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

  Published:1997-12-01
 Printed: Dec 1997
  • Jenna P. Carpenter
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Abstract

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.
MSC Classifications: 11E12, 11E08 show english descriptions Quadratic forms over global rings and fields
Quadratic forms over local rings and fields
11E12 - Quadratic forms over global rings and fields
11E08 - Quadratic forms over local rings and fields
 

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