Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields. Issue 2 (3rd April 2019)

• Record Type: Journal Article
• Authors: Holmin, S.; Jones, N.; Kurlberg, P.; McLeman, C.; Petersen, K.
• Is Part Of: Experimental mathematics. Volume 28:Issue 2(2019)
• Page Start: 233
• Page End: 254
• Publication Date: 2019-04-03
• Subjects: class numbers -- class groups -- Cohen–Lenstra heuristics; 11R29 -- 11Y40; Mathematics -- Periodicals; Mathematics -- Research -- Periodicals; 510.724
• DOI: 10.1080/10586458.2017.1383952
• Languages: English
• ISSNs: 1058-6458

Abstract

ABSTRACT: The number F ( h ) of imaginary quadratic fields with class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F ( h ) . The unconditional computation of F ( h ) for h ⩽ 100 was completed by Watkins, using ideas of Goldfeld and Gross–Zagier; Soundararajan has more recently made conjectures about the order of magnitude of F ( h ) as h → ∞ and determined its average order. In the present paper, we refine Soundararajan's conjecture to a conjectural asymptotic formula for odd h by amalgamating the Cohen–Lenstra heuristic with an archimedean factor, and obtain an adelic, or global, refinement of the Cohen–Lenstra heuristic. We also consider the problem of determining the number F ( G ) of imaginary quadratic fields with class group isomorphic to a given finite abelian group G . Using Watkins' tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance, ( Z / 3 Z ) 3 does not). This observation is explained in part by the Cohen–Lenstra heuristics, which have often been used to study the distribution of the p -part of an imaginary quadratic class group. We combine heuristics of Cohen–Lenstra together with our prediction for the asymptotic behavior of F ( h ) to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of "missing" class groups, for the case of p -groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating F ( h ) for odd h ⩽ 10 6 and F ( G ) for G a p -group of odd order with | G | ⩽ 10 6 . (In order to do this, we need to examine the class numbers of all negative prime fundamental discriminants − q, for q ⩽ 1.1881 × 10 15 .) The numerical evidence matches quite well with our conjectures, though there appears to be a small "bias" for class number divisible by powers of 3.