Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication

Abstract

Since Gauss, ideal class groups of imaginary quadratic fields have been the focus of many investigations, and recently there have been many investigations regarding TateShafarevich groups of elliptic curves. In both cases the literature is quite extensive, but little is known. Throughout D will denote a fundamental discriminant of a quadratic field. Let CL(D) denote the class group of Q( √ D), and let h(D) denote its order, i.e. the usual class number of primitive positive binary quadratic forms with discriminant D. One of the main problems deals with the structure of CL(D), and so one naturally studies the divisibility of h(D) by primes. Here we consider imaginary quadratic fields. Gauss’ genus theory precisely determines the parity of h(D), but the divisibility of h(D) by odd primes ` is much less well understood. In view of these difficulties, Cohen and Lenstra [C-L] gave heuristics describing the “expected” behavior of CL(D), and in particular they predicted that the probability that ` h(D) is