Logarithms of formal $A$-modules in the case of small ramification
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S. S. Afanas′eva and R. P. Vostokova
Translated by: N. B. Lebedinskaya - St. Petersburg Math. J. 27 (2016), 863-868
- DOI: https://doi.org/10.1090/spmj/1423
- 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.References
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Bibliographic Information
- S. S. Afanas′eva
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petrodvorets, St. Petersburg 198504, Russia
- Email: cheery_sonya@mail.ru
- R. P. Vostokova
- Affiliation: D. F. Ustinov Baltic State Technical University “Voenmekh”, 1-ya Krasnoarmeiskaya ul. 1, St. Petersburg 198005, Russia
- Email: rvostokova@yandex.ru
- 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. - © Copyright 2016 American Mathematical Society
- Journal: St. Petersburg Math. J. 27 (2016), 863-868
- MSC (2010): Primary 20G25
- DOI: https://doi.org/10.1090/spmj/1423
- MathSciNet review: 3589219
Dedicated: To Sergeĭ Vladimirovich Vostokov on the occasion of his 70th anniversary