Rigidity in Elliptic Curve Local-Global Principles

Abstract

Let ℓ be a prime number. Katz established a local-global principle for elliptic curves over a number field that have nontrivial ℓ-torsion locally everywhere. Sutherland gave an analogous local-global principle for elliptic curves that admit a rational ℓ-isogeny locally everywhere. By analyzing the subgroups of GL 2 (F ℓ ), we show that a failure of either of these "locally everywhere" conditions must be rather significant. More specifically, we prove that if an elliptic curve over a number field fails one of the above "locally everywhere" conditions, then its reductions at no more than 75% of primes ideals may satisfy the underlying local condition. In the appendix, we give for (conjecturally) all elliptic curve over the rationals without complex multiplication, the densities of prime that satisfy the local conditions mentioned above.

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].…”