On a probabilistic local-global principle for torsion on elliptic curves

Cullinan, John; Kenney, Meagan; Voight, John

Journal de théorie des nombres de Bordeaux, Tome 34 (2022) no. 1, pp. 41-90.

Soit $m$ un entier positif et soit $E$ une courbe elliptique sur $\mathbb{Q}$ telle que $m\mid \#E\left({\mathbf{F}}_p\right)$ pour un ensemble de densité $1$ de nombres premiers $p$. En nous appuyant sur les travaux de Katz et Harron–Snowden, nous étudions la probabilité que $m\mid \#E{\left(\mathbb{Q}\right)}_{tor}$ : nous trouvons qu’elle est non nulle pour tout $m\in \{1,2,\cdots ,10\}\cup \{12,16\}$ et nous la calculons explicitement pour $m\in \{1,2,3,4,5,7\}$. En complément, nous donnons un comptage asymptotique des courbes elliptiques avec une structure de niveau supplémentaire lorsque la courbe modulaire paramétrant ces structures provient du quotient par un groupe sans torsion de genre zéro.

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid \#E\left({\mathbf{F}}_p\right)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron–Snowden, we study the probability that $m\mid \#E{\left(\mathbb{Q}\right)}_{tor}$: we find it is nonzero for all $m\in \{1,2,\cdots ,10\}\cup \{12,16\}$ and we compute it exactly when $m\in \{1,2,3,4,5,7\}$. As a supplement, we give an asymptotic count of elliptic curves with extra level structure when the parametrizing modular curve arises from the quotient by a torsion-free group of genus zero.

Classification : 11G05, 14H52
Mots clés : Elliptic curves, torsion subgroups, arithmetic statistics

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