Integral points on a certain family of elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, p. 353-373

The Thue-Siegel method is used to obtain an upper bound for the number of primitive integral solutions to a family of quartic Thue’s inequalities. This will provide an upper bound for the number of integer points on a family of elliptic curves with j-invariant equal to 1728.

Nous utilisons la méthode de Thue-Siegel pour obtenir une borne supérieure du nombre de solutions entières primitives d’une famille d’inégalités quartic de Thue. Cela permet de donner une borne supérieure du nombre de points entiers pour une famille de courbes elliptiques d’invariant j égal à 1728.

DOI : https://doi.org/10.5802/jtnb.905
Classification:  11D25,  11J86
Keywords: Elliptic Curvers, Quartic Thue equations
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     author = {Akhtari, Shabnam},
     title = {Integral points on a certain family of elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
     number = {2},
     year = {2015},
     pages = {353-373},
     doi = {10.5802/jtnb.905},
     mrnumber = {3393158},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2015__27_2_353_0}
}
Akhtari, Shabnam. Integral points on a certain family of elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 2, pp. 353-373. doi : 10.5802/jtnb.905. http://www.numdam.org/item/JTNB_2015__27_2_353_0/

[1] S. Akhtari, The method of Thue-Siegel for binary quartic forms, Acta Arith. 141, 1 (2010), 1–31. | MR 2570336 | Zbl 1219.11053

[2] S. Akhtari, The representation of small integers by binary forms, to appear.

[3] E. Bombieri and W. M. Schmidt, On Thue’s equation, Invent. Math. 88 (1987), 69–81. | MR 877007 | Zbl 0614.10018

[4] G.  V.  Chudnovsky, On the method of Thue-Siegel, Annals of Math. 117 (1983), 325–382. | MR 690849 | Zbl 0518.10038

[5] J. E. Cremona, Reduction of binary cubic and quartic forms, LMS JMC 2 (1999), 62–92. | MR 1693411 | Zbl 0927.11020

[6] J. H. Evertse and J. H. Silverman, Uniform bounds for the number of solutions to Y n =f(x), Math. Proc. Camb. Phil. Soc. 100 (1986), 237–248. | MR 848850 | Zbl 0611.10009

[7] H. A. Helfgott and A. Venkatesh, Integral points on elliptic curves and 3-torsion in class groups, J. Amer. Math. Soc. 19, 3 (2006), 527–550. | MR 2220098 | Zbl 1127.14029

[8] H. W.  Lenstra, JR, Solving the Pell equation, Notices Amer. Math. Soc. 49 (2002), 182–192. | MR 1875156 | Zbl 1126.11312

[9] D. J. Lewis, Diophantine equations: p-adic methods, Studies in Number Theory, M.A.A. Studies in Mathematics, Math. Assoc. of America 6 (1969). | Zbl 0218.10035

[10] L. J.   Mordell, Diophantine Equations, Academic Press, London, New York, (1969). | MR 249355 | Zbl 0188.34503

[11] T. Nagell, Introduction to Number Theory, Chelsea, New York, (1964). | MR 174513 | Zbl 0042.26702

[12] W. M. Schmidt, Integer points on curve on genus 1, Compositio Math. 81 (1992), 33–59. | Numdam | MR 1145607 | Zbl 0747.11026

[13] J. H.  Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag. | MR 1329092 | Zbl 0585.14026

[14] A. Thue, Berechnung aller Lösungen gewisser Gleichungen von der form ax r -by r =f, Vid. Skrifter I Mat.-Naturv. Klasse (1918), 1–9. | JFM 46.1448.07

[15] N.  Tzanakis, On the Diophantine equation x 2 -Dy 4 =k, Acta Arith. 46 (1986), 257–269. | MR 864261 | Zbl 0543.10017

[16] G. P. Walsh, On the number of large integer points on elliptic curves, Acta Arith. 138, 4 (2009), 317–327. | MR 2534137 | Zbl 1254.11035