2024/10/28 by Nic Fellini, Fellini, Nic
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2410.20934
openalex publication_date 2024/10/28 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/28
In 1951, Ankeny, Artin, and Chowla published a brief note containing four congruence relations involving the class number of ℚ(√(d)) for positive squarefree integers d≡ 1 \bmod4. 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≡ 2,3\bmod4 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\unicodex2013Leopoldt p-adic L-functions. Aside from the classical congruence involving Bernoulli numbers, we derive congruences involving quadratic residues and non-residues in ℤ/pℤ by relating these values to a well known expression for Lp(1, χ). We conclude with a discussion of known counterexamples to the so-called Composite Ankeny\unicodex2013Artin\unicodex2013Chowla conjecture and relate these to special dihedral extensions of ℚ.