We consider an irreducible curve in , where is an elliptic curve and and are both defined over . Assuming that is not contained in any translate of a proper algebraic subgroup of , we show that the points of the union , where ranges over all proper algebraic subgroups of , form a set of bounded canonical height. Furthermore, if has Complex Multiplication then the set , for ranging over all algebraic subgroups of of codimension at least , is finite. If has no Complex Multiplication then the set for ranging over all proper algebraic subgroups of of codimension at least , is finite.
@article{ASNSP_2003_5_2_1_47_0, author = {Viada, Evelina}, title = {The intersection of a curve with algebraic subgroups in a product of elliptic curves}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {47--75}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {1}, year = {2003}, mrnumber = {1990974}, zbl = {1170.11314}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_1_47_0/} }
TY - JOUR AU - Viada, Evelina TI - The intersection of a curve with algebraic subgroups in a product of elliptic curves JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2003 SP - 47 EP - 75 VL - 2 IS - 1 PB - Scuola normale superiore UR - http://www.numdam.org/item/ASNSP_2003_5_2_1_47_0/ LA - en ID - ASNSP_2003_5_2_1_47_0 ER -
%0 Journal Article %A Viada, Evelina %T The intersection of a curve with algebraic subgroups in a product of elliptic curves %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2003 %P 47-75 %V 2 %N 1 %I Scuola normale superiore %U http://www.numdam.org/item/ASNSP_2003_5_2_1_47_0/ %G en %F ASNSP_2003_5_2_1_47_0
Viada, Evelina. The intersection of a curve with algebraic subgroups in a product of elliptic curves. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 1, pp. 47-75. http://www.numdam.org/item/ASNSP_2003_5_2_1_47_0/
[1] âIntersecting a Curve with Algebraic Subgroups of Multiplicative Groupsâ, International Mathematics Research Notices 20, 1999. | MR | Zbl
- - ,[2] On Siegel's Lemma, Invent. Math. 73 (1983), 11-32. | EuDML | MR | Zbl
- ,[3] Addendum to: On Siegel's Lemma, Invent. Math. 75 (1984), 377. | EuDML | MR | Zbl
- ,[4] âTheory of Groups of Finite Orderâ, 2 ed., Dover Publ., New York, 1955. | JFM | MR | Zbl
,[5] âAn Introduction to the Geometry of Numbersâ, Springer-Verlag, 1971. | MR | Zbl
,[6] Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), 104-129. | MR | Zbl
,[7] Minoration de la hauteur de Néron-Tate sur le variétés abéliennes de type C.M., J. Reine Angew. Math. 529 (2000) 1-74. | MR | Zbl
- ,[8] EndlichkeitssĂ€tze fĂŒr abelsche VarietĂ€ten ĂŒber Zahlkörpern, Invent. Math. 73 (1983), 349-366. | EuDML | MR | Zbl
,[9] Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), 570-603. | EuDML | MR | Zbl
,[10] âFundamentals of Diophantine Geometryâ, Springer-Verlag, 1993. | MR | Zbl
,[11] Equations diophantiennes exponentielles, Invent. Math. 78 (1984), 299-327. | MR | Zbl
,[12] Counting points of small height on elliptic curves, Bull. Soc. Math. France 117, 1989, no. 2, 247-265. | Numdam | MR | Zbl
,[13] Fields of Large Transcendence Degree Generated by Values of Elliptic Functions, Invent. Math. 72 (1983), 407-464. | MR | Zbl
- ,[14] Abelian Varieties, In: âArithmetic Geometryâ, G. Cornell - J. Silverman (eds), Springer-Verlag, 1986. | MR | Zbl
,[15] Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. | MR | Zbl
,[16] Sous-variĂ©tĂ©s d'une variĂ©tĂ© abĂ©lienne et points de torsion, In: âArithmetic and Geometryâ, (dĂ©diĂ© Ă Shafarevich), BirkhĂ€user, 1, 1983, 327-352. | MR | Zbl
,[17] Lower bounds for heights on finitely generated groups, Monatsh. Math. 123 (1997), 171-178. | MR | Zbl
,[18] Proprieté Galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. | MR | Zbl
,[19] âCorps locauxâ, Hermann Paris, 1968. | MR | Zbl
,[20] Local class field theory, In: âAlgebraic Number Theoryâ, J. W. S. Cassels - A. Fröhlich (eds.), Academic Press, London, 1967, 129-162. | MR
,[21] âLectures on the Mordell-Weil Theoremâ, Friedr. Vieweg & Sohn, 1989. | MR | Zbl
,[22] âAdvanced Topics in the Arithmetic of Elliptic Curvesâ, Springer-Verlag, 1994. | MR | Zbl
,[23] âThe Arithmetic of Elliptic Curvesâ, Springer-Verlag, 1986. | MR | Zbl
,