The 3-Part of Class Numbers of Quadratic Fields

In 1801, Gauss published Disquisitiones Arithmeticae, which, among many other things, develops genus theory, describing the divisibility by 2 of class numbers of quadratic fields. In the centuries since this work, the divisibility properties of class numbers by integers g ≥ 3 have largely remained mysterious. In particular, the problem of bounding the g-part hg(D) of class numbers of quadratic fields Q( √ D) for g ≥ 3 has remained unsolved. This thesis provides three nontrivial bounds for h3(D), giving the first improvement on the previously known trivial bound h3(D) |D| . This thesis approaches the problem via analytic number theory, phrasing the problem of bounding h3(D) in terms of counting the number of integer points in a bounded region on the cubic surface 4x = y + dz, for a positive square-free integer d. We obtain our first two nontrivial bounds for h3(D) by regarding this as the congruence 4x ≡ y modulo d. Using exponential sum techniques, we prove two nontrivial bounds for the number of solutions to a congruence of the more general form x ≡ y (mod q), for a positive square-free integer q and nonzero integers a, b. As results of these bounds, we show that h3(D) |D| if D has a divisor of size |D|, and h3(D) |D| in general. We obtain a third nontrivial bound of h3(D) |D| by counting the number of integer points on the cubic surface directly. Specifically, we estimate the number of squares of the form 4x − dz, using the square sieve and the q-analogue of Van der Corput’s method. Each of our three bounds for h3(D) also gives a corresponding improvement on the previously known bound for the number of elliptic curves over Q with fixed conductor.