Variation of Class Groups of Abelian Extensions of Imaginary Quadratic Fields

Abstract

In number theory, two important objects of study associated to a number field are its class group and its abelian extensions. In the case where the ground field is imaginary quadratic, the abelian extensions are well-understood: they are essentially obtained by adjoining torsion points of elliptic curves with complex multiplication. Through this explicit characterization, a rich theory involving Heegner points of elliptic curves, \(p\)-adic \(L\)-functions, and finite index subgroups of units enables one to study the class groups of these abelian extensions. In this thesis, we count the sizes of isotypical components of certain ray class groups of imaginary quadratic number fields as one varies over an infinite collection, concluding that the number of trivial components grows with the discriminant of the imaginary quadratic field \(K\). We do this by combining a known horizontal variation result for Heegner points with a congruence linking Heegner points with these class numbers.