The intersection of a curve with algebraic subgroups in a product of elliptic curves

Viada, Evelina

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 1, pp. 47-75

We consider an irreducible curve $𝒞$ in $E^n$, where $E$ is an elliptic curve and $𝒞$ and $E$ are both defined over $\overline{\mathbb{Q}}$. Assuming that $𝒞$ is not contained in any translate of a proper algebraic subgroup of $E^n$, we show that the points of the union $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, 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 $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$, 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 $\bigcup 𝒞\cap A\left(\overline{\mathbb{Q}}\right)$ for $A$ ranging over all proper algebraic subgroups of $E^n$ of codimension at least $\frac{n}{2}+2$, is finite.

Classification : 11D45, 11G50

[1] E. Bombieri - D. Masser - U. Zannier, “Intersecting a Curve with Algebraic Subgroups of Multiplicative Groups”, International Mathematics Research Notices 20, 1999. | Zbl | MR

[2] E. Bombieri - J. D. Vaaler, On Siegel's Lemma, Invent. Math. 73 (1983), 11-32. | Zbl | MR | EuDML

[3] E. Bombieri - J. D. Vaaler, Addendum to: On Siegel's Lemma, Invent. Math. 75 (1984), 377. | Zbl | MR | EuDML

[4] W. Burnside, “Theory of Groups of Finite Order”, 2 ed., Dover Publ., New York, 1955. | Zbl | MR | JFM

[5] J. W. S. Cassels, “An Introduction to the Geometry of Numbers”, Springer-Verlag, 1971. | Zbl | MR

[6] S. David, Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), 104-129. | Zbl | MR

[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. | Zbl | MR

[8] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366. | Zbl | MR | EuDML

[9] M. Hindry, Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), 570-603. | Zbl | MR | EuDML

[10] S. Lang, “Fundamentals of Diophantine Geometry”, Springer-Verlag, 1993. | Zbl | MR

[11] M. Laurent, Equations diophantiennes exponentielles, Invent. Math. 78 (1984), 299-327. | Zbl | MR

[12] D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117, 1989, no. 2, 247-265. | Zbl | MR | Numdam

[13] D. Masser - G. Wüstholz, Fields of Large Transcendence Degree Generated by Values of Elliptic Functions, Invent. Math. 72 (1983), 407-464. | Zbl | MR

[14] J. S. Milne, Abelian Varieties, In: “Arithmetic Geometry”, G. Cornell - J. Silverman (eds), Springer-Verlag, 1986. | Zbl | MR

[15] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. | Zbl | MR

[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. | Zbl | MR

[17] H. P. Schlickewei, Lower bounds for heights on finitely generated groups, Monatsh. Math. 123 (1997), 171-178. | Zbl | MR

[18] J-P. Serre, Proprieté Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. | Zbl | MR

[19] J-P. Serre, “Corps locaux”, Hermann Paris, 1968. | Zbl | MR

[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. | MR

[21] J-P. Serre, “Lectures on the Mordell-Weil Theorem”, Friedr. Vieweg & Sohn, 1989. | Zbl | MR

[22] J. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Springer-Verlag, 1994. | Zbl | MR

[23] J. Silverman, “The Arithmetic of Elliptic Curves”, Springer-Verlag, 1986. | Zbl | MR