An Investigation of Bounds for the Regulator of Quadratic Fields

Abstract

It is well known that the nontorsion part of the unit group of a real quadratic field K is cyclic. With no loss of generality we may assume that it has a generator ∊₀ > 1, called the fundamental unit of K. The natural logarithm of ∊₀ is called the regulator R of K. This paper considers the following problems: How large, and how small, can R get? And how often?

The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large R can get. In order to investigate this, we conducted several large-scalenumerical experiments, involving the ExtendedRiemann Hypothesis and the Cohen-Lenstra class number heuristics. These experiments provide numerical confirmation for what is currently believed about the magnitude of R.