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A SIMPLE PROOF OF CHEBOTAREV’S DENSITY THEOREM OVER FINITE FIELDS

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  12 July 2018

STEVE MEAGHER Open the ORCID record for STEVE MEAGHER [Opens in a new window]
STEVE MEAGHER*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia email [email protected]
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Abstract

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We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.

Keywords

Chebotarev density theoremquasi-projective varietyFrobenius conjugacy classLang–Weil bounds

MSC classification

Primary: 11G25: Varieties over finite and local fields
Secondary: 14G15: Finite ground fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 196 - 202
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Chavdarov, N., ‘The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy’, Duke Math. J. 87 (1997), 151–180.Google Scholar
Fried, M., ‘The nonregular analogue of Tchebotarev’s theorem’, Pacific J. Math. 112 (1984), 303–311.Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique II. Étude globale élémentaire de quelques classes de morphismes, Publications Mathématiques, 8 (IHES, Bures-sur-Yvette, 1961).Google Scholar
Kosters, M., ‘A short proof of the Chebotarev density theorem for function fields’, Math. Commun. 22 (2017), 1–7.Google Scholar
Kowalski, E., The Large Sieve and its Applications: Arithmetic Geometry, Random Walks and Discrete Groups (Cambridge University Press, Cambridge, 2008).Google Scholar
Lang, S., ‘Sur les séries L d’une variété algébrique’, Bull. Soc. Math. France 84 (1956), 335–407.Google Scholar
Lang, S., Algebra, revised third edn (Springer, New York, 2002).Google Scholar
Lang, S. and Weil, A., ‘Number of points of varieties in finite fields’, Amer. J. Math. 76(4) (1954), 819–827.Google Scholar
Mumford, D., Lectures on Curves on an Algebraic Surface (Princeton University Press, Princeton, NJ, 1966).Google Scholar
Mumford, D., Abelian Varieties (Hindustan Book Agency, New Delhi, 2008).Google Scholar
Serre, J.-P., Local Fields (Springer, New York, 1979).Google Scholar
Serre, J.-P., Cohomologie Galoisienne, cinquième éd., Lecture Notes in Mathematics, 5 (Springer, Berlin, 1997).Google Scholar
The Stacks Project, http://stacks.math.columbia.edu, 2017.Google Scholar