JOINT PRINCETON UNIVERSITY/INSTITUTE FOR ADVANCED STUDY NUMBER THEORY SEMINAR

4/19/2004

Harald Helfgott
Yale University


The behavior of the root numbers in families of elliptic curves

Abstract

Let E be a one-parameter family of elliptic curves over Q. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain special condition. We will see that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we will show that any generic family E has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behavior of the root number in any family E not in the generic class will be seen to be rather regular and non-random; we will show how to find expressions for the average root number in this case.