Title: "The Ankeny-Artin-Chowla conjecture in real and false real-quadratic orders "
Abstract : Algebraic number fields and their orders (subrings) are a research focus in number theory. The simplest examples are the quadratic number orders which, despite their relatively simple structure, are in many respects mysterious and the subject of a number of unproven conjectures. The structural and algebraic properties of quadratic orders are fundamentally different depending on whether they are subrings of the real or merely of the complex numbers. In an unpublished note from 2014, the French number theorist Henri Cohen made the surprising observation that the adjunction of a prime ideal to an imaginary-square order forms a number ring that has exactly the characteristics of a real-square order. Cohen described these rings as "fake real quadratic orders", i.e. as false (as opposed to real) real quadratic orders. This raises the natural question of which of the assumptions mentioned above also apply to fake real quadratic orders. The fundamental units of real-quadratic orders of prime discriminants are the subject of such a relatively controversial conjecture, which goes back to Ankeny-Artin-Chowla. In this talk, which assumes only basic algebra knowledge, we will present a version of this conjecture in false real-quadratic orders and report on the results of our very extensive numerical computations. This work was carried out jointly with my colleague Mike Jacobson and our former Masters student Hongyan Wang at the University of Calgary in Canada. The lecture will take place on Wednesday, 18 May 2022 at 17:15 in room W01 0-006 Interested parties are cordially invited.