Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

ABSTRACT

The number $\mathcal{F}\left(h\right)$ 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 $\mathcal{F}\left(h\right)$. The unconditional computation of $\mathcal{F}\left(h\right)$ 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 $\mathcal{F}\left(h\right)$ 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 $\mathcal{F}\left(G\right)$ 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, ${(\mathbb{Z}/3\mathbb{Z})}^3$ 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 $\mathcal{F}\left(h\right)$ 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 $\mathcal{F}\left(h\right)$ for odd h ⩽ 10⁶ and $\mathcal{F}\left(G\right)$ 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.

KEYWORDS:

• class numbers
• class groups
• Cohen–Lenstra heuristics

2010 AMS Subject Classification:

• 11R29
• 11Y40

Acknowledgments

The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC Centre for High Performance Computing (PDC-HPC). We thank the PDC support (Magnus Helmersson, Jonathan Vincent, Radovan Bast, Peter Gille, Jin Gong, Mattias Claesson) for their assistance concerning technical and implementational aspects in making the code run on the PDC-HPC resources.

We would like to thank Andrew Booker, Pete Clark, Henri Cohen, Noam Elkies, Farshid Hajir, Hendrik W. Lenstra, Steve Lester, and Peter Sarnak for enlightening conversations on the topic. We would also like to thank Tobias Magnusson and the anonymous referee for carefully reading of the manuscript and suggesting improvements.

Funding

P.K. was partially supported by grants from the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine, and the Swedish Research Council (621-2011-5498). S.H. was partially supported by a grant from the Swedish Research Council (621-2011-5498). K.P. was partially supported by Simons Foundation grant # 209266.

Notes

1 The complete list of computed values of $\mathcal{F}\left(h\right)$ is given in [Holmin and Kurlberg XX].

2 In fact, a negative second-order correction to (2–1(2--1) $$\begin{array}{c}\hfill \sum _{\begin{array}{c}-X<d<0\\ d\text{fund.}\text{disc.}\end{array}}\left|H\left(d\right)\left[3\right]\right|\sim CX,\end{array}$$(2--1) ) of size X^5/6 was recently obtained by [Taniguchi and Thorne 13], and by [Bhargava et al. 13].

3 The complete list of all $\mathcal{F}\left(G_{\lambda }\left(p\right)\right)$ is given in [Holmin and P. Kurlberg XX], and a complete list of all corresponding discriminants d and groups H(d) is given in [Holmin and P. Kurlberg XX].

4 For techniques to quickly compute these constant with very high precision, see [Cohen Preprint, Section 2].

5 Using that h is odd, a well-known refinement of Shanks’ algorithm gives us a speedup factor of $\sqrt{2}$.

6 We expect the class group to be cyclic more than 97.7% of the time, and class groups containing $\mathbb{Z}/q\mathbb{Z}\times \mathbb{Z}/q\mathbb{Z}$ for prime q > 5 are very rare (cf. [Cohen and Lenstra 84, p. 56]).