2025/12/12 by Kopp, Gene S., Lagarias, Jeffrey C.
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Finite Group Theory Research #Cryptography and Residue Arithmetic
paper · doi:10.48550/arxiv.2512.11311
Unit-generated orders of a quadratic field are orders of the form O = ℤ[ε], where ε is a unit in the quadratic field. If the order O is a maximal order of a real quadratic field, then the quadratic number field is necessarily of a restricted form, being of narrow Richaud--Degert type. However, every real quadratic field contains infinitely many distinct unit-generated orders. They are parametrized as O = On± having quadratic discriminants Δ(O) = Δn+ = n2 - 4 (for n ≥ 3) and Δ(O) = Δn- = n2 + 4 (for n ≥ 1). We show the (wide or narrow) class numbers of unit-generated orders satisfy log |\rm Cl(O)| ∼ log (1)/(2)|Δ(O)| as |Δ(O)| → ∞, using a result of L.-K. Hua. We deduce that there are finitely many unit-generated quadratic orders of class number one and finitely many unit-generated quadratic orders whose class group is 2-torsion. We classify all unit-generated real quadratic orders having class number one. We provide numerical lists of quadratic unit-generated orders whose class groups are 2-torsion for Δ≤ 1010, for both wide and narrow class groups. These lists are conjecturally complete for all Δ.