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 $h \le 100$ was completed by M. Watkins, and K. Soundararajan has more recently made conjectures about the order of magnitude of $F(h)$ as $h \to \infty$ and determined its average order.
For odd $h$ we refine Soundararajan's conjecture to a conjectural asymptotic formula and also consider the subtler 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 $(\mathbb{Z}/3\mathbb{Z})^3$ 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 the Cohen-Lenstra heuristics with a refinement of Soundararajan's conjecture 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 families of $p$-groups as $p$ tends to infinity. For instance, it appears that no groups of the form $(\mathbb{Z}/p\mathbb{Z})^3$ and $p$ prime occurs as a class group of a quadratic imaginary field.
Conditionally on the Generalized Riemann Hypothesis, we extend Watkins' data, tabulating $F(h)$ for odd $h \le 10^6$ and $F(G)$ for $G$ a $p$-group of odd order with $|G| \le 10^6$. (To do this, we examine the class numbers of all negative prime fundamental discriminants $-q$, for $q \le 1.1881 \cdot 10^{15}.$) The numerical evidence matches quite well with our conjectures.
This is joint work with S. Holmin, N. Jones, C. McLeman, and K. Petersen.[-]

We discuss Arithmetic Statistics as a 'new' branch of number theory by briefly sketching its development in the last 50 years. The non-triviality of the meaning of `random behaviour' and the problematic absence of good probability measures on countably infinite sets are illustrated by the example of the 1983 Cohen-Lenstra heuristics for imaginary quadratic class groups. We then focus on the Negative Pell equation, of which the random behaviour in the case of fundamental discriminants (Stevenhagen's conjecture) has now been established after 30 years.
We explain the open conjecture for the general case, which is based on equidistribution results for units over residue classes that remain to be proved.[-]