Computing the average root number of an elliptic surface

TL;DR

This paper investigates the average root number of a specific family of elliptic curves over $Q$, demonstrating that the family exhibits parity bias infinitely often, extending previous work on root number distribution.

Contribution

It computes the average root number for elliptic curves over $Q(T)$ with limited places of multiplicative reduction, revealing parity bias in a particular family.

Findings

Average root number computed for explicit family

Family shows parity bias infinitely often

Extends Helfgott's work to new elliptic surface cases

Abstract

By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the variation of the root number for many elliptic surfaces. In this paper, we are concerned with elliptic curves defined over $\mathbb{Q}(T)$ with no place of multiplicative reduction over $\mathbb{Q}(T)$, except possibly at $-$deg. More precisely, we will use the work of Helfgott to compute the average root number of an explicit family of elliptic curves defined over $\mathbb{Q}$ and show that this family is "parity-biased" infinitely-often.