Reciprocity laws through formal groups
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- by Oleg Demchenko and Alexander Gurevich PDF
- Proc. Amer. Math. Soc. 141 (2013), 1591-1596 Request permission
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.References
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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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1591-1596
- MSC (2010): Primary 11A15, 14L05
- DOI: https://doi.org/10.1090/S0002-9939-2012-11632-6
- MathSciNet review: 3020846