Canadian
Mathematical
Society
Dr. Renate Scheidler
University of Calgary
Dr. Renate Scheidler is a Professor at the University of Calgary, with a joint appointment in the Department of Mathematics & Statistics and the Department of Computer Science. She received her Master’s degree in mathematics from the University of Cologne and her PhD in computer science from the University of Manitoba under the supervision of Hugh Williams. In 2022, she held a one-year Helene Lange Visiting Professorship at the University of Oldenburg in Germany. Dr. Scheidler’s research straddles the interface between mathematics and computer science, centering on the design and analysis of algorithms and computations in global fields in the context of algebraic number theory, arithmetic geometry and cryptography. She is a Fellow of the Association for Women in Mathematics, and a co-founder and steering committee member of the Women in Numbers network.
The Ankeny-Artin-Chowla Conjecture in Actual and Fake Real Quadratic Orders
Quadratic orders exhibit vastly different structural and invariant properties, depending on whether the ambient quadratic field is real or imaginary. In an unpublished note from 2014, Henri Cohen made the surprising observation that a certain subring of an imaginary quadratic order where denominators are restricted to powers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 structures.
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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.
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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).
Krieger-Nelson Prize Winner
