Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-18T22:27:32.968Z Has data issue: false hasContentIssue false
  • English
  • Français

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: jenna@math.latech.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Carpenter, J., Finiteness theorems for forms over global fields, Math. Zeit. 209 (1992), 153–166.Google Scholar
2. Conner, P. E., Local root numbers associated to real quadratic characters and the Witt ring, personal communication.Google Scholar
3. Conner, P. E. and Yui, N., The additive characters of the Witt ring of an Algebraic Number Field, Canad. J. Math. 40 (1988), 546–588.Google Scholar
4. Czogala, A., On reciprocity equivalence of quadratic number fields, Acta Arith. 58 (1991), 27–46.Google Scholar
5. O’Meara, O. T., Introduction to Quadratic Forms, Springer-Verlag, Berlin, 1973.Google Scholar
6. Perlis, R., Szymiczek, K., Conner, P. E. and Litherland, R., Matching Witts with global fields, Proc. RAGSQUADConf. (Special year in Real AlgebraicGeometry and Quadratic Forms, Berkeley, 1990/1991), 1993.Google Scholar
7. Scharlau, W., Quadratic and Hermitian Forms, Springer-Verlag, Berlin, 1985.Google Scholar
8. Tate, J., Local constants. In: Algebraic number fields, (ed. A. Frohlich), Academic Press, New York, 1977, 89–131.Google Scholar