Rigidity in Elliptic Curve Local-Global Principles
We study the rigidity of the local conditions in two well-known local-global principles for elliptic curves over number fields. In particular, we consider a local-global principle for torsion due to Serre and Katz, and one for isogenies due to Sutherland. For each of these local-global principles, we prove that if an elliptic curve $E$ over a number field $K$ is such that it fails to satisfy the local condition for at least one prime ideal of $K$ of good reduction, then $E$ can satisfy the local condition at no more than 75% of prime ideals. We also give for (conjecturally) all elliptic curves over the rationals without complex multiplication, the densities of primes that satisfy the local conditions mentioned above.