maass form, with later work generalizing this construction to a class of functions which are mock Maass theta functions, and in special cases Maass forms.

In this talk, I will discuss the some background in each of these areas, and the history of research on this problem, as well as my current approach working towards generating further identities in other fields.

Holly Swisher