Character sums for primitive root densities

TL;DR

This paper provides a transparent, character sum-based framework for understanding and computing primitive root densities and their correction factors, applicable to Galois representations and elliptic curves.

Contribution

It introduces a character sum interpretation of entanglement correction factors, simplifying explicit calculations and non-vanishing criteria in primitive root and elliptic curve contexts.

Findings

Explicit formulas for correction factors as character sums

Facilitates computations of primitive root densities

Applies to Galois representations and elliptic curves

Abstract

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 $\prod_p \delta_p$ of local factors $\delta_p$ reflecting the degree of the splitting field of $X^p-r$ 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 GL$_2$-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.