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 Mosc. Math. J., 2016, Volume 16, Number 1, Pages 45–93 (Mi mmj594)

An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic

Marc Hindrya, Amílcar Pachecob

a Université Paris Diderot, Institut de Mathématiques de Jussieu, UFR de Mathématiques, bâtiment Sophie Germain, 5 rue Thomas Mann, 75205 Paris Cedex 13, France
b Universidade Federal do Rio de Janeiro, Instituto de Matemática. Rua Alzira Brandão 355/404, Tijuca, 20550-035 Rio de Janeiro, RJ, Brasil

Abstract: Consider a family of abelian varieties $A_i$ of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevich–Tate group of $A_i$. We ask then when does the product of the order of the Shafarevich–Tate group by the regulator of $A_i$ behave asymptotically like the exponential height of the abelian variety. We give examples of families of abelian varieties for which this analogue of the Brauer–Siegel theorem can be proved unconditionally, but also hint at other situations, where the behaviour is different. We also prove interesting inequalities between the degree of the conductor, the height and the number of components of the Néron model of an abelian variety.

Key words and phrases: abelian varieties, global fields, function fields, $L$-function, Birch and Swinnerton-Dyer conjecture, heights, torsion points, Néron models, Brauer–Siegel theorem.

 Funding Agency Grant Number ANR HAMOT National Council for Scientific and Technological Development (CNPq) 300419/2009-0201663/2011-2 Centre National de la Recherche Scientifique Paris Science Foundation Marc Hindry was supported in part by the ANR HAMOT; Amílcar Pacheco was supported in part by CNPq research grant number 300419/2009-0, CNPq senior grant 201663/2011-2, Poste Rouge CNRS and Paris Science Foundation.

DOI: https://doi.org/10.17323/1609-4514-2016-16-1-45-93

Full text: http://www.mathjournals.org/.../2016-016-001-003.html
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Bibliographic databases:

MSC: 11G05, 11G10, 11G40, 11G50, 11R58, 14G10, 14G25, 14G40, 14K15
Received: April 2, 2014; in revised form July 4, 2015
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Citation: Marc Hindry, Amílcar Pacheco, “An analogue of the Brauer–Siegel theorem for abelian varieties in positive characteristic”, Mosc. Math. J., 16:1 (2016), 45–93

Citation in format AMSBIB
\Bibitem{HinPac16}
\by Marc~Hindry, Am{\'\i}lcar~Pacheco
\paper An analogue of the Brauer--Siegel theorem for abelian varieties in positive characteristic
\jour Mosc. Math.~J.
\yr 2016
\vol 16
\issue 1
\pages 45--93
\mathnet{http://mi.mathnet.ru/mmj594}
\crossref{https://doi.org/10.17323/1609-4514-2016-16-1-45-93}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3470576}

• http://mi.mathnet.ru/eng/mmj594
• http://mi.mathnet.ru/eng/mmj/v16/i1/p45

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. R. Griffon, “Analogue of the Brauer-Siegel theorem for Legendre elliptic curves”, J. Number Theory, 193 (2018), 189–212
2. R. Griffon, “A Brauer-Siegel theorem for Fermat surfaces over finite fields”, J. Lond. Math. Soc.-Second Ser., 97:3 (2018), 523–549
3. R. Griffon, “Explicit $L$-functions and a Brauer-Siegel theorem for Hessian elliptic curves”, J. Theor. Nr. Bordx., 30:3 (2018), 1059–1084