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Explicit form of the Hilbert symbol for polynomial formal groups


Authors: S. Vostokov and V. Volkov
Translated by: V. Volkov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 5.
Journal: St. Petersburg Math. J. 26 (2015), 785-796
MSC (2010): Primary 11S31; Secondary 14L05
DOI: https://doi.org/10.1090/spmj/1358
Published electronically: July 27, 2015
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be a local field, $ c$ a unit in $ K$, and $ F_c (X,Y) = X + Y + cXY$ a polynomial formal group that gives rise to a formal module $ F_c(\mathfrak{M})$ on the maximal ideal in the ring of integers of $ K$. Assume that $ K$ contains the group $ \mu _{F_c, n}$ of the roots of isogeny $ [p^n]_c(X)$. The natural Hilbert symbol $ (\,\cdot \,,\,\cdot \,)_c \colon K^*\times F_c(\mathfrak{M}) \to \mu _{F_c, n}$ is defined over the module $ F(\mathfrak{M})$. An explicit formula for $ (\,\cdot \,,\,\cdot \,)_c$ is constructed.


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Additional Information

S. Vostokov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: sergei.vostokov@gmail.com

V. Volkov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
Email: vladvolkov239@gmail.com

DOI: https://doi.org/10.1090/spmj/1358
Keywords: Polynomial formal groups, formal groups, Hilbert symbol, local rings
Received by editor(s): January 10, 2014
Published electronically: July 27, 2015
Additional Notes: Supported by RFBR (grant no. 14-01-00393)
Article copyright: © Copyright 2015 American Mathematical Society

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