Abstract

The root number $w$ of an elliptic curve defined over $\Q$ has an intrinsic definition as an infinite product of \textit{local root numbers} $w_p$, over all places $p$ of $\Q$, with $w_p=\pm 1$ for all $p$ and such that $w_p=1$ for all but finitely-many $p$. By considering a one-parameter family of elliptic curves defined over $\Q$, 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 in his Ph.D. thesis, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\Q(T)$ is zero as soon as there is a place of multiplicative reduction over $\Q(T)$ other than $-\deg$.

In this paper, we are concerned with elliptic curves defined over $\Q(T)$ with no place of multiplicative reduction over $\Q(T)$, except possibly at $-\deg$. More precisely, we use the work of Helfgott to compute the average root number of an explicit family of elliptic curves defined over $\Q$ and show that this family is "parity-biased" infinitely-often.