On uniform lower bound of the Galois images associated to elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, p. 23-43

Let p be a prime and let K be a number field. Let ρ E,p :G K Aut(T p E)GL 2 ( p ) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the image of ρ E,p is open if E has no complex multiplication. For an elliptic curve E over K whose j-invariant does not appear in an exceptional finite set (which is non-explicit however), we give an explicit uniform lower bound of the size of the image of ρ E,p .

Soit p un nombre premier et K un corps de nombres. Soit ρ E,p :G K Aut(T p E)GL 2 ( p ) la représentation Galoisienne donnée par l’action du groupe de Galois sur le module de Tate p-adique d’une courbe elliptique E définie sur K. Serre a prouvé que l’image de ρ E,p est ouverte si E n’a pas de multiplication complexe. Pour E une courbe elliptique définie sur K et dont l’invariant j n’appartient pas à un ensemble fini exceptionnel (qui est non explicite cependant), nous donnons une minoration uniforme et explicite de la taille de l’image de ρ E,p .

@article{JTNB_2008__20_1_23_0,
     author = {Arai, Keisuke},
     title = {On uniform lower bound of the Galois images associated to elliptic curves},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {1},
     year = {2008},
     pages = {23-43},
     doi = {10.5802/jtnb.614},
     mrnumber = {2434156},
     zbl = {pre05543189},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_1_23_0}
}
Arai, Keisuke. On uniform lower bound of the Galois images associated to elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 1, pp. 23-43. doi : 10.5802/jtnb.614. http://www.numdam.org/item/JTNB_2008__20_1_23_0/

[1] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. Modular functions of one variable, II, 143–316. Lecture Notes in Math. 349. Springer, Berlin, 1973. | MR 337993 | Zbl 0281.14010

[2] B. Edixhoven, Rational torsion points on elliptic curves over number fields (after Kamienny and Mazur). Séminaire Bourbaki, Vol. 1993/94. Astérisque No. 227 (1995), Exp. No. 782, 4, 209–227. | Numdam | MR 1321648 | Zbl 0832.14024

[3] G. Faltings, Finiteness theorems for abelian varieties over number fields. Translated from the German original [Invent. Math. 73 (1983), no. 3, 349–366; ibid. 75 (1984), no. 2, 381] by Edward Shipz. Arithmetic geometry (Storrs, Conn., 1984), 9–27. Springer, New York, 1986. | MR 718935 | Zbl 0602.14044

[4] S. Kamienny, Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109 (1992), no. 2, 221–229. | MR 1172689 | Zbl 0773.14016

[5] N. Katz, B. Mazur, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108. Princeton University Press, Princeton, NJ, 1985. | MR 772569 | Zbl 0576.14026

[6] D.-S. Kubert, Universal bounds on the torsion of elliptic curves. Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. | MR 434947 | Zbl 0331.14010

[7] J. Manin, The p-torsion of elliptic curves is uniformly bounded. Translated from the Russian original [Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 459–465]. Mathematics of the USSR-Izvestija 3 (1969), No. 3-4, 433–438. | MR 272786 | Zbl 0205.25002

[8] B. Mazur, Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Études Sci. 47 (1977), 33–186. | Numdam | MR 488287 | Zbl 0394.14008

[9] B. Mazur, Rational points on modular curves. Modular functions of one variable V, 107–148. Lecture Notes in Math. 601. Springer, Berlin, 1977. | MR 450283 | Zbl 0357.14005

[10] B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld). Invent. Math. 44 (1978), no. 2, 129–162. | MR 482230 | Zbl 0386.14009

[11] L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124 (1996), no. 1-3, 437–449. | MR 1369424 | Zbl 0936.11037

[12] F. Momose, Rational points on the modular curves X split (p). Compositio Math. 52 (1984), no. 1, 115–137. | Numdam | MR 742701 | Zbl 0574.14023

[13] F. Momose, Isogenies of prime degree over number fields. Compositio Math. 97 (1995), no. 3, 329–348. | Numdam | MR 1353278 | Zbl 1044.11582

[14] K. Nakata, On the 2-adic representation associated to an elliptic curve defined over . (Japanese), Number Theory Symposium in Kinosaki, December 1979, 221–235.

[15] P. Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres. J. Reine Angew. Math. 506 (1999), 85–116. | MR 1665681 | Zbl 0919.11040

[16] P. Parent, Towards the triviality of X 0 + (p r )() for r>1. Compositio Math. 141 (2005), no. 3, 561–572. | MR 2135276 | Zbl pre02183028

[17] M. Rebolledo, Module supersingulier, formule de Gross-Kudla et points rationnels de courbes modulaires. To appear in Pacific J. Math. | MR 2375318

[18] J.-P. Serre, Abelian l-adic representations and elliptic curves. Lecture at McGill University. W. A. Benjamin Inc., New York-Amsterdam, 1968. | MR 263823 | Zbl 0186.25701

[19] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259–331. | Zbl 0235.14012

[20] J.-P. Serre, Représentations l-adiques. Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976), 177–193. Japan Soc. Promotion Sci., Tokyo, 1977. | MR 476753 | Zbl 0406.14015

[21] J.-P. Serre, Points rationnels des courbes modulaires X 0 (N) [d’après B. Mazur]. Séminaire Bourbaki, 30e année (1977/78), Exp. No. 511, 89–100. Lecture Notes in Math. 710. Springer, Berlin, 1979. | Numdam | Zbl 0411.14005

[22] G. Shimura, Introduction to the arithmetic theory of automorphic functions. Princeton University Press, Princeton, NJ, 1994. | MR 1291394 | Zbl 0872.11023

[23] J. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106. Springer-Verlag, New York, 1986. | MR 817210 | Zbl 0585.14026