Root numbers of a family of elliptic curves and two applications

TL;DR

This paper investigates the root numbers of a specific family of elliptic curves over rationals, applying these results to classify certain elliptic curves and to identify rational numbers expressible as products of slopes of rational right triangles.

Contribution

It provides a formula for root numbers of the family and applies conjectures to classify elliptic curves with particular Mordell-Weil group structures and to characterize certain rational numbers.

Findings

Classified elliptic curves with specific Mordell-Weil group structures.

Identified rational numbers as products of slopes of rational right triangles.

Derived a formula for root numbers as a function of parameter t.

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 Mordell-Weil 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.