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On the probabilities of local behaviors in abelian field extensions

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  08 December 2009

Melanie Matchett Wood
Melanie Matchett Wood*
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA (email: melanie.wood@math.princeton.edu)
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Abstract

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For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these probabilities are nicely behaved and mostly independent. This is in analogy to Chebotarev’s density theorem, which gives the probability that in a fixed extension a random prime of K splits into r primes in the extension. We also give the asymptotics for the number of G-extensions with bounded conductor. In fact, we give a class of extension invariants, including conductor, for which we obtain the same counting and probabilistic results. In contrast, we prove that neither the analogy with the Chebotarev probabilities nor the independence of probabilities holds when extensions are ordered by discriminant.

Keywords

abelian extensionsdensity of discriminantslocal-to-globallocal behavior

MSC classification

Secondary: 11R20: Other abelian and metabelian extensions 11R45: Density theorems
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 102 - 128
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Artin, E. and Tate, J., Class field theory (W. A. Benjamin Inc., New York, 1968).Google Scholar
[2]Bhargava, M. and Wood, M. M., The density of discriminants of S 3-sextic number fields, Proc. Amer. Math. Soc. 136 (2008), 1581–1587.CrossRefGoogle Scholar
[3]Bhargava, M., The density of discriminants of quartic rings and fields, Ann. of Math. (2) 162 (2005), 1031–1063.CrossRefGoogle Scholar
[4]Bhargava, M., The density of discriminants of quintic rings and fields, Ann. of Math., to appear.Google Scholar
[5]Bhargava, M., Mass formulae for extensions of local fields, and conjectures on the density of number field discriminants, Int. Math. Res. Not. (2007), Art. ID rnm052, 20 pp.CrossRefGoogle Scholar
[6]Cohen, H., Constructing and counting number fields, in Proceedings of the International congress of mathematicians, Beijing, 2002, vol. II (Higher Ed. Press, Beijing, 2002), 129–138.Google Scholar
[7]Cohen, H., Diaz y diaz, F. and Olivier, M., Counting discriminants of number fields, J. Théor. Nombres Bordeaux 18 (2006), 573–593.CrossRefGoogle Scholar
[8]Cohen, H., Diaz y Diaz, F. and Olivier, M., A survey of discriminant counting, in Algorithmic number theory, Sydney, 2002, Lecture Notes in Computer Science, vol. 2369 (Springer, Berlin, 2002), 80–94.Google Scholar
[9]Cohen, H., Diaz y Diaz, F. and Olivier, M., Enumerating quartic dihedral extensions of ℚ, Compositio Math. 133 (2002), 65–93.CrossRefGoogle Scholar
[10]Datskovsky, B. and Wright, D. J., The adelic zeta function associated to the space of binary cubic forms. II. Local theory, J. Reine Angew. Math. 367 (1986), 27–75.Google Scholar
[11]Davenport, H. and Heilbronn, H., On the density of discriminants of cubic fields. II, Proc. Roy. Soc. London Ser. A 322 (1971), 405–420.Google Scholar
[12]Del Corso, I. and Dvornicich, R., A converse of Artin’s density theorem: the case of cubic fields, J. Number Theory 45 (1993), 28–44.Google Scholar
[13]Del Corso, I. and Dvornicich, R., Uniformity over primes of unramified splittings, Mathematika 45 (1998), 177–189.Google Scholar
[14]Del Corso, I. and Dvornicich, R., Uniformity over primes of tamely ramified splittings, Manuscripta Math. 101 (2000), 239–266.CrossRefGoogle Scholar
[15]Ellenberg, J. S. and Venkatesh, A., Counting extensions of function fields with bounded discriminant and specified Galois group, in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser, Boston, MA, 2005), 151–168.CrossRefGoogle Scholar
[16]Grunwald, W., Ein allgemeines Existenztheorem für algebraische Zahlkörper, J. Reine Angew. Math. 169 (1933), 103–107.CrossRefGoogle Scholar
[17]Mäki, S., On the density of abelian number fields, Ann. Acad. Sci. Fenn. Ser. A I Math. Diss. 54 (1985), 104.Google Scholar
[18]Mäki, S., The conductor density of abelian number fields, J. London Math. Soc. (2) 47 (1993), 18–30.CrossRefGoogle Scholar
[19]Malle, G., On the distribution of Galois groups. II, Experiment. Math. 13 (2004), 129–135.CrossRefGoogle Scholar
[20]Narkiewicz, W., Number theory (World Scientific, Singapore, 1983). (Translated by S. Kanemitsu.)Google Scholar
[21]Neukirch, J., Algebraic number theory (Springer, Berlin, 1999). (Translated from the 1992 German original and with a note by Norbert Schappacher.)Google Scholar
[22] PARI/GP, version 2.3.2, Bordeaux, 2006, http://pari.math.u-bordeaux.fr/.Google Scholar
[23]Taylor, M. J., On the equidistribution of Frobenius in cyclic extensions of a number field, J. London Math. Soc. (2) 29 (1984), 211–223.Google Scholar
[24]van der Ploeg, C. E., On a converse to the Tschebotarev density theorem, J. Aust. Math. Soc. Ser. A 44 (1988), 287–293.Google Scholar
[25]Wang, S., On Grunwald’s theorem, Ann. of Math. (2) 51 (1950), 471–484.Google Scholar
[26]Wood, M. M., Mass formulas for local Galois representations to wreath products and cross products, Algebra Number Theory 2 (2008), 391–405.CrossRefGoogle Scholar
[27]Wright, D. J., Distribution of discriminants of abelian extensions, Proc. London Math. Soc. (3) 58 (1989), 17–50.Google Scholar