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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

The product of symbols of $ p^n$th power residues as an Abelian integral


Author: M. A. Ivanov
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 24 (2012), nomer 2.
Journal: St. Petersburg Math. J. 24 (2013), 275-281
MSC (2010): Primary 11A15
Published electronically: January 22, 2013
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Abstract: In accordance with the Hilbert-Shafarevich ideology, the reciprocity law must be an analog of an integral theorem asserting that the Abelian integral of a differential form on a Riemann surface is equal to the sum of residues at singular points. In the present paper, it is shown that the product of the symbols of $ p^n$th power residues is the integral of a certain function.


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Additional Information

M. A. Ivanov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Petrodvorets, St. Petersburg 198904, Russia
Email: micliva@gmail.com

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01238-6
PII: S 1061-0022(2013)01238-6
Keywords: Power residue symbol, Shnirel′man’s integral
Received by editor(s): March 25, 2011
Published electronically: January 22, 2013
Additional Notes: Supported by RFBR (grant no. 11-01-00588-a)
Article copyright: © Copyright 2013 American Mathematical Society