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

Evelina Viada

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 1, page 47-75
  • ISSN: 0391-173X

Abstract

top
We consider an irreducible curve 𝒞 in E n , where E is an elliptic curve and 𝒞 and E are both defined over 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 ⋃ 𝒞 ∩ A ( 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 ⋃ 𝒞 ∩ A ( 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 ⋃ 𝒞 ∩ A ( ℚ ¯ ) for A ranging over all proper algebraic subgroups of E n of codimension at least n 2 + 2 , is finite.

How to cite

top

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 2.1 (2003): 47-75. <http://eudml.org/doc/84499>.

@article{Viada2003,
abstract = {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 $\bigcup \{\mathcal \{C\}\} \cap A(\overline\{\mathbb \{Q\}\})$, 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 $\bigcup \{\mathcal \{C\}\}\cap A(\overline\{\mathbb \{Q\}\})$, 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 $\bigcup \{\mathcal \{C\}\} \cap A(\overline\{\mathbb \{Q\}\})$ for $A$ ranging over all proper algebraic subgroups of $E^n$ of codimension at least $\frac\{n\}\{2\}+2$, is finite.},
author = {Viada, Evelina},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {1},
pages = {47-75},
publisher = {Scuola normale superiore},
title = {The intersection of a curve with algebraic subgroups in a product of elliptic curves},
url = {http://eudml.org/doc/84499},
volume = {2},
year = {2003},
}

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
PB - Scuola normale superiore
VL - 2
IS - 1
SP - 47
EP - 75
AB - 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 $\bigcup {\mathcal {C}} \cap A(\overline{\mathbb {Q}})$, 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 $\bigcup {\mathcal {C}}\cap A(\overline{\mathbb {Q}})$, 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 $\bigcup {\mathcal {C}} \cap A(\overline{\mathbb {Q}})$ for $A$ ranging over all proper algebraic subgroups of $E^n$ of codimension at least $\frac{n}{2}+2$, is finite.
LA - eng
UR - http://eudml.org/doc/84499
ER -

References

top
  1. [1] E. Bombieri – D. Masser – U. Zannier, “Intersecting a Curve with Algebraic Subgroups of Multiplicative Groups”, International Mathematics Research Notices 20, 1999. Zbl0938.11031MR1728021
  2. [2] E. Bombieri – J. D. Vaaler, On Siegel’s Lemma, Invent. Math. 73 (1983), 11-32. Zbl0533.10030MR707346
  3. [3] E. Bombieri – J. D. Vaaler, Addendum to: On Siegel’s Lemma, Invent. Math. 75 (1984), 377. Zbl0533.10030MR732552
  4. [4] W. Burnside, “Theory of Groups of Finite Order”, 2 ed., Dover Publ., New York, 1955. Zbl0064.25105MR69818JFM42.0151.02
  5. [5] J. W. S. Cassels, “An Introduction to the Geometry of Numbers”, Springer-Verlag, 1971. Zbl0209.34401MR306130
  6. [6] S. David, Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), 104-129. Zbl0873.11035MR1450488
  7. [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. Zbl0993.11034MR1799933
  8. [8] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349-366. Zbl0588.14026MR718935
  9. [9] M. Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), 570-603. Zbl0638.14026MR969244
  10. [10] S. Lang, “Fundamentals of Diophantine Geometry”, Springer-Verlag, 1993. Zbl0528.14013MR715605
  11. [11] M. Laurent, Equations diophantiennes exponentielles, Invent. Math. 78 (1984), 299-327. Zbl0554.10009MR767195
  12. [12] D. Masser, Counting points of small height on elliptic curves, Bull. Soc. Math. France 117, 1989, no. 2, 247-265. Zbl0723.14026MR1015810
  13. [13] D. Masser – G. Wüstholz, Fields of Large Transcendence Degree Generated by Values of Elliptic Functions, Invent. Math. 72 (1983), 407-464. Zbl0516.10027MR704399
  14. [14] J. S. Milne, Abelian Varieties, In: “Arithmetic Geometry”, G. Cornell – J. Silverman (eds), Springer-Verlag, 1986. Zbl0604.14028MR861974
  15. [15] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207-233. Zbl0564.14020MR688265
  16. [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. Zbl0581.14031MR717600
  17. [17] H. P. Schlickewei, Lower bounds for heights on finitely generated groups, Monatsh. Math. 123 (1997), 171-178. Zbl0973.11067MR1430503
  18. [18] J-P. Serre, Proprieté Galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. Zbl0235.14012MR387283
  19. [19] J-P. Serre, “Corps locaux”, Hermann Paris, 1968. Zbl0137.02601MR354618
  20. [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. [21] J-P. Serre, “Lectures on the Mordell-Weil Theorem”, Friedr. Vieweg & Sohn, 1989. Zbl0676.14005MR1002324
  22. [22] J. Silverman, “Advanced Topics in the Arithmetic of Elliptic Curves”, Springer-Verlag, 1994. Zbl0911.14015MR1312368
  23. [23] J. Silverman, “The Arithmetic of Elliptic Curves”, Springer-Verlag, 1986. Zbl0585.14026MR817210

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.