Average root numbers in two isotrivial families of elliptic curves

Abstract

We study root numbers in the isotrivial families $y^2=x^3+a$ and $y^2=x^3+ax$. For a broad class of fixed binary forms, we prove that the average root number over primitive pairs exists. Assuming finiteness of the relevant Tate--Shafarevich groups, we deduce Zariski density for certain del Pezzo surfaces of degree $1$ arising from separable binary sextics. We also establish explicit averages of root numbers for almost all polynomials of each fixed degree $d\geq2$, ordered by coefficient height. The proof develops a new quantitative transference principle for polynomial values.