Computing the average root number of an elliptic surface

Abstract

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 [8], 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. In [1], the authors classify all such one-parameter families of elliptic curves whose coefficients, in the parameter T, have degree less than or equal to 2; they also use the work of Helfgott to compute the average root number of two particular subfamilies. We complement the work in [1] by computing the average root number of one of these “potentially parity-biased” families and show that it is “parity-biased” infinitely-often.