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Character sums for primitive root densities

Published online by Cambridge University Press:  05 November 2014

H. W. LENSTRA JR. ,
P. STEVENHAGEN  and
P. MOREE
H. W. LENSTRA JR.
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands. e-mail: hwl@math.leidenuniv.nl, psh@math.leidenuniv.nl
P. STEVENHAGEN
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands. e-mail: hwl@math.leidenuniv.nl, psh@math.leidenuniv.nl
P. MOREE
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: moree@mpim-bonn.mpg.de
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Abstract

It follows from the work of Artin and Hooley that, under assumption of the generalised 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 ∏p δp 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 several of its generalisations 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 157 , Issue 3 , November 2014 , pp. 489 - 511
Copyright
Copyright © Cambridge Philosophical Society 2014 

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