Exponents of Class Groups of Quadratic Fields

Abstract

Given a positive integer g ≥ 2, we would like to study the number of real and imaginary quadratic fields that have an element of order g in their ideal class group. Conjectures of Cohen and Lenstra predict a positive probability for such an event. Our goal here is to derive quantitative results in this direction. We establish for g ≥ 3, the number of imaginary quadratic fields whose absolute discriminant is ≤ x and whose class group has an element of order g is \(\gg {x^{\frac{1}{2} + \frac{1}{g}}}\). For g odd we show that the number of real quadratic fields whose discriminant is ≤ x and whose class group has an element of order g is » x 1/2g-e for any e > 0. (The implied constant may depend on e.)