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

Tyler Genao

doi:10.5802/jtnb.1292

Tyler Genao ¹

¹ Department of Mathematics The Ohio State University 231 W. 18th Ave., Columbus, OH 43210, USA

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

Abstract (VO)
Résumé

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 $E (F)$ has a point of order $N$ then

N \leq c_{ϵ} \cdot {[F : ℚ]}^{1 / 2 + ϵ},

and one also has

# E (F) [tors] \leq C_{ϵ} \cdot {[F : 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 $E (F)$ a un point d’ordre $N$ alors

N \leq c_{ϵ} \cdot {[F : ℚ]}^{1 / 2 + ϵ},

et on a aussi

# E (F) [tors] \leq C_{ϵ} \cdot {[F : ℚ]}^{1 + ϵ} .

Reçu le : 2023-01-01
Révisé le : 2023-07-17
Accepté le : 2023-09-15
Publié le : 2024-11-13

MR Zbl

DOI : 10.5802/jtnb.1292

Classification : 11G05
Mots-clés : Elliptic curve, Galois representation, isogeny, torsion subgroup

Affiliations des auteurs :

Tyler Genao ¹

¹ Department of Mathematics The Ohio State University 231 W. 18th Ave., Columbus, OH 43210, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Tyler Genao},
     title = {Polynomial {Bounds} on {Torsion} {From} a {Fixed} {Geometric} {Isogeny} {Class} of {Elliptic} {Curves}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {661--670},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {2},
     year = {2024},
     doi = {10.5802/jtnb.1292},
     mrnumber = {4830946},
     zbl = {07948981},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1292/}
}

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Tyler Genao. Polynomial Bounds on Torsion From a Fixed Geometric Isogeny Class of Elliptic Curves. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 2, pp. 661-670. doi : 10.5802/jtnb.1292. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1292/

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