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arXiv · 2201.04708

Root numbers of a family of elliptic curves and two applications

Jonathan Love
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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.

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Jonathan Love. 2022-01-12. Root numbers of a family of elliptic curves and two applications. https://arxiv.org/abs/2201.04708

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