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

http://dx.doi.org/10.4153/CMB-1997-048-2
Canad. Math. Bull. 40(1997), 402-415

  Published:1997-12-01
 Printed: Dec 1997

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.