The class number value problem

Abstract

It was ascertained in previous chapters that upper bounds for the solutions of diophantine equations under our consideration depend essentially on the regulators of certain algebraic number fields related to the equation. Now we concentrate our attention on this phenomenon and relate it to the general problem of the magnitude of ideal class numbers. We show that algebraic number fields with ‘small’ regulator (hence ‘large’ class number) occur very frequently and in some sense constitute the majority of fields. Bounds for the solutions of the corresponding diophantine equations, e.g., Thue equations, are much better than the general bounds.