Low rank specializations of elliptic surfaces

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Abstract

Let $E/\mathbb{Q}(T)$ be a non-isotrivial elliptic curve of rank $r$. A theorem due to Silverman implies that the rank $r_t$ of the specialization $E_t/\mathbb{Q}$ is at least $r$ for all but finitely many $t \in \mathbb{Q}$. Moreover, it is conjectured that $r_t \leq r+2$, except for a set of density $0$. In this article, when $E/\mathbb{Q}(T)$ has a torsion point of order $2$, under an assumption on the discriminant of a Weierstrass equation for $E/\mathbb{Q}(T)$, we produce an upper bound for $r_t$ that is valid for infinitely many $t$. We also present two examples of non-isotrivial elliptic curves $E/\mathbb{Q}(T)$ such that $r_t \leq r+1$ for infinitely many $t$.
Original languageUndefined
PublisherArXiv.org
DOIs
Publication statusPublished - 5 Aug 2024

Keywords

  • math.NT

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