Peter Stevenhagen: Primitivity problems for number fields and elliptic curves

Artin’s primitive root conjectures (1927) predict densities of sets of primes for which a fixed element of a number field is a primitive root. They can be proved under GRH. Elliptic analogues, which go under the name of Lang–Trotter, still remain unproved. We will show that providing explicit densities in both cases is a non-trivial matter, and that the vanishing of the density, which can be proved unconditionally, is a phenomenon that leads to interesting questions on the Galois representations associated to elliptic curves.