Elliptic curves with abelian division fields. (English) Zbl 1410.11051
Summary: Let \(E\) be an elliptic curve over \(\mathbb Q\), and let \(n\geq 1\). The central object of study of this article is the division field \(\mathbb Q(E[n])\) that results by adjoining to \(\mathbb Q\) the coordinates of all \(n\)-torsion points on \(E(\overline{\mathbb Q})\). In particular, we classify all curves \(E/\mathbb Q\) such that \(\mathbb Q(E[n])\) is as small as possible, that is, when \(\mathbb Q(E[n])=\mathbb Q(\zeta _n)\), and we prove that this is only possible for \(n=2,3,4\), or 5. More generally, we classify all curves such that \(\mathbb Q(E[n])\) is contained in a cyclotomic extension of \(\mathbb Q\) or, equivalently (by the Kronecker-Weber theorem), when \(\mathbb Q(E[n])/\mathbb Q\) is an abelian extension. In particular, we prove that this only happens for \(n=2,3,4,5,6\), or 8, and we classify the possible Galois groups that occur for each value of \(n\).
References:
| [1] | Bilu, Y., Parent, P.: Serre’s uniformity problem in the split Cartan case. Ann. Math. 173(1), 569-584 (2011) · Zbl 1278.11065 · doi:10.4007/annals.2011.173.1.13 |
| [2] | Birch, B.J., Kuyk, W. (eds.): Modular functions of one variable IV. In: Lecture Notes in Mathematics, vol. 476. Springer, Berlin (1975) · Zbl 0315.14014 |
| [3] | Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235-265 (1997) · Zbl 0898.68039 · doi:10.1006/jsco.1996.0125 |
| [4] | Cojocaru, A.C.: On the surjectivity of the Galois representations associated to non-CM elliptic curves (with an appendix by Ernst Kani). Can. Math. Bull. 48, 16-31 (2005) · Zbl 1062.11031 · doi:10.4153/CMB-2005-002-x |
| [5] | Cojocaru, A.C., Hall, C.: Uniform results for Serre’s theorem for elliptic curves. In: International Mathematics Research Notices, pp. 3065-3080 (2005) · Zbl 1178.11045 |
| [6] | Cremona, J.E.: Elliptic curve data for conductors up to 350.000. http://homepages.warwick.ac.uk/masgaj/ftp/data/ (2015) · Zbl 0862.11037 |
| [7] | Daniels, H.: Siegel functions, modular curves, and Serre’s uniformity problem. Alban. J. Math. 9(1), 3-29 (2015) · Zbl 1326.11027 |
| [8] | Dokchitser, T., Dokchitser, V.: Surjectivity of mod \[2^n2\] n representations of elliptic curves. Math. Z. 272(3-4), 961-964 (2012) · Zbl 1315.11046 · doi:10.1007/s00209-011-0967-7 |
| [9] | Elkies, N.: Elliptic and modular curves over finite fields and related computational issues. In: Buell, D.A., Teitelbaum, J.T. (eds.) Computational Perspectives on Number Theory: Proceedings of a Conference in Honor of A.O.L. Atkin, pp. 21-76. AMS/International Press, Providence, RI (1998) · Zbl 0915.11036 |
| [10] | Elkies, N. : Explicit modular towers. In: Basar, T., Vardy, A. (eds.) Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing, 1997, pp. 23-32. University of Illinois at Urbana-Champaign (math.NT/0103107 on the arXiv) (1998) · Zbl 0898.68039 |
| [11] | Fricke, R., Klein, F.: Vorlesungen ber die Theorie der elliptischen Modulfunctionen (Volumes 1 and 2), p. 1892. B. G. Teubner, Leipzig (1890) · Zbl 0386.14009 |
| [12] | Fricke, R.: Die elliptischen Funktionen und ihre Anwendungen. Leipzig-Berlin, Teubner (1922) · JFM 48.0432.01 |
| [13] | Fujita, Y.: Torsion subgroups of elliptic curves in elementary abelian 2-extensions of \[\mathbb{Q}\] Q. J. Number Theory 114, 124-134 (2005) · Zbl 1087.11038 · doi:10.1016/j.jnt.2005.03.005 |
| [14] | Ishii, N.: Rational expression for J-invariant function in terms of generators of modular function fields. In: International Mathematical Forum, vol. 2 , no. 38, pp. 1877-1894 (2007) · Zbl 1161.11013 |
| [15] | Kamienny, S.: Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109, 129-133 (1992) · Zbl 0807.14022 · doi:10.1007/BF01232025 |
| [16] | Kenku, M.A.: On the number of \[\mathbb{Q}\] Q-isomorphism classes of elliptic curves in each \[\mathbb{Q}\] Q-isogeny class. J. Number Theory 15, 199-202 (1982) · Zbl 0493.14017 · doi:10.1016/0022-314X(82)90025-7 |
| [17] | Kenku, M.A., Momose, F.: Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109, 125-149 (1988) · Zbl 0647.14020 |
| [18] | Klein, F.: Lectures on the icosahedron and the solution of equations of the fifth degree. Trubner and Co., London (1888) |
| [19] | Knapp, A.W.: Elliptic Curves Mathematical Notes, vol. 40. Princeton University Press, Princeton (1992) · Zbl 0804.14013 |
| [20] | Kraus, A.: Une remarque sur les points de torsion des courbes elliptiques. C. R. Acad. Sci. Paris Ser. I Math. 321, 1143-1146 (1995) · Zbl 0862.11037 |
| [21] | Laska, M., Lorenz, M.: Rational points on elliptic curves over \[\mathbb{Q}\] Q in elementary abelian 2-extensions of \[\mathbb{Q}\] Q. J. Reine Angew. Math. 355, 163-172 (1985) · Zbl 0586.14013 |
| [22] | Lozano-Robledo, Á.: On the field of definition of \[p\] p-torsion points on elliptic curves over the rationals. Math. Ann. 357(1), 279-305 (2013) · Zbl 1277.14028 · doi:10.1007/s00208-013-0906-5 |
| [23] | Lozano-Robledo, Á.: Division fields of elliptic curves with minimal ramification. Preprint (to appear in Revista Matemática Iberoamericana) · Zbl 1331.11042 |
| [24] | Maier, R.: On rationally parametrized modular equations. J. Ramanujan Math. Soc. 24, 1-73 (2009) · Zbl 1214.11049 |
| [25] | Masser, D.W., Wüstholz, G.: Galois properties of division fields of elliptic curves. Bull. Lond. Math. Soc. 25, 247-254 (1993) · Zbl 0809.14026 · doi:10.1112/blms/25.3.247 |
| [26] | Mazur, B.: Rational isogenies of prime degree. Invent. Math. 44, 129-162 (1978) · Zbl 0386.14009 · doi:10.1007/BF01390348 |
| [27] | Merel, L.: Sur la nature non-cyclotomique des points d’ordre fini des courbes elliptiques (Appendice de E. Kowalski et P. Michel). Duke Math. J. 110(1), 81-119 (2001) · Zbl 1020.11041 · doi:10.1215/S0012-7094-01-11013-2 |
| [28] | Merel, L., Stein, W.: The field generated by the points of small prime order on an elliptic curve. IMRN 20, 1075-1082 (2001) · Zbl 1027.11041 · doi:10.1155/S1073792801000514 |
| [29] | Najman, F.: Torsion of rational elliptic curves over cubic fields and sporadic points on \[X_1(n)\] X1(n). Math. Res. Lett. (to appear) · Zbl 1416.11084 |
| [30] | Paladino, L.: Elliptic curves with \[\mathbb{Q}(E[3])=\mathbb{Q}(\zeta_3)\] Q(E[3])=Q(ζ3) and counterexamples to local-global divisibility by \[99\]. Journal de Théorie des Nombres de Bordeaux 22, 139-160 (2010) · Zbl 1216.11064 · doi:10.5802/jtnb.708 |
| [31] | Pellarin, F.: Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques. Acta Arith. 100, 203-243 (2001) · Zbl 0986.11046 · doi:10.4064/aa100-3-1 |
| [32] | Rebolledo, M.: Corps engendré par les points de 13-torsion des courbes elliptiques. Acta Arith. 109(3), 219-230 (2003) · Zbl 1049.11058 · doi:10.4064/aa109-3-2 |
| [33] | Rouse, J., Zureick-Brown, D.: Elliptic curves over \[\mathbb{Q}\] Q and \[22\]-adic images of Galois. Res. Number Theory 1, 12 (2015). (Data files and subgroup descriptions available at: http://users.wfu.edu/rouseja/2adic/) · Zbl 1397.11095 |
| [34] | Rubin, K., Silverberg, A.: Families of elliptic curves with constant mod \[p\] p representations. In: Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number Theory, I. Int. Press, Cambridge, MA, pp. 148-161 (1995) · Zbl 0856.11027 |
| [35] | Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086 |
| [36] | Shanks, D.: The Simplest Cubic Fields, Mathematics of Computation, vol. 28, no. 128, pp. 1137-1152 (Oct. 1974) · Zbl 0307.12005 |
| [37] | Silverman, J.H.: The Arithmetic of Elliptic Curves, 2nd edn. Springer, New York (2009) · Zbl 1194.11005 · doi:10.1007/978-0-387-09494-6 |
| [38] | Vélu, J.: Isogénies entre courbes elliptiques. C. R. Acad. Sci. Paris, Sér. A. 273, 238-241 (1971) · Zbl 0225.14014 |
| [39] | Zywina, D.: On the possible images of the mod \[\ell\] ℓ representations associated to elliptic curves over \[\mathbb{Q}\] Q. (arXiv:1508.07660) · Zbl 0986.11046 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
