DISTRIBUTION OF PRIME ORDER IDEAL CLASSES OF QUADRATIC CLASS GROUPS

Abstract

The Cohen-Lenstra heuristics predicts that for any odd prime k, the k-part of quadraticclass groups occur randomly on the space of k-groups with respect to a natural probability measure. However, apart from the first moments of the 3-torsion part of quadratic class groups, consequences of these heuristics still remain highly conjectural. The quadratic ideal classes have geometric representations on the modular curve as CM-points in the case of negative discriminants and as closed primitive geodesics in the case of positive discriminants. We mainly focus on the asymptotic distribution of these geometrical objects. As motivation, it is seen that in the case of imaginary quadratic fields, knowledge on the (joint) distribution of k-order CM-points leads in-general to the resolution of the Cohen-Lenstra conjectures on moments of the k-part of class groups. As a first step, inspired by the works of Duke, Hough conjectures that the k-order CM-points are equidistributed on the modular curve. Although the case with k = 3 was resolved by Hough himself, k > 3 remains unresolved. In this dissertation, we revisit Hough’s conjectures, with empirical evidence. We were able to reprove the conjecture for k = 3, and even stronger to show that the result holds along certain subfamilies of imaginary quadratic fields defined by local behaviors of their discriminants. In addition, we study the case for k > 3. We introduce a heuristics model, and show that this model agrees with Hough’s conjectures. We also show that the difference between the actual asymptotics and the heuristic model reduces down to the distribution of solutions to certain quadratic congruences. We, then again inspired by Duke’s work, investigate an analog for the real quadratic fields. Backed by empirical evidence, we go on to conjecture the asymptotic behavior of the length of k-order geodesics on the modular curve. In addition, based on a theorem and its proof by Siegel, we prove certain results that may shed light on a probable proof direction of these conjectures.