Siman WongFarshid HajirTom WestonKucuksakalli, Omer2024-04-262024-04-262009-0510.7275/6c2c-fv53https://hdl.handle.net/20.500.14394/39160Let K be an imaginary quadratic field with class number one and let [Special characters omitted.] be a degree one prime ideal of norm p not dividing 6 d K . In this thesis we generalize an algorithm of Schoof to compute the class number of ray class fields [Special characters omitted.] heuristically. We achieve this by using elliptic units analytically constructed by Stark and the Galois action on them given by Shimura's reciprocity law. We have discovered a very interesting phenomena where p divides the class number of [Special characters omitted.] . This is a counterexample to the elliptic analogue of a well-known conjecture, namely the Vandiver's conjecture.Class numbersComplex multiplicationElliptic curvesQuadratic fieldsRay class fieldsImaginary quadratic fieldsMathematicsdissertation