S -integral points on elliptic curves - Notes on a paper of B. M. M. de Weger

Emanuel Herrmann; Attila Pethö

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 2, page 443-451
  • ISSN: 1246-7405

Abstract

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In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and p -adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.

How to cite

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Herrmann, Emanuel, and Pethö, Attila. "$S$-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger." Journal de théorie des nombres de Bordeaux 13.2 (2001): 443-451. <http://eudml.org/doc/248713>.

@article{Herrmann2001,
abstract = {In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.},
author = {Herrmann, Emanuel, Pethö, Attila},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {-integral point; elliptic curve linear form in elliptic logarithms; -adic elliptic logarithm},
language = {eng},
number = {2},
pages = {443-451},
publisher = {Université Bordeaux I},
title = {$S$-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger},
url = {http://eudml.org/doc/248713},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Herrmann, Emanuel
AU - Pethö, Attila
TI - $S$-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 2
SP - 443
EP - 451
AB - In this paper we give a much shorter proof for a result of B.M.M de Weger. For this purpose we use the theory of linear forms in complex and $p$-adic elliptic logarithms. To obtain an upper bound for these linear forms we compare the results of Hajdu and Herendi and Rémond and Urfels.
LA - eng
KW - -integral point; elliptic curve linear form in elliptic logarithms; -adic elliptic logarithm
UR - http://eudml.org/doc/248713
ER -

References

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  2. [2] S. David, Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France(N.S.) 62 (1995). Zbl0859.11048MR1385175
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  4. [4] J. Gebel, A. Peth, H.G. Zimmer, Computing S-integral points on elliptic curves. Algorithmic number theory (Talence, 1996), 157-171, Lecture Notes in Comput. Sci.1122, Springer, Berlin, 1996. Zbl0899.11012MR1446509
  5. [5] A. Peth, H.G. Zimmer, J. Gebel, E. Herrmann, Computing all S-integral points on elliptic curves. Math. Proc. Cambr. Phil. Soc.127 (1999), 383-402. Zbl0949.11033MR1713117
  6. [6] G. Rémond, F. Urfels, Approximation diophantienne de logarithmes elliptiques p-adiques. J. Numb. Th.57 (1996), 133-169. Zbl0853.11055MR1378579
  7. [7] J.H. Silverman, The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics106, Springer-Verlag, New York, 1986. Zbl0585.14026MR817210
  8. [8] N.P. Smart, S-integral Points on elliptic curves. Math. Proc. Cambr. Phil. Soc.116 (1994), 391-399. Zbl0817.11031MR1291748
  9. [9] J.T. Tate, Algorithm for determining the type of a singular fibre in an elliptic pencil. Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52, Lecture Notes in Math.476, Springer, Berlin, 1975. Zbl1214.14020MR393039
  10. [10] N. Tzanakis, Solving elliptic Diophantine equations by estimating linear forms in elliptic logarithms. The case of quartic equations. Acta Arith.75 (1996), 165-190. Zbl0858.11016MR1379397
  11. [11] B.M.M. De Weger, Algorithms for Diophantine equations. PhD Thesis, Centr. for Wiskunde en Informatica, Amsterdam, 1987. Zbl0687.10013
  12. [12] B.M.M. De Weger, S-integral solutions to a Weierstrass equation, J. Théor. Nombres Bordeaux9 (1997), 281-301. Zbl0898.11009MR1617399
  13. [13] Apecs, Arithmetic of plane elliptic curves, ftp://ftp.math.mcgill.ca/pub/apecs. 
  14. [14] mwrank, a package to compute ranks of elliptic curves over the rationals. http://www.maths.nott.ac.uk/personal/jec/ftp/progs. 
  15. [15] Simath, a computer algebra system for algorithmic number theory. http://simath.math.unisb.de. 

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