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On the Preservation of Root Numbers and the Behavior of Weil Characters under Reciprocity Equivalence

Published online by Cambridge University Press:  20 November 2018

Jenna P. Carpenter
Jenna P. Carpenter*
Affiliation:
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, Louisiana, USA 71272, e-mail: [email protected]
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Abstract

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This paper studies how the local root numbers and the Weil additive characters of the Witt ring of a number field behave under reciprocity equivalence. Given a reciprocity equivalence between two fields, at each place we define a local square class which vanishes if and only if the local root numbers are preserved. Thus this local square class serves as a local obstruction to the preservation of local root numbers. We establish a set of necessary and sufficient conditions for a selection of local square classes (one at each place) to represent a global square class. Then, given a reciprocity equivalence that has a finite wild set, we use these conditions to show that the local square classes combine to give a global square class which serves as a global obstruction to the preservation of all root numbers. Lastly, we use these results to study the behavior of Weil characters under reciprocity equivalence.

Keywords

Primary: 11E12Secondary:11E08
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 4 , 01 December 1997 , pp. 402 - 415
Copyright
Copyright © Canadian Mathematical Society 1997

References

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