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St. Petersburg Mathematical Journal

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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
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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  • 1. Z. I. Borevič and I. R. Šafarevič, The theory of numbers, Nauka, Moscow, 1964; English transl., Newcomb Greenleaf Pure Appl. Math., vol. 20, Acad. Press, New York, 1966. MR 0195803
  • 2. S. Lang, Cyclotomic fields. I, II, Comb. 2nd ed., Grad. Texts in Math., vol. 121, Springer-Verlag, New York, 1990. MR 1029028
  • 3. L. V. Kuz'min, Some remarks on an $ \ell $-adic Dirichlet theorem and $ \ell $-adic regulator, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 6, 1203-1240; English transl., Math. USSR-Izv. 19 (1982), no. 3, 445-478. MR 641800
  • 4. -, On a new type of $ \ell $-adic regulator for algebraic number fields (the $ \ell $-adic regulator without logarithms), Izv. Ross. Akad. Nauk Ser. Mat. 79 (2015), no. 1, 115-152; English transl., Izv. Math. 79 (2015), no. 1, 109-144. MR 3352584
  • 5. -, Some remarks on the $ \ell $-adic regulator. V. The growth on the $ \ell $-adic regulator in the cyclotomic $ \mathbb{Z}_\ell $-extension of an algebraic number field, Izv. Ross. Akad. Nauk Ser. Mat. 73 (2009), no. 5, 105-170; English transl., Izv. Math. 73 (2009), no. 5, 959-1021. MR 2584230

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Additional Information

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

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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