Congruence relations of Ankeny–Artin–Chowla type for real quadratic fields

In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of [Formula: see text] for positive square-free integers [Formula: see text]. Many of the ideas present in their paper can be seen as the precursors to the now developed theory of cyclotomic fields. Curiously, little attention has been paid to the cases of [Formula: see text] in the literature. In this work, we show that the congruences of the type proven by Ankeny, Artin, and Chowla can be seen as a special case of a more general methodology using Kubota–Leopoldt p-adic L-functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in [Formula: see text] by relating these values to a well known expression for [Formula: see text]. We conclude with a discussion of known counterexamples to the so-called Composite Ankeny–Artin–Chowla conjecture and relate these to special dihedral extensions of [Formula: see text].