It follows from the work of Artin and Hooley that, under assumption of the
generalized Riemann hypothesis, the density of the set of primes q for which
a given non-zero rational number r is a primitive root modulo q can be
written as an infinite product ∏pδp of local factors δp
reflecting the degree of the splitting field of Xp−r at the primes p,
multiplied by a somewhat complicated factor that corrects for the
`entanglement' of these splitting fields. We show how the correction factors
arising in Artin's original primitive root problem and some of its
generalizations can be interpreted as character sums describing the nature of
the entanglement. The resulting description in terms of local contributions is
so transparent that it greatly facilitates explicit computations, and naturally
leads to non-vanishing criteria for the correction factors. The method not only
applies in the setting of Galois representations of the multiplicative group
underlying Artin's conjecture, but also in the GL2-setting arising for
elliptic curves. As an application, we compute the density of the set of primes
of cyclic reduction for Serre curves.Comment: 23 pages. This version is to appear in the Mathematical Proceedings
of the Cambridge Philosophical Societ