Some remarks on almost rational torsion points
Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 13-28.

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.

Lorsque G désigne un groupe algébrique sur un corps parfait k, Ribet a défini l’ensemble des points de torsion presque rationnels G tors,k ar de G sur k. Si d, g désignent des entiers positifs, nous montrons qu’il existe un entier U d,g tel que, pour tout tore T de dimension au plus d sur un corps de nombres de degré au plus g, on ait T tors,k ar T[U d,g ]. Nous montrons le résultat analogue pour les variétés abéliennes à multiplication complexe puis, sous une hypothèse supplémentaire, pour les courbes elliptiques sans multiplication complexe. Enfin, nous montrons que, à l’exception d’un ensemble fini explicite de variétés semi-abéliennes G sur un corps fini, G tors,k ar est infini et nous utilisons ce résultat pour montrer que pour toute variété abélienne sur un corps p-adique k, il existe une extension finie de k sur laquelle A tors,k ar est infini.

DOI: 10.5802/jtnb.531
Keywords: Elliptic curves, torsion, almost rational.
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
@article{JTNB_2006__18_1_13_0,
     author = {John Boxall and David Grant},
     title = {Some remarks on almost rational torsion points},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {13--28},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     doi = {10.5802/jtnb.531},
     mrnumber = {2245873},
     zbl = {05070445},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.531/}
}
TY  - JOUR
AU  - John Boxall
AU  - David Grant
TI  - Some remarks on almost rational torsion points
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2006
DA  - 2006///
SP  - 13
EP  - 28
VL  - 18
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.531/
UR  - https://www.ams.org/mathscinet-getitem?mr=2245873
UR  - https://zbmath.org/?q=an%3A05070445
UR  - https://doi.org/10.5802/jtnb.531
DO  - 10.5802/jtnb.531
LA  - en
ID  - JTNB_2006__18_1_13_0
ER  - 
%0 Journal Article
%A John Boxall
%A David Grant
%T Some remarks on almost rational torsion points
%J Journal de théorie des nombres de Bordeaux
%D 2006
%P 13-28
%V 18
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.531
%R 10.5802/jtnb.531
%G en
%F JTNB_2006__18_1_13_0
John Boxall; David Grant. Some remarks on almost rational torsion points. Journal de théorie des nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 13-28. doi : 10.5802/jtnb.531. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.531/

[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. | Numdam | MR | Zbl

[2] Z. I. Borevich, I. R. Shafarevich, Number Theory. Academic Press, New York, San Francisco and London, (1966). | MR | Zbl

[3] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21. Springer-Verlag, Berlin, 1990. | MR | Zbl

[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. | MR | Zbl

[5] J. Boxall, D. Grant, Singular torsion points on elliptic curves. Math. Res. Letters 10 (2003), 847–866. | MR | Zbl

[6] J. Boxall, D. Grant, Examples of torsion points on genus two curves. Trans. AMS 352 (2000), 4533–4555. | MR | Zbl

[7] F. Calegari, Almost rational torsion points on semistable elliptic curves. Intern. Math. Res. Notices (2001), 487–503. | MR | Zbl

[8] R. Coleman, Torsion points on curves and p-adic abelian integrals. Ann. of Math. (2) 121 (1985), no. 1, 111–168. | MR | Zbl

[9] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics 150. Springer-Verlag, New York, 1995. | MR | Zbl

[10] S. Lang, Complex Multiplication. Springer-Verlag, New York, (1983). | MR | Zbl

[11] D. Masser, G. Wüstholz, Galois properties of division fields. Bull. London Math. Soc. 25 (1993), 247–254. | MR | Zbl

[12] B. Mazur, Rational isogenies of prime degree. Invent. Math. 44 (1978), 129–162. | MR | Zbl

[13] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), 437-449. | MR | Zbl

[14] M. Newman, Integral matrices. Academic Press, New York and London (1972). | MR | Zbl

[15] T. Ono, Arithmetic of algebraic tori. Ann. Math. 74 (1961), 101–139. | MR | Zbl

[16] P. Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. reine angew. Math. 506 (1999), 85–116. | MR | Zbl

[17] F. Pellarin, Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques. Acta Arith. 100 (2001), 203–243. | MR | Zbl

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

[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). | MR | Zbl

[21] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), 259–332. | MR | Zbl

[22] J.-P. Serre, Algèbre et géométrie. Ann. Collège de France (1985–1986), 95–100. | MR

[23] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Iwanami Shoten Publishers and Princeton University Press (1971). | MR | Zbl

[24] A. Silverberg, Fields of definition for homomorphisms of abelian varieties. J. Pure and Applied Algebra 77 (1992), 253–272. | MR | Zbl

[25] A. Silverberg, Yu. G. Zarhin, Étale cohomology and reduction of abelian varieties. Bull. Soc. Math. France. 129 (2001), 141–157. | Numdam | MR | Zbl

[26] J. Tate, Endomorphisms of abelian varieties over finite fields. Invent. Math. 2 (1966), 134–144. | MR | Zbl

[27] E. Viada, Bounds for minimal elliptic isogenies. Preprint.

Cited by Sources: