Polynomial Bounds on Torsion From a Fixed Geometric Isogeny Class of Elliptic Curves

Genao, Tyler

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 661-670

We show there exist polynomial bounds on torsion of elliptic curves which come from a fixed geometric isogeny class. More precisely, for an elliptic curve $E_0$ defined over a number field $F_0$, for each $ϵ>0$ there exist constants $c_{ϵ}:=c_{ϵ}(E_0,F_0),C_{ϵ}:=C_{ϵ}(E_0,F_0)>0$ such that for any elliptic curve $E_{/F}$ geometrically isogenous to $E_0$, if $E\left(F\right)$ has a point of order $N$ then

$$N\le c_{ϵ}·{[F:\mathbb{Q}]}^{1/2+ϵ},$$

and one also has

$$\#E\left(F\right)\left[\mathrm{tors}\right]\le C_{ϵ}·{[F:\mathbb{Q}]}^{1+ϵ}.$$

Nous montrons qu’il existe des bornes polynomiales pour la torsion des courbes elliptiques qui proviennent d’une classe d’isogénie géométrique fixe. Plus précisément, si $E_0$ est une courbe elliptique définie sur un corps de nombres $F_0$, alors pour chaque $ϵ>0$ il existe des constantes $c_{ϵ}:=c_{ϵ}(E_0,F_0)$ et $C_{ϵ}:=C_{ϵ}(E_0,F_0)>0$ telles que pour toute courbe elliptique $E_{/F}$ géométriquement isogène à $E_0$, si $E\left(F\right)$ a un point d’ordre $N$ alors

$$N\le c_{ϵ}·{[F:\mathbb{Q}]}^{1/2+ϵ},$$

et on a aussi

$$\#E\left(F\right)\left[\mathrm{tors}\right]\le C_{ϵ}·{[F:\mathbb{Q}]}^{1+ϵ}.$$

Classification : 11G05
Keywords: Elliptic curve, Galois representation, isogeny, torsion subgroup

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