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Corrigendum: On the probabilities of local behaviors in abelian field extensions

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  15 May 2020

Melanie Matchett Wood
Melanie Matchett Wood*
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3840, Berkeley, CA94720-3840, USA email mmwood@berkeley.edu
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Abstract

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Keywords

abelian extensionsdensity of discriminantslocal-to-globallocal behavior

MSC classification

Primary: 11R20: Other abelian and metabelian extensions 11R45: Density theorems
Type
Corrigendum
Information
Compositio Mathematica , Volume 156 , Issue 5 , May 2020 , pp. 1078
Copyright
© The Author 2020

The right-hand side of Theorem 3.1 of [Reference WoodWoo10] should contain an additional factor of $(\operatorname{Res}_{s=1}\unicode[STIX]{x1D701}_{K}(s))^{w_{K,C}}$. The equality $\lim _{s\rightarrow 1}\unicode[STIX]{x1D701}_{K}(s)=1$ was mistakenly used, which holds for $K=\mathbb{Q}$ but does not hold in general (the residue is given by the analytic class number formula).

Thank you to Rachel Newton, Christopher Frei, and Daniel Loughran for pointing this out.

References

Wood, M. M., On the probabilities of local behaviors in abelian field extensions, Compos. Math. 146 (2010), 102–128, doi:10.1112/S0010437X0900431X.CrossRefGoogle Scholar