Bicyclotomic polynomials and impossible intersections

David Masser; Umberto Zannier

doi:10.5802/jtnb.851

David Masser ; Umberto Zannier

Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 635-659.

Résumé
Abstract

Nous avons déjà démontré qu’il n’existe qu’un nombre fini de nombres complexes $t \neq 0, 1$ tels que les points $(2, \sqrt{2 (2 - t)})$ et $(3, \sqrt{6 (3 - t)})$ soient d’ordre fini sur la courbe elliptique de Legendre définie par $y^{2} = x (x - 1) (x - t)$ . Nous avons généralisé ensuite ce résultat aux couples de points algébriques quelconques sur $C (t)$ . Nous revenons ici aux points $(u, \sqrt{u (u - 1) (u - t)})$ et $(v, \sqrt{v (v - 1) (v - t)})$ avec des nombres complexes $u$ et $v$ quelconques.

In a recent paper we proved that there are at most finitely many complex numbers $t \neq 0, 1$ such that the points $(2, \sqrt{2 (2 - t)})$ and $(3, \sqrt{6 (3 - t)})$ are both torsion on the Legendre elliptic curve defined by $y^{2} = x (x - 1) (x - t)$ . In a sequel we gave a generalization to any two points with coordinates algebraic over the field $Q (t)$ and even over $C (t)$ . Here we reconsider the special case $(u, \sqrt{u (u - 1) (u - t)})$ and $(v, \sqrt{v (v - 1) (v - t)})$ with complex numbers $u$ and $v$ .

Reçu le : 2011-08-31
Publié le : 2014-02-17

MR 3179679 | Zbl 06291370

DOI : https://doi.org/10.5802/jtnb.851
Classification : 11G05, 14H52

@article{JTNB_2013__25_3_635_0,
     author = {David Masser and Umberto Zannier},
     title = {Bicyclotomic polynomials and impossible intersections},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {635--659},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {25},
     number = {3},
     year = {2013},
     doi = {10.5802/jtnb.851},
     zbl = {06291370},
     mrnumber = {3179679},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.851/}
}

David Masser; Umberto Zannier. Bicyclotomic polynomials and impossible intersections. Journal de Théorie des Nombres de Bordeaux, Tome 25 (2013) no. 3, pp. 635-659. doi : 10.5802/jtnb.851. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.851/

Bibliographie

[1] E. Bombieri, D. Masser and U. Zannier, Finiteness results for multiplicatively dependent points on complex curves, Michigan Math. J. 51 (2003), 451–466. | MR 2021000 | Zbl 1048.11056

[2] F. Beukers and C. J. Smyth, Cyclotomic points on curves, Number theory for the millennium I (Urbana 2000), A.K. Peters Ltd. (2002), 67–85. | MR 1956219 | Zbl 1029.11009

[3] S. Lang, Elliptic functions, Addison-Wesley (1973). | MR 409362 | Zbl 0316.14001

[4] D. Masser, Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc. 311 (1989), 413–424. | MR 974783 | Zbl 0673.14016

[5] D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves. C. R. Acad. Sci. Paris, Ser. I 346 (2008), 491–494. | MR 2412783 | Zbl 1197.11066

[6] D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), 1677–1691. | MR 2766181 | Zbl 1225.11078

[7] D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Annalen 352 (2012), 453–484. | MR 2874963 | Zbl pre06006368

[8] J. Pila, Integer points on the dilation of a subanalytic surface, Quart. J. Math. 55 (2004), 207–223. | MR 2068319 | Zbl 1111.32004

[9] J. Pila, Counting rational points on a certain exponential-algebraic surface, Annales Institut Fourier 60 (2010), 489–514. | Numdam | MR 2667784 | Zbl 1210.11074

[10] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Inventiones Math. 71 (1983), 207–233. | MR 688265 | Zbl 0564.14020

[11] J.B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. | MR 137689 | Zbl 0122.05001

[12] J.H. Silverman, The arithmetic of elliptic curves, Springer-Verlag (1986). | MR 817210 | Zbl 1194.11005

[13] A. Weil, Foundations of algebraic geometry, American Math. Soc. Colloquium Pub. XXIX (1946). | MR 23093 | Zbl 0168.18701

[14] U. Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Math. Studies, 181, Princeton (2012). | MR 2918151 | Zbl 1246.14003