The Department of Mathematics

College of Charleston

in conjunction with

The South Eastern Regional Meeting On Numbers (SERMON) conference

proudly present a talk

"Class-Numbers
of CM fields and Siegel Zeros."

by

Harold Stark

University of California, San Diego

on

Friday, April 16, 2004

4:30 P.M. Maybank Hall, Room 100

**Abstract**

The class-number of a quadratic field is a measure of how close the integers
of the field come to having the unique factorization property. Class-number
one is equivalent to unique factorization. Quadratic fields come in two
flavors, real and complex. Real quadratic fields are those which are subsets
of the real numbers and complex quadratic fields are the rest. CM fields
generalize the concept of complex quadratic fields. A CM field is a totally
complex quadratic extension of a totally real field. In the 1970s, the conjecture
was put forward that there are only finitely many CM fields (over all degrees)
with a given class-number. This generalizes the two hundred year old conjecture
of Gauss that there are only a finite number of complex quadratic fields
with any fixed class-number. The original Gauss conjecture was proved in
the 1930s ineffectively and only effectivelyproved in the 1990s by Gross
and Zagier. The more general conjecture for CM fields of fixed degree has
also been proved effectively (and indeed, for fixed degree six or more, was
effectively proved in 1974!). However, the degree independent conjecture
remains open. It has long been thought that the obstruction is the possible
existence of "Siegel zeros", but we will see that this is not completely
accurate.