Constant root number on integer fibres of elliptic surfaces
Abstract
Rizzo showed that the family of elliptic curves $\mathcal{W}(t) :y^2=x^3+tx^2-(t+3)x+1$, 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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2020
- DOI:
- 10.48550/arXiv.2011.02386
- arXiv:
- arXiv:2011.02386
- Bibcode:
- 2020arXiv201102386C
- Keywords:
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- Mathematics - Number Theory;
- 11G05;
- 11G40
- E-Print:
- 35 pages, comments welcome