S-integral points on elliptic curves - Notes on a paper of B. M. M. de Weger
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 443-451.

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.

Nous donnons une nouvelle preuve beaucoup plus courte d’un résultat de B. M. M de Weger. Cette preuve est basée sur la théorie des formes linéaires de logarithmes complexes, p-adiques et elliptiques, pour lesquelles nous obtenons une majoration en confrontant les résultats de Hajdu et Herendi à ceux de Rémond et Urfels.

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     author = {Herrmann, Emanuel and Peth\"o, Attila},
     title = {$S$-integral points on elliptic curves - {Notes} on a paper of {B.} {M.} {M.} de {Weger}},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux I},
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Herrmann, Emanuel; 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, Volume 13 (2001) no. 2, pp. 443-451. http://archive.numdam.org/item/JTNB_2001__13_2_443_0/

[1] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language. J. Symb. Comp., 24, 3/4 (1997), 235-265. (See also the Magma home page at http://www.maths.usyd.edu.au:8000/u/magma/) | MR | Zbl

[2] S. David, Minorations de formes linéaires de logarithmes elliptiques. Mém. Soc. Math. France(N.S.) 62 (1995). | Numdam | MR | Zbl

[3] J. Gebel, A. Peth, H.G. Zimmer, Computing integral points on elliptic curves. Acta Arith. 68 (1994), 171-192. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[6] G. Rémond, F. Urfels, Approximation diophantienne de logarithmes elliptiques p-adiques. J. Numb. Th. 57 (1996), 133-169. | MR | Zbl

[7] J.H. Silverman, The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics 106, Springer-Verlag, New York, 1986. | MR | Zbl

[8] N.P. Smart, S-integral Points on elliptic curves. Math. Proc. Cambr. Phil. Soc. 116 (1994), 391-399. | MR | Zbl

[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. | MR

[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. | MR | Zbl

[11] B.M.M. De Weger, Algorithms for Diophantine equations. PhD Thesis, Centr. for Wiskunde en Informatica, Amsterdam, 1987. | Zbl

[12] B.M.M. De Weger, S-integral solutions to a Weierstrass equation, J. Théor. Nombres Bordeaux 9 (1997), 281-301. | Numdam | MR | Zbl

[13] Apecs, Arithmetic of plane elliptic curves, ftp://ftp.math.mcgill.ca/pub/apecs.

[14] mwrank, a package to compute ranks of elliptic curves over the rationals. http://www.maths.nott.ac.uk/personal/jec/ftp/progs.

[15] Simath, a computer algebra system for algorithmic number theory. http://simath.math.unisb.de.