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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Reciprocity laws through formal groups


Authors: Oleg Demchenko and Alexander Gurevich
Journal: Proc. Amer. Math. Soc. 141 (2013), 1591-1596
MSC (2010): Primary 11A15, 14L05
Published electronically: November 8, 2012
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Abstract: A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let $ \xi $ denote an $ m$th primitive root of unity. For a character $ \chi $ of order $ m$, we define two one-dimensional formal groups over $ \mathbb{Z}[\xi ]$ and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of $ \chi $. This allows us to deduce a reciprocity formula for the $ m$th residue symbol which, in particular, implies the cubic reciprocity law.


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Additional Information

Oleg Demchenko
Affiliation: Department of Mathematics and Mechanics, Saint Petersburg State University, Universitetsky pr. 28, Stary Petergof, 198504 Saint Petersburg, Russia
Email: vasja@eu.spb.ru

Alexander Gurevich
Affiliation: Einstein Institute of Mathematics, Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel
Email: gurevich@math.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11632-6
PII: S 0002-9939(2012)11632-6
Received by editor(s): September 7, 2011
Published electronically: November 8, 2012
Additional Notes: The first author was partially supported by RFBR grant 11-01-00588a, by Saint Petersburg State University research grant 6.38.75.2011, and by Grant-in-Aid (No. S-23224001) for Scientific Research, JSPS
The second author was partially supported by ISF Center of Excellency grant 1691/10
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.