Pär Kurlberg: The distribution of class groups for binary quadratic forms

Gauss made the remarkable discovery that the set of integral binary
quadratic forms of fixed discriminant carries a composition law, i.e.,
two forms can be "glued together" into a third form. Moreover, as two
quadratic forms related to each other via an integral linear change of
variables can be viewed as equivalent, it is natural to consider
equivalence classes of quadratic forms. Amazingly, Gauss' composition
law makes these equivalence classes into a finite abelian group – in a sense it is the first abstract group "found in nature". The fine scale structure of these groups is rather mysterious - in many ways the
groups "behave randomly"; even modeling their cardinality remains a
challenge.

Extensive calculations led Gauss and others to conjecture that the
number h(d) of equivalence classes of such forms of negative
discriminant d tends to infinity with |d|, and that the class number is
h(d) = 1 in exactly 13 cases: d is in {-3, -4, -7, -8, -11, -12, -16,
-19, -27, -28, -43, -67, -163}. While this was known assuming the
Generalized Rieman Hypothesis, it was only in the 1960's that the
problem was solved by Alan Baker and by Harold Stark.

We will outline the resolution of Gauss' class number one problem and
survey some known results regarding the growth of h(d). We will also consider how often a fixed abelian group occur as a class group --- is there a natural probability measure on the set of abelian groups? Using a probabilistic model we will address the question: do all abelian groups occur, or are there "missing" class groups?