Uniform polynomial bounds on torsion from rational geometric isogeny classes
Abbey Bourdon, Tyler Genao
TL;DR
The paper proves polynomial-in-degree uniform bounds on torsion for elliptic curves within the rational-geometric-isogeny framework $\\mathcal{I}_{\\mathbb{Q}}$, extending previous polynomial bounds known for curves with rational $j$-invariants. By combining modular-curve geometry (points on $X_1(n)$ and $X_0(n)$), Galois-representation theory, and ramification techniques, the authors show that for any $\\varepsilon>0$, non-CM curves in $\\mathcal{I}_{\\mathbb{Q}}$ satisfy $\\exp E(F)[tors] \\le c_{\\varepsilon} [F:\\mathbb{Q}]^{2+\\varepsilon}$ and $\\#E(F)[tors] \\le c_{\\varepsilon} [F:\\mathbb{Q}]^{3+\\varepsilon}$, with sharper bounds when $j(E) \\in \\mathbb{Q}$, namely $\\exp E(F)[tors] \\le c_{\\varepsilon} [F:\\mathbb{Q}]^{1+\\varepsilon}$ and $\\#E(F)[tors] \\le c_{\\varepsilon} [F:\\mathbb{Q}]^{3/2+\\varepsilon}$. The key strategy reduces polynomial-torsion questions to a combination of (i) uniform control over small-prime torsion via established bounds, (ii) uniform large-prime torsion bounds through isogeny arguments and $\\ell$-adic representations, and (iii) an isogeny-relations step that ties any $E/F$ in $\\mathcal{I}_{\\mathbb{Q}}$ back to a rational-base curve. The results close gaps between CM and non-CM cases and have implications for uniform torsion growth across broad families of elliptic curves.
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 $ε>0$, there exists $c_ε>0$ such that for any elliptic curve $E/F\in \mathcal{I}_{\mathbb{Q}}$, one has \[ E(F)[\textrm{tors}]\leq c_ε\cdot [F:\mathbb{Q}]^{3+ε}. \] 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.