Polynomial Bounds on Torsion From a Fixed Geometric Isogeny Class of Elliptic Curves
Tyler Genao1
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.

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≤c ϵ ·[F:ℚ] 1/2+ϵ ,

et on a aussi

#E(F)[tors]≤C ϵ ·[F:ℚ] 1+ϵ .

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≤c ϵ ·[F:ℚ] 1/2+ϵ ,

and one also has

#E(F)[tors]≤C ϵ ·[F:ℚ] 1+ϵ .

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1292
Classification : 11G05
Mots clés : Elliptic curve, Galois representation, isogeny, torsion subgroup
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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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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