Low rank specializations of elliptic surfaces

TL;DR

This paper investigates the ranks of specializations of non-isotrivial elliptic curves over $Q(T)$, providing new bounds for the rank of specialized curves under certain conditions and presenting examples with notably low ranks.

Contribution

It introduces bounds on the specialization rank $r_t$ when the elliptic curve has a torsion point of order 2, under specific discriminant conditions, and provides explicit examples with low ranks.

Findings

Bound for $r_t$ valid for infinitely many $t$ under certain conditions

Existence of examples with $r_t \,\leq r+1$ for infinitely many $t$

Extension of Silverman's specialization theorem with additional constraints

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 $t$.