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Logarithms of formal $ A$-modules in the case of small ramification

Authors: S. S. Afanas′eva and R. P. Vostokova
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 863-868
MSC (2010): Primary 20G25
Published electronically: September 30, 2016
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Abstract: Formal $ \mathcal {O}_0$-modules over the ring of integers $ \mathcal {O}$ of a local field, i.e., formal groups over $ \mathcal {O}$ with endomorphism ring including a fixed ring $ \mathcal {O}_0$ are studied. A complete description of the logarithms of all such modules is obtained in the case of small ramification. Earlier it was shown that in the case of small ramification ( $ e(\mathcal {O}/\mathcal {O}_0)<q$), any $ \mathcal {O}_0$-module is strictly isomorphic to an $ \mathcal {O}_0$-module the logarithm of which can be represented in the form $ vu^{-1}(X)$, where $ u$ and $ v$ are certain matrices over the ring of operators described in the paper. The result obtained in the present paper enables one to determine the type ($ u$ and $ v$) of a formal $ \mathcal {O}_0$-module by the form of its logarithm, and provides a way for constructing all formal $ \mathcal {O}_0$-modules.

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Additional Information

S. S. Afanas′eva
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia

R. P. Vostokova
Affiliation: D. F. Ustinov Baltic State Technical University “Voenmekh”, 1-ya Krasnoarmeiskaya ul. 1, St. Petersburg 198005, Russia

Keywords: Formal grous, formal modules, multidimensional formal groups
Received by editor(s): June 10, 2015
Published electronically: September 30, 2016
Additional Notes: Supported by RFBR (grant no. 14-01-00393).
The first author thanks Saint Petersburg State University for support.
Dedicated: To Sergeĭ Vladimirovich Vostokov on the occasion of his 70th anniversary
Article copyright: © Copyright 2016 American Mathematical Society