On the behaviour of root numbers in families of elliptic curves

TL;DR

This paper investigates the average root number in families of elliptic curves over Q, showing it is zero under broad conditions and analyzing its behavior in various cases, contingent on classical conjectures.

Contribution

It establishes that the average root number is zero for many families of elliptic curves, especially those with points of multiplicative reduction, under certain conjectural assumptions.

Findings

Average root number is zero for large classes of families.

Families with at least one point of multiplicative reduction have average root number 0 under conjectures.

In families without multiplicative reduction, the root number behavior is regular and explicitly characterized.

Abstract

Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root number in this case.