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.