SPEAKER:     Lillian Pierce

TITLE:    A Bound for the 3-Part of Class Numbers of Quadratic Fields via the Square Sieve

ABSTRACT:

Since Gauss's publication of Disquisitiones Arithmeticae in 1801, mathematicians have been interested in the divisibility properties of class numbers. However, still today many of the properties of class numbers remain mysterious. In this talk we investigate the divisibility by 3 of class numbers of quadratic fields. It is conjectured that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded above by an arbitrarily small power of |D|. However, until recently, the only known bound was the trivial bound O(|D|^{1/2 + epsilon}).

We use a variant of the square sieve and the q-analogue of van der Corput's method to count the number of squares of the form 4x^3 - dz^2, where d is a square-free positive integer and x and z lie in the ranges x << d^{1/2}, z<< d^{1/4}. As a result, we show that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded by O(|D|^{27/56 + epsilon}). This gives a corresponding bound for the number of elliptic curves over the rationals with conductor N.