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

Abstract

Let m be a positive integer and let E be an elliptic curve over Q with the property that m | #E(F p ) for a density 1 set of primes p. Building upon work of Katz and Harron-Snowden, we study the probability that m | #E(Q) tor : we find it is nonzero for all m ∈ {1, 2, . . . , 10} ∪ {12, 16} and we compute it exactly when m ∈ {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 is torsion free of genus zero.

Counting elliptic curves with a rational $N$-isogeny for small $N$

“…The case of N = 3 was recently worked out by Pizzo, Pomerance and Voight [PPV20]. The case N = 4 was completed by Pomerance and Schaefer in [PS20], building off of work by Cullinan, Keeney and Voight in [CKV20]. The case N = 4 also follows from recent work of Bruin and Najman in [BN20], which we say more about in Remark 1.2.…”

Examples of abelian surfaces failing the local-global principle for isogenies

“…Etropolski [Etr15] considers a local-global question for arbitrary subgroups of GL 2 (F ℓ ), and Vogt [Vog20] generalises the prime-degree-isogeny problem to composite degree isogenies. Very recently Mayle [May20] bounds by 3 4 the density of prime ideals for elliptic curves E/K which do not satisfy either of the "everywherelocal" conditions for torsion or isogenies, and Cullinan, Kenney and Voight study a probabilistic version of the torsion local-global principle for elliptic curves [CKV20].…”

Rational Points of Bounded Height on Some Genus Zero Modular Curves over Number Fields

“…The method of Harron and Snowden allowed them to give a uniform proof of the G(1, 1), G(1, 2), and G(1, 3) cases. Cullian, Kenny, and Voight showed that the method of Harron and Snowden could be applied more widely, and doing so were able to count isomorphism classes in all cases for which w = (1, 1) and also for the cases of G(2, 2)-level structure and G 0 (4)level structure [CKV21]. Over general number fields the only previously known result is that the asymptotic growth rate is w 0 +w 1 12e G , due to work of Bruin and Najman [BN20].…”