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

Nic Fellini

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 $L_p(1, χ)$. 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 $\mathbb{Q}$.

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

Abstract

In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of for positive squarefree integers . 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 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 KubotaLeopoldt -adic -functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in by relating these values to a well known expression for . We conclude with a discussion of known counterexamples to the so-called Composite AnkenyArtinChowla conjecture and relate these to special dihedral extensions of .

Paper Structure

This paper contains 10 sections, 21 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. $p$-adic preliminaries
  4. The $p$-adic Logarithm
  5. The Teichmüller Character
  6. The Kubota--Leopoldt $L$-function
  7. Proof of Theorem \ref{['thm: 1']}
  8. Proof of Theorem \ref{['thm: 2']}
  9. Counterexamples to the Ankeny--Artin--Chowla conjecture
  10. Dihedral extensions of $\mathbb{Q}$

Key Result

Theorem 1.1

Let $d=pm$ be a positive squarefree integer, $p$ an odd prime, and $k=\mathbb{Q}(\sqrt{d})$. We write $D=\delta^2d$, $h$, and $\varepsilon = \frac{\delta}{2}(t+u\sqrt{d} )$ for the discriminant, class number and fundamental unit of $k$ respectively.

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2: $\text{Ankeny--Artin--Chowla}$ and Kiselev
  • Corollary 1.3: Kiselev, Mordell
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Remark
  • Remark
  • Theorem 1.8
  • ...and 23 more