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On the essential minimum of Faltings' height

Authors: José Ignacio Burgos Gil, Ricardo Menares and Juan Rivera-Letelier
Journal: Math. Comp. 87 (2018), 2425-2459
MSC (2010): Primary 11F11, 11G50, 14G40, 37P30
Published electronically: January 18, 2018
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Abstract: We study the essential minimum of the (stable) Faltings' height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum.

One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions.

Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.

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Additional Information

José Ignacio Burgos Gil
Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UCM-UCM3), Calle Nicolás Cabrera 15, Campus UAB, Cantoblanco, 28049 Madrid, Spain

Ricardo Menares
Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
Address at time of publication: Pontificia Universidad Católica de Chile, Avda. Vicuña Mackenna 4860, Santiago, Chile

Juan Rivera-Letelier
Affiliation: Department of Mathematics, University of Rochester, Hylan Building, Rochester, New York 14627

Received by editor(s): December 20, 2016
Received by editor(s) in revised form: April 6, 2017
Published electronically: January 18, 2018
Additional Notes: The first author was partially supported by the MINECO research projects MTM2013-42135-P, MTM2016-79400P, and ICMAT Severo Ochoa SEV-2015-0554
The second author was partially supported by FONDECYT grant 1171329
The third author was partially supported by FONDECYT grant 1141091 and NSF grant DMS-1700291
Article copyright: © Copyright 2018 American Mathematical Society

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