23rd November 2004 <br>
Lillian Pierce, Princeton<br><br>
"A Bound for the 3-Part of Class Numbers of Quadratic Fields
via the Square Sieve" 
<br><br>
ABSTRACT<br>
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.