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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2009, Issue 628

Issues

Linear forms in elliptic logarithms

Sinnou David
  • Théorie des nombres, Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, Place Jussieu, 75005 Paris, France. e-mail:
  • Other articles by this author:
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/ Noriko Hirata-Kohno
  • Department of Mathematics, College of Science and Technology, Nihon University, Kanda, Chiyoda, 101-8308 Tokyo, Japan. e-mail:
  • Other articles by this author:
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Published Online: 2009-01-21 | DOI: https://doi.org/10.1515/CRELLE.2009.018

Abstract

We prove a new lower bound for linear forms in elliptic logarithms. As far as the height of the linear forms is concerned, our result is the first optimal one. This had been achieved so far only in the CM case, by M. Ably [Formes linéaires de logarithmes de points algébriques sur une courbe elliptique de type CM, Ann. Inst. Fourier, 2000]. We thus solve a conjecture of S. Lang dating back to the sixties, cf. [Adv. Math. 17: 281–336, 1975]. Our general result includes a “simultaneous version” which is totally effective: it takes into account the height of the point, the height of the elliptic curves and the degree of the field of definition of the given data (it is not fully explicit but the numerical constant would probably not be much better than the one provided in [David, Supplément au Bull. Soc. Math. 123: 1995]). The previously best known estimate in this context (in fact valid for a general commutative algebraic group) was due to the second author (cf. [Hirata-Kohno, Invent. Math. 104: 401–433, 1991], [Hirata-Kohno, Compos. Math. 86: 69–96, 1993]) and goes back to the early nineties. Our work, has been subsequently extended to the case of a general commutative algebraic group by E. Gaudron in [Invent. Math. 162: 137–188, 2005]. Beside the classical machinery of Baker's theory, our approach relies on the arithmetic properties of the formal logarithm. This is nothing but a more conceptual presentation of the good old “variable change trick” of G. V. Chudnovsky, as is explained in [David and Hirata-Kohno, Recent progress on linear forms in elliptic logarithms: 26–37, Cambridge University Press, 2002].

About the article

Received: 2005-07-09

Revised: 2007-10-30

Published Online: 2009-01-21

Published in Print: 2009-03-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2009, Issue 628, Pages 37–89, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2009.018.

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