Constant root number on integer fibres of elliptic surfaces

TL;DR

This paper explores families of elliptic curves with constant root number over integer fibers, generalizing a known example and identifying conditions for constant root number in specific small degree families.

Contribution

It generalizes a known elliptic curve family with constant root number and characterizes conditions for this phenomenon in new small degree families.

Findings

Identified conditions for constant root number in family _s(t)

Provided partial results for family _{w,s,v}(t)

Examples of subfamilies with rank elevation at integer fibers

Abstract

Rizzo showed that the family of elliptic curves $\mathcal{W}(t) :y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, has root number $W(\mathcal{W}(t))=-1$ for all $t\in\mathbb{Z}$. In this paper we generalize this example and identify the families of small degree on which this phenomenon happens. Motivated by results from David, Bettin and Delaunay (arXiv:1612.03095) and Desjardins (arXiv:1810.12787), we study in detail the two families $\mathcal{F}_s(t):y^2=x^3+3tx^2+3sx+st$ and $\mathcal{L}_{w,s,v}(t): wy^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ and describe necessary and sufficient conditions for which subfamilies of $\mathcal{F}_s(t)$ have constant root number on integer fibres. We further prove similar but partial results on $\mathcal{L}_{w,s,v}(t)$. Our results give examples of subfamilies for which there is rank elevation at integer fibres.