Some remarks on almost rational torsion points

John Boxall[1]; David Grant[2]

  • [1] Laboratoire de Mathématiques Nicolas Oresme, CNRS – UMR 6139 Université de Caen boulevard Maréchal Juin BP 5186, 14032 Caen cedex, France
  • [2] Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395 USA

Journal de Théorie des Nombres de Bordeaux (2006)

  • Volume: 18, Issue: 1, page 13-28
  • ISSN: 1246-7405

Abstract

top
For a commutative algebraic group G over a perfect field k , Ribet defined the set of almost rational torsion points G tors , k ar of G over k . For positive integers d , g , we show there is an integer U d , g such that for all tori T of dimension at most d over number fields of degree at most g , T tors , k ar T [ U d , g ] . We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite set of semi-abelian varieties G over a finite field k , G tors , k ar is infinite, and use this to show for any abelian variety A over a p -adic field k , there is a finite extension of k over which A tors , k ar is infinite.

How to cite

top

Boxall, John, and Grant, David. "Some remarks on almost rational torsion points." Journal de Théorie des Nombres de Bordeaux 18.1 (2006): 13-28. <http://eudml.org/doc/249655>.

@article{Boxall2006,
abstract = {For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G^\{\operatorname\{ar\}\}_\{\operatorname\{tors\},k\}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_\{d,g\}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T^\{\operatorname\{ar\}\}_\{\operatorname\{tors\},k\}\subseteq T[U_\{d,g\}]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite set of semi-abelian varieties $G$ over a finite field $k$, $G^\{\operatorname\{ar\}\}_\{\operatorname\{tors\},k\}$ is infinite, and use this to show for any abelian variety $A$ over a $p$-adic field $k$, there is a finite extension of $k$ over which $A^\{\operatorname\{ar\}\}_\{\operatorname\{tors\},k\}$ is infinite.},
affiliation = {Laboratoire de Mathématiques Nicolas Oresme, CNRS – UMR 6139 Université de Caen boulevard Maréchal Juin BP 5186, 14032 Caen cedex, France; Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395 USA},
author = {Boxall, John, Grant, David},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Elliptic curves; torsion; almost rational; elliptic curves},
language = {eng},
number = {1},
pages = {13-28},
publisher = {Université Bordeaux 1},
title = {Some remarks on almost rational torsion points},
url = {http://eudml.org/doc/249655},
volume = {18},
year = {2006},
}

TY - JOUR
AU - Boxall, John
AU - Grant, David
TI - Some remarks on almost rational torsion points
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2006
PB - Université Bordeaux 1
VL - 18
IS - 1
SP - 13
EP - 28
AB - For a commutative algebraic group $G$ over a perfect field $k$, Ribet defined the set of almost rational torsion points $G^{\operatorname{ar}}_{\operatorname{tors},k}$ of $G$ over $k$. For positive integers $d$, $g,$ we show there is an integer $U_{d,g}$ such that for all tori $T$ of dimension at most $d$ over number fields of degree at most $g$, $T^{\operatorname{ar}}_{\operatorname{tors},k}\subseteq T[U_{d,g}]$. We show the corresponding result for abelian varieties with complex multiplication, and under an additional hypothesis, for elliptic curves without complex multiplication. Finally, we show that except for an explicit finite set of semi-abelian varieties $G$ over a finite field $k$, $G^{\operatorname{ar}}_{\operatorname{tors},k}$ is infinite, and use this to show for any abelian variety $A$ over a $p$-adic field $k$, there is a finite extension of $k$ over which $A^{\operatorname{ar}}_{\operatorname{tors},k}$ is infinite.
LA - eng
KW - Elliptic curves; torsion; almost rational; elliptic curves
UR - http://eudml.org/doc/249655
ER -

References

top
  1. M. H. Baker, K. A. Ribet, Galois theory and torsion points on curves. Journal de Théorie des Nombres de Bordeaux 15 (2003), 11–32. Zbl1065.11045MR2018998
  2. Z. I. Borevich, I. R. Shafarevich, Number Theory. Academic Press, New York, San Francisco and London, (1966). Zbl0145.04902MR195803
  3. S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21. Springer-Verlag, Berlin, 1990. Zbl0705.14001MR1045822
  4. J. Boxall, D. Grant, Theta functions and singular torsion on elliptic curves, in Number Theory for the Millenium, Bruce Berndt, et. al. editors. A K Peters, Natick, (2002), 111–126. Zbl1195.11076MR1956221
  5. J. Boxall, D. Grant, Singular torsion points on elliptic curves. Math. Res. Letters 10 (2003), 847–866. Zbl1130.11322MR2025060
  6. J. Boxall, D. Grant, Examples of torsion points on genus two curves. Trans. AMS 352 (2000), 4533–4555. Zbl1007.11038MR1621721
  7. F. Calegari, Almost rational torsion points on semistable elliptic curves. Intern. Math. Res. Notices (2001), 487–503. Zbl1002.14004MR1832537
  8. R. Coleman, Torsion points on curves and p -adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168. Zbl0578.14038MR782557
  9. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics 150. Springer-Verlag, New York, 1995. Zbl0819.13001MR1322960
  10. S. Lang, Complex Multiplication. Springer-Verlag, New York, (1983). Zbl0536.14029MR713612
  11. D. Masser, G. Wüstholz, Galois properties of division fields. Bull. London Math. Soc. 25 (1993), 247–254. Zbl0809.14026MR1209248
  12. B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129–162. Zbl0386.14009MR482230
  13. L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437-449. Zbl0936.11037MR1369424
  14. M. Newman, Integral matrices. Academic Press, New York and London (1972). Zbl0254.15009MR340283
  15. T. Ono, Arithmetic of algebraic tori. Ann. Math. 74 (1961), 101–139. Zbl0119.27801MR124326
  16. P. Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. reine angew. Math. 506 (1999), 85–116. Zbl0919.11040MR1665681
  17. F. Pellarin, Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques. Acta Arith. 100 (2001), 203–243. Zbl0986.11046MR1865384
  18. M. Raynaud, Courbes sur une variété abélienne et points de torsion. Invent. Math. 71 (1983), 207–233. Zbl0564.14020MR688265
  19. K. Ribet, M. Kim, Torsion points on modular curves and Galois theory. Notes of a series of talks by K. Ribet in the Distinguished Lecture Series, Southwestern Center for Arithmetic Algebraic Geometry, May 1999. arXiv:math.NT/0305281 
  20. J.-P. Serre, Abelian l -adic representations and elliptic curves. Benjamin, New York (1968). Zbl0186.25701MR263823
  21. J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–332. Zbl0235.14012MR387283
  22. J.-P. Serre, Algèbre et géométrie. Ann. Collège de France (1985–1986), 95–100. MR965792
  23. G. Shimura, Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten Publishers and Princeton University Press (1971). Zbl0221.10029MR314766
  24. A. Silverberg, Fields of definition for homomorphisms of abelian varieties. J. Pure and Applied Algebra 77 (1992), 253–272. Zbl0808.14037MR1154704
  25. A. Silverberg, Yu. G. Zarhin, Étale cohomology and reduction of abelian varieties. Bull. Soc. Math. France. 129 (2001), 141–157. Zbl1037.11042MR1871981
  26. J. Tate, Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134–144. Zbl0147.20303MR206004
  27. E. Viada, Bounds for minimal elliptic isogenies. Preprint. Zbl1201.11080

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.