Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

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/3Z)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 ⩽ 10⁶ and F(G) for G a p-group of odd order with |G| ⩽ 10⁶. (In order to do this, we need to examine the class numbers of all negative prime fundamental discriminants − q, for q ⩽ 1.1881 × 10¹⁵.) 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.