Variations on a theme of Runge: effective determination of integral points on certain varieties
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, p. 385-417

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our method, we completely solve certain equations involving squares in products of terms in an arithmetic progression.

Nous considérons quelques variations sur la méthode classique de Runge pour déterminer effectivement les points entiers sur certaines courbes. Nous prouvons d’abord une version du théorème de Runge valide pour des variétés de dimension supérieure, généralisant une version uniforme du théorème de Runge due à Bombieri. Nous étudions alors comment la méthode de Runge peut être étendue en utilisant certains revêtements. Nous prouvons un résultat pour les courbes arbitraires et un résultat plus explicite pour les courbes superelliptic. Comme application de notre méthode, nous résolvons complètement certaines équations impliquant des carrés dans les produits des termes dans une progression arithmétique.

@article{JTNB_2008__20_2_385_0,
     author = {Levin, Aaron},
     title = {Variations on a theme of Runge: effective determination of integral points on certain varieties},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     pages = {385-417},
     doi = {10.5802/jtnb.634},
     mrnumber = {2477511},
     zbl = {pre05543169},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_2_385_0}
}
Levin, Aaron. Variations on a theme of Runge: effective determination of integral points on certain varieties. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 385-417. doi : 10.5802/jtnb.634. http://www.numdam.org/item/JTNB_2008__20_2_385_0/

[1] A. Baker, Transcendental number theory, second ed. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1990. | MR 1074572 | Zbl 0715.11032

[2] M. A. Bennett, N. Bruin, K. Győry, L. Hajdu, Powers from products of consecutive terms in arithmetic progression. Proc. London Math. Soc. (3) 92 (2006), no. 2, 273–306. | MR 2205718 | Zbl pre05014379

[3] E. Bombieri, On Weil’s “théorème de décomposition”. Amer. J. Math. 105 (1983), no. 2, 295–308. | Zbl 0516.12009

[4] E. Bombieri, W. Gubler, Heights in Diophantine geometry. New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. | MR 2216774 | Zbl 1115.11034

[5] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765–770. | MR 808103 | Zbl 0588.14015

[6] P. Erdős, J. L. Selfridge, The product of consecutive integers is never a power. Illinois J. Math. 19 (1975), 292–301. | MR 376517 | Zbl 0295.10017

[7] D. L. Hilliker, E. G. Straus, Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge’s theorem. Trans. Amer. Math. Soc. 280 (1983), no. 2, 637–657. | Zbl 0528.10011

[8] M. Hindry, J. H. Silverman, Diophantine geometry. Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. | MR 1745599 | Zbl 0948.11023

[9] N. Hirata-Kohno, S. Laishram, T. N. Shorey, R. Tijdeman, An extension of a theorem of Euler. Acta Arith. 129 (2007), no. 1, 71–102. | MR 2326488 | Zbl 1137.11022

[10] S. Laishram, T. N. Shorey, Squares in products in arithmetic progression with at most two terms omitted and common difference a prime power. Acta Arith. (to appear). | Zbl pre05376844

[11] S. Laishram, T. N. Shorey, S. Tengely, Squares in products in arithmetic progression with at most one term omitted and common difference a prime power. (to appear). | MR 2453529 | Zbl pre05376844

[12] A. Levin, Ideal class groups and torsion in Picard groups of varieties. (submitted).

[13] A. Levin, Ideal class groups, Hilbert’s irreducibility theorem, and integral points of bounded degree on curves. J. Théor. Nombres Bordeaux 19 (2007), no. 2, 485–499. | Numdam | Zbl pre05302786

[14] D. W. Masser, G. Wüstholz, Fields of large transcendence degree generated by values of elliptic functions. Invent. Math. 72 (1983), no. 3, 407–464. | MR 704399 | Zbl 0516.10027

[15] A. Mukhopadhyay, T. N. Shorey, Almost squares in arithmetic progression. II Acta Arith. 110 (2003), no. 1, 1–14. | MR 2007540 | Zbl 1030.11010

[16] A. Mukhopadhyay, T. N. Shorey, Almost squares in arithmetic progression. III. Indag. Math. (N.S.) 15 (2004), no. 4, 523–533. | MR 2114935 | Zbl pre02192843

[17] A. Mukhopadhyay, T. N. Shorey, Square free part of products of consecutive integers. Publ. Math. Debrecen 64 (2004), no. 1-2, 79–99. | MR 2035890 | Zbl 1049.11037

[18] R. Obláth, Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe. Publ. Math. Debrecen 1 (1950), 222–226. | MR 39745 | Zbl 0038.17901

[19] C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen. J. Reine Angew. Math. 100 (1887), 425–435.

[20] N. Saradha, T. N. Shorey, Almost squares and factorisations in consecutive integers. Compositio Math. 138 (2003), no. 1, 113–124. | MR 2002956 | Zbl 1038.11020

[21] N. Saradha, T. N. Shorey, Almost squares in arithmetic progression. Compositio Math. 138 (2003), no. 1, 73–111. | MR 2002955 | Zbl 1036.11007

[22] A. Schinzel, W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers. Acta Arith. 4 (1958), 185–208; erratum 5 (1958), 259. | MR 106202 | Zbl 0082.25802

[23] T. N. Shorey, Exponential Diophantine equations involving products of consecutive integers and related equations. Number theory, Trends Math., Birkhäuser, Basel, 2000, pp. 463–495. | MR 1764814 | Zbl 0958.11026

[24] T. N. Shorey, Powers in arithmetic progressions. III. The Riemann zeta function and related themes: papers in honour of Professor K. Ramachandra, Ramanujan Math. Soc. Lect. Notes Ser., vol. 2, Ramanujan Math. Soc., Mysore, 2006, pp. 131–140. | MR 2335192 | Zbl 1127.11027

[25] T. N. Shorey, R. Tijdeman, Some methods of Erdős applied to finite arithmetic progressions. The mathematics of Paul Erdős, I, Algorithms Combin., vol. 13, Springer, Berlin, 1997, pp. 251–267. | MR 1425190 | Zbl 0874.11035

[26] V. G. Sprindžuk, Reducibility of polynomials and rational points on algebraic curves. Dokl. Akad. Nauk SSSR 250 (1980), no. 6, 1327–1330. | MR 564338 | Zbl 0447.12010

[27] W. Stein, Sage: Open Source Mathematical Software (Version 2.10.2). The Sage Group, 2008, http://www.sagemath.org.

[28] S. Tengely, Note on a paper “An extension of a theorem of Euler” by Hirata-Kohno et al. arXiv:0707.0596v1 [math.NT]. | Zbl pre05354500

[29] The PARI Group, Bordeaux, PARI/GP, version 2.3.3, 2005, available from http://pari.math.u-bordeaux.fr/.

[30] P. G. Walsh, A quantitative version of Runge’s theorem on Diophantine equations. Acta Arith. 62 (1992), no. 2, 157–172. | Zbl 0769.11017