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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: [email protected])
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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

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