Local imprimitivity of points on elliptic curves

Contact: Pieter Moree (moree@mpim-bonn.mpg.de)

An element in a number field K that is primitive, i.e., not a k-th power for k>1, is a primitive root modulo infinitely many primes of K, at least under GRH. Primitive points on elliptic curves E/K may fail to have this reasonable property even in cases where E has infinitely many primes of cyclic reduction. We discuss the `anomalous' behaviour of the Galois representations underlying the failure of this local-global principle.
This concerns joint work (at MPIM) with Francesco Campagna and Francesco Pappalardi.