Class Numbers of Quadratic Fields

Abstract

International audience We present a survey of some recent results regarding the class numbers of quadratic fields

NOTE ON THE p-DIVISIBILITY OF CLASS NUMBERS OF AN INFINITE FAMILY OF IMAGINARY QUADRATIC FIELDS

“…Later, in 1988, Mollin and Williams [2] proved this conjecture under the assumption of the generalized Riemann hypothesis. Chowla [3] also formulated a conjecture analogous to this for a broader family of real quadratic fields. Specifically, he conjectured the following:…”

Cotorsion of anti-cyclotomic Selmer groups on average

“…As for the last property, the Cohen-Lenstra heuristics predict that among all imaginary quadratic fields, the proportion for which p divides the class number is [BM19], [CL84, Section 9.I] (4.1)…”

A brief survey of recent results on Pólya groups

“…Leriche classified sextic Polya fields that contain a pure cubic field as a subfield. In other words, she characterized Pólya fields of the form K = Q(ω, 3 √ m), where ω is a complex cube root of unity.…”

“…The interested reader is encouraged to look at [5], [6], [11], [24], [33], [43] and the references listed therein. Recently, Bhand and Murty [3] have written an elaborate article on the class numbers of quadratic fields and it gives a very nice overview of the recent results on this theme.…”