Uniform polynomial bounds on torsion from rational geometric isogeny classes

TL;DR

This paper establishes polynomial bounds on torsion groups for elliptic curves geometrically isogenous to rational elliptic curves, extending previous results and improving bounds for certain families, with implications for understanding torsion in number fields.

Contribution

It proves the existence of polynomial bounds on torsion for elliptic curves in the family al_{\u00a0}, generalizing prior work and providing improved bounds for curves with rational j-invariant.

Findings

Bound (F)[ extrm{tors}] _psilon imes [F:b]^{3+psilon} for curves in al_{b}

Bounds on torsion are optimal for curves with complex multiplication without restrictions

Improved bounds for elliptic curves with rational j-invariant compared to previous work

Abstract

In 1996, Merel showed there exists a function $B\colon \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for any elliptic curve $E/F$ defined over a number field of degree $d$, one has the torsion group bound $\# E(F)[\textrm{tors}]\leq B(d)$. Based on subsequent work, it is conjectured that one can choose $B$ to be polynomial in the degree $d$. In this paper, we show that such bounds exist for torsion from the family $\mathcal{I}_{\mathbb{Q}}$ of elliptic curves which are geometrically isogenous to at least one rational elliptic curve. More precisely, we show that for each $\epsilon>0$, there exists $c_\epsilon>0$ such that for any elliptic curve $E/F\in \mathcal{I}_{\mathbb{Q}}$, one has \[ E(F)[\textrm{tors}]\leq c_\epsilon\cdot [F:\mathbb{Q}]^{3+\epsilon}. \] This generalizes work of the second author for elliptic curves within a fixed rational geometric isogeny class. For the family of elliptic curves with rational $j$-invariant, we also obtain bounds which improve those of Clark and Pollack. In this case, our bounds on the exponent of $E(F)[\textrm{tors}]$ are optimal if one does not exclude elliptic curves with complex multiplication.