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.