Mathematics Colloquium

Tel Aviv University

Monday, November 13, 2017
Schreiber 006
Pär Kurlberg
KTH Stockholm
Class numbers and class groups for definite 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".

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 some recent conjectures regarding how often a fixed abelian group occur as a class group, and how often an integer occurs as a class number. In particular: do all abelian groups occur, or are there "missing" class groups?

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