The distribution of class numbers in families of quadratic fields

Improving a result of Montgomery and Weinberger, we establish 
the existence of infinitely many real quadratic fields for which the 
class numbers are as large as possible. These values are achieved using 
a special family of fields, first studied by Chowla. In a subsequent 
work, joint with A. Dahl, we investigate the distribution of class 
numbers in Chowla’s family, and show a strong similarity between this 
distribution and that of class numbers of imaginary quadratic fields, 
previously studied by Granville and Soundararajan. As an application of 
these methods, we determine the average order of the number of quadratic 
fields with class number $h$ in several families including Chowla's 
family of real quadratic fields, and the family of imaginary quadratic 
fields with prime discriminants.