Congruence relations of Ankeny$\unicode{x2013}$Artin$\unicode{x2013}$Chowla type for real quadratic fields

TL;DR

This paper generalizes classical congruences related to class numbers of real quadratic fields using $p$-adic $L$-functions, extending previous results and discussing counterexamples to related conjectures.

Contribution

It introduces a new methodology using Kubota–Leopoldt $p$-adic $L$-functions to unify and extend congruence relations for real quadratic fields, including cases previously overlooked.

Findings

Derived new congruences involving quadratic residues and non-residues.

Connected classical congruences to $p$-adic $L$-functions.

Discussed counterexamples to the Composite Ankeny–Artin–Chowla conjecture.

Abstract

In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of $\mathbb{Q}(\sqrt{d})$ for positive squarefree integers $d\equiv 1 \bmod{4}$. 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 $d\equiv 2,3\bmod{4}$ in the literature. In the present 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$\unicode{x2013}$Leopoldt $p$-adic $L$-functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in $\mathbb{Z}/p\mathbb{Z}$ by relating these values to a well known expression for $L_p(1, \chi)$. We conclude with a discussion of known counterexamples to the so-called Composite Ankeny$\unicode{x2013}$Artin$\unicode{x2013}$Chowla conjecture and relate these to special dihedral extensions of $\mathbb{Q}$.