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q-series Identities Connected to Ideals in Quadratic Fields
STAG 163
REU ColloquiumSpeaker: Lucas Perryman-Deskins
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… Read more.