The intersection of a curve with algebraic subgroups in a product of elliptic curves. (English) Zbl 1170.11314

Summary: We consider an irreducible curve \({\mathcal C}\) in \(E^n\), where \(E\) is an elliptic curve and \({\mathcal C}\) and \(E\) are both defined over \(\overline \mathbb{Q}\). Assuming that \({\mathcal C}\) is not contained in any translate of a proper algebraic subgroup of \(E^n\), we show that the points of the union \(\cup{\mathcal C}\cap A(\overline \mathbb{Q})\), where \(A\) ranges over all proper algebraic subgroups of \(E^n\), form a set of bounded canonical height. Furthermore, if \(E\) has Complex Multiplication then the set \(\cup{\mathcal C}\cap A(\overline\mathbb{Q})\), for \(A\) ranging over all algebraic subgroups of \(E^n\) of codimension at least 2, is finite. If \(E\) has no Complex Multiplication then the set \(\cup{\mathcal C}\cap A(\mathbb{Q})\) for \(A\) ranging over all proper algebraic subgroups of \(E^n\) of codimension at least \(\frac n2+2\), is finite.


11G10 Abelian varieties of dimension \(> 1\)
11G50 Heights
Full Text: EuDML


[1] E. Bombieri - D. Masser - U. Zannier, “Intersecting a Curve with Algebraic Subgroups of Multiplicative Groups”, International Mathematics Research Notices 20, 1999. Zbl0938.11031 MR1728021 · Zbl 0938.11031 · doi:10.1155/S1073792899000628
[2] E. Bombieri - J. D. Vaaler, On Siegel’s Lemma, Invent. Math. 73 (1983), 11-32. Zbl0533.10030 MR707346 · Zbl 0533.10030 · doi:10.1007/BF01393823
[3] E. Bombieri - J. D. Vaaler, Addendum to: On Siegel’s Lemma, Invent. Math. 75 (1984), 377. Zbl0533.10030 MR732552 · Zbl 0533.10030 · doi:10.1007/BF01393823
[4] W. Burnside, “Theory of Groups of Finite Order”, 2 ed., Dover Publ., New York, 1955. Zbl0064.25105 MR69818 JFM42.0151.02 · Zbl 0064.25105
[5] J. W. S. Cassels, “An Introduction to the Geometry of Numbers”, Springer-Verlag, 1971. Zbl0209.34401 MR306130 · Zbl 0209.34401
[6] S. David, Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), 104-129. Zbl0873.11035 MR1450488 · Zbl 0873.11035 · doi:10.1006/jnth.1997.2100
[7] S. David - M. Hindry, Minoration de la hauteur de Néron-Tate sur le variétés abéliennes de type C.M., J. Reine Angew. Math. 529 (2000) 1-74. Zbl0993.11034 MR1799933 · Zbl 0993.11034 · doi:10.1515/crll.2000.096
[8] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366. Zbl0588.14026 MR718935 · Zbl 0588.14026 · doi:10.1007/BF01388432
[9] M. Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), 570-603. Zbl0638.14026 MR969244 · Zbl 0638.14026 · doi:10.1007/BF01394276
[10] S. Lang, “Fundamentals of Diophantine Geometry”, Springer-Verlag, 1993. Zbl0528.14013 MR715605 · Zbl 0528.14013
[11] M. Laurent, Equations diophantiennes exponentielles, Invent. Math. 78 (1984), 299-327. Zbl0554.10009 MR767195 · Zbl 0554.10009 · doi:10.1007/BF01388597
[12] D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117, 1989, no. 2, 247-265. Zbl0723.14026 MR1015810 · Zbl 0723.14026
[13] D. Masser - G. Wüstholz, Fields of Large Transcendence Degree Generated by Values of Elliptic Functions, Invent. Math. 72 (1983), 407-464. Zbl0516.10027 MR704399 · Zbl 0516.10027 · doi:10.1007/BF01398396
[14] J. S. Milne, Abelian Varieties, In: “Arithmetic Geometry”, G. Cornell - J. Silverman (eds), Springer-Verlag, 1986. Zbl0604.14028 MR861974 · Zbl 0604.14028
[15] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. Zbl0564.14020 MR688265 · Zbl 0564.14020 · doi:10.1007/BF01393342
[16] M. Raynaud, Sous-variétés d’une variété abélienne et points de torsion, In: “Arithmetic and Geometry”, (dédié à Shafarevich), Birkhäuser, 1, 1983, 327-352. Zbl0581.14031 MR717600 · Zbl 0581.14031
[17] H. P. Schlickewei, Lower bounds for heights on finitely generated groups, Monatsh. Math. 123 (1997), 171-178. Zbl0973.11067 MR1430503 · Zbl 0973.11067 · doi:10.1007/BF01305970
[18] J-P. Serre, Proprieté Galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. Zbl0235.14012 MR387283 · Zbl 0235.14012 · doi:10.1007/BF01405086
[19] J-P. Serre, “Corps locaux”, Hermann Paris, 1968. Zbl0137.02601 MR354618 · Zbl 0137.02601
[20] J-P. Serre, Local class field theory, In: “Algebraic Number Theory”, J. W. S. Cassels - A. Fröhlich (eds.), Academic Press, London, 1967, 129-162. MR220701
[21] J-P. Serre, “Lectures on the Mordell-Weil Theorem”, Friedr. Vieweg & Sohn, 1989. Zbl0676.14005 MR1002324 · Zbl 0676.14005
[22] J. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Springer-Verlag, 1994. Zbl0911.14015 MR1312368 · Zbl 0911.14015
[23] J. Silverman, “The Arithmetic of Elliptic Curves”, Springer-Verlag, 1986. Zbl0585.14026 MR817210 · Zbl 0585.14026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.