Numerical Tests of Two Conjectures in Fake Real Quadratic Orders
Abstract
A fake real quadratic order is defined based on an imaginary quadratic field and a prime p but behaves similarly to real quadratic orders. Two conjectures regarding fake real quadratic orders are discussed in the thesis. The first one is the Cohen-Lenstra heuristic. Our computation showed that for fixed p, the proportion of fake real quadratic orders for which the odd part of the class number is one converges to C=0.754458..., which equals exactly the proportion of real quadratic fields for which the odd part of the class number is one. The second one is the Ankeny-Artin-Chowla conjecture, which states that D∤b where b is the second coefficient of the fundamental unit in the real quadratic field Q(√D). No counterexamples have been found in real quadratic fields but we found numerous counterexamples in fake real quadratic orders and this is evidence that the conjecture is false for real quadratic fields.
Description
Keywords
Education--Mathematics
Citation
Wang, H. (2017). Numerical Tests of Two Conjectures in Fake Real Quadratic Orders (Master's thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/27146