Colloquium Mathematics - Renate Scheidler, University of Calgary and University of Oldenburg

Title: The Ankeny-Artin-Chowla Conjecture in Actual and Fake Real Quadratic Orders

Abstract:

Algebraic number fields, along with their subrings (called orders) are among the most intensely researched objects in number theory. Even the simplest case of quadratic fields – which are subfields of the complex numbers as well as two-dimensional vector spaces over the rational numbers – is subject to many interesting open questions and computational challenges. Quadratic fields come in two flavours: the real fields are contained in the real numbers, whereas the imaginary fields are not, and the two exhibit entirely different structural and invariant properties.

In an unpublished note from 2014, the renowned computational number theorist Henri Cohen made the surprising observation that a certain subring of an imaginary quadratic field where denominators are restricted to powers of one fixed prime behaves very much like a real quadratic order. Cohen coined the term "fake real quadratic order" for these special imaginary subrings. This invites the tantalizing question of whether certain unproved conjectures formulated for actual real quadratic orders also hold in their fake cousins. One such conjecture is the somewhat controversial Ankeny-Artin-Chowla conjecture about fundamental units in real quadratic orders of prime discriminant. Although no counterexamples to have been found to date despite extensive computations, number theorists are divided over whether this conjecture is true or false.

In this talk, which is aimed at an audience of non-specialists, we present extensive numerical data that speak to this conjecture in the settings of both actual and fake real quadratic orders. This is joint work with my colleague Mike Jacobson (University of Calgary), our jointly supervised former graduate student Hongyan Wang and, as of very recently, Florian Hess (University of Oldenburg).