Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.02419 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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 $t$.