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

MathSciNet review:
3020846

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let denote an th primitive root of unity. For a character of order , we define two one-dimensional formal groups over and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of . This allows us to deduce a reciprocity formula for the th residue symbol which, in particular, implies the cubic reciprocity law.

**[CS1]**Nancy Childress and Jeffrey Stopple,*Formal groups and Dirichlet 𝐿-functions. I, II*, J. Number Theory**41**(1992), no. 3, 283–294, 295–302. MR**1168989**, 10.1016/0022-314X(92)90127-B**[CS2]**Nancy Childress and Jeffrey Stopple,*Formal groups and Dirichlet 𝐿-functions. I, II*, J. Number Theory**41**(1992), no. 3, 283–294, 295–302. MR**1168989**, 10.1016/0022-314X(92)90127-B**[CG]**Nancy Childress and David Grant,*Formal groups of twisted multiplicative groups and 𝐿-series*, 𝐾-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 89–102. MR**1327292****[G]**David Grant,*Geometric proofs of reciprocity laws*, J. Reine Angew. Math.**586**(2005), 91–124. MR**2180602**, 10.1515/crll.2005.2005.586.91**[H1]**Taira Honda,*On the theory of commutative formal groups*, J. Math. Soc. Japan**22**(1970), 213–246. MR**0255551****[H2]**Taira Honda,*Invariant differentials and 𝐿-functions. Reciprocity law for quadratic fields and elliptic curves over 𝑄*, Rend. Sem. Mat. Univ. Padova**49**(1973), 323–335. MR**0360593****[IR]**Kenneth Ireland and Michael Rosen,*A classical introduction to modern number theory*, 2nd ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990. MR**1070716**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
11A15,
14L05

Retrieve articles in all journals with MSC (2010): 11A15, 14L05

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

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.