Non-isotrivial elliptic surfaces with non-zero average root number

TL;DR

This paper classifies certain elliptic curve families with non-zero average root number, computes their ranks, and explores the range of possible average root numbers, revealing rich arithmetic structures.

Contribution

It classifies all degree ≤ 2 elliptic curve families with non-zero average root number and computes their ranks and explicit average root numbers.

Findings

Classified all degree ≤ 2 families with non-zero average root number.

Computed ranks over (t) for these families.

Showed all rational numbers in [-1,1] can be average root numbers.

Abstract

We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbb{Z}$. We classify all such families when the degree of the coefficients (in the parameter $t$) is less than or equal to $2$ and we compute the rank over $\mathbb{Q}(t)$ of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in $[-1,1]$ are average root numbers for some $1$-parameter family.