Root numbers of a family of elliptic curves and two applications
Abstract
For each $t\in\mathbb{Q}\setminus\{1,0,1\}$, define an elliptic curve over $\mathbb{Q}$ by \begin{align*} E_t:y^2=x(x+1)(x+t^2). \end{align*} Using a formula for the root number $W(E_t)$ as a function of $t$ and assuming some standard conjectures about ranks of elliptic curves, we determine (up to a set of density zero) the set of isomorphism classes of elliptic curves $E/\mathbb{Q}$ whose MordellWeil group contains $\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$, and the set of rational numbers that can be written as a product of the slopes of two rational right triangles.
 Publication:

arXiv eprints
 Pub Date:
 January 2022
 DOI:
 10.48550/arXiv.2201.04708
 arXiv:
 arXiv:2201.04708
 Bibcode:
 2022arXiv220104708L
 Keywords:

 Mathematics  Number Theory
 EPrint:
 15 pages, 1 figure. Updated to reference prior work that proved the root number formula (Lemma 5.1)