23rd November 2004
Lillian Pierce, Princeton

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

Since Gauss's publication of Disquisitiones Arithmeticae in 1801, mathematicians have been interested in the divisibility properties of class numbers. However, still today little has been proved about such divisibility properties. In this talk we investigate the divisibility by 3 of class numbers of quadratic fields. 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 << 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.