2022/01/27 by Creutz, Brendan, Lu, Sheng · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2201.11839
We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime p we give sharp lower bounds on the degree d of a number field over which there exists a CM elliptic curve which gives a counterexample to the local-global principle for divisibility by a power of p. As a corollary we deduce that there are at most finitely many elliptic curves (with or without CM) which are counterexamples with p > 2d+1. We also deduce that the local-global principle for divisibility by powers of 7 holds over quadratic fields.