On local-global divisibility by p n in elliptic curves

Abstract

Let p be a prime number and let k be a number field, which does not contain the field ℚ ( ζ p + ζ p ¯ ) . Let ℰ be an elliptic curve defined over k. We prove that if there are no k-rational torsion points of exact order p on ℰ, then the local–global principle holds for divisibility by pn, with n a natural number. As a consequence of the deep theorem of Merel, for p larger than a constant depending only on the degree of k, there are no counterexamples to the local–global divisibility principle. Explicit small constants for elliptic curves defined over a number field of degree at most 5 over ℚ are provided in several deep works of different authors.