The somewhat controversial Ankeny-Artin-Chowla (AAC) conjecture asserts a certain divisibility condition about fundamental units in real quadratic orders of prime discriminant. Although no counterexamples have been found despite extensive computations, number theorists are divided over the truth of this conjecture. A closely related conjecture, due to Mordell, was recently established to be false by Reinhart who found a counterexample.

In the hopes that an investigation of a “fake” AAC analogue might shed light on the original AAC conjecture, we investigated AAC in fake real quadratic orders. In this talk, which is aimed at a general math audience, I report our findings, consisting of extensive numerical computations, heuristics and asymptotic results. This is joint work with Hongyan Wang (a former Master's student at University of Calgary), Florian Hess (University of Oldenburg, Germany) and Mike Jacobson (University of Calgary).