Constant root number on integer fibres of elliptic surfaces

Abstract

Rizzo showed that the family of elliptic curves $\mathcal{W}(t):y^2=x^3+t{x}^2-(t+3)x+1$, 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 ${\mathcal{F}}_s(t):y^2=x^3+3t{x}^2+3sx+st$ and ${\mathcal{L}}_{w,s,v}(t):w{y}^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ 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.

35 pages, comments welcome

References in corpus (1)

• On the behaviour of root numbers in families of elliptic curves