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On a new type of $ \ell$-adic regulator for algebraic number fields. II


Author: L. V. Kuz′min
Translated by: the author
Original publication: Algebra i Analiz, tom 27 (2015), nomer 6.
Journal: St. Petersburg Math. J. 27 (2016), 977-984
MSC (2010): Primary 11R04
DOI: https://doi.org/10.1090/spmj/1430
Published electronically: September 30, 2016
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Abstract: In a preceding paper of the author, a new type of an $ \ell $-adic regulator $ \mathfrak{R}_\ell (K)$ was introduced for an algebraic number field $ K$ such that the prime $ \ell $ splits completely in $ K$. Nevertheless, the element $ \mathfrak{R}_\ell (K)\in \mathbb{Z}_\ell $ is defined only up to an arbitrary factor in $ (\mathbb{Z}_\ell ^\times )^2$. In the present paper, under the assumption of the validity of the Shanuel conjecture (both Archimedean and $ \ell $-adic), the definition of $ \mathfrak{R}_\ell (K)$ as a certain number in $ \mathbb{Z}_\ell $ is given. For a real quadratic field $ K$, such a defition can be obtained without using any additional conjectures.


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Additional Information

L. V. Kuz′min
Affiliation: National Research Center “Kurchatovskii Institute”, Akad. Kurchatov Sq. 1, 123182 Moscow, Russia
Email: lvkuzmin@mail.ru

DOI: https://doi.org/10.1090/spmj/1430
Keywords: $\ell$-adic regulator, $S$-units, global universal norms, Shanuel conjecture, Iwasawa theory
Received by editor(s): June 25, 2015
Published electronically: September 30, 2016
Additional Notes: The author was supported by RFBR (grant no.14-01-00393)
Dedicated: Dedicated to S. V. Vostokov on the occasion of his 70th anniversary.
Article copyright: © Copyright 2016 American Mathematical Society

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