Riv. Mat. Univ. Parma, Vol. 9, No. 2, 2018

On 5-torsion of CM elliptic curves

Pages: 329-350
Accepted in revised form: 17 December 2018
Mathematics Subject Classification (2010): 11G05, 11F80, 11G18.
Keywords: Elliptic curves, complex multiplication, torsion points.
[a]: University of Calabria, Ponte Bucci, Cubo 30B, Rende, 87036, Italy

Abstract: Let $$\mathcal{E}$$ be an elliptic curve defined over a number field $$K$$. Let $$m$$ be a positive integer. We denote by $$\mathcal{E}[m]$$ the $$m$$-torsion subgroup of $$\mathcal{E}$$ and by $$K_m:=K(\mathcal{E}[m])$$ the field obtained by adding to $$K$$ the coordinates of the points of $$\mathcal{E}[m]$$. We describe the fields $$K_5$$, when $$\mathcal{E}$$ is a CM elliptic curve defined over $$K$$, with Weiestrass form either $$y^2=x^3+bx$$ or $$y^2=x^3+c$$. In particular we classify the fields $$K_5$$ in terms of generators, degrees and Galois groups. Furthermore we show some applications of those results to the Local-Global Divisibility Problem and to modular curves.

References
[1]
C. Adelmann, The decomposition of primes in torsion point fields, Lecture Notes in Math., 1761, Springer-Verlag, Berlin, 2001. MR1836119
[2]
A. Bandini, Three-descent and the Birch and Swinnerton-Dyer conjecture, Rocky Mountain J. Math. 34 (2004), 13–27. MR2061115
[3]
A. Bandini, $$3$$-Selmer groups for curves $$y^2=x^3+a$$, Czechoslovak Math. J. 58 (2008), 429–445. MR2411099
[4]
A. Bandini and L. Paladino, Number fields generated by the $$3$$-torsion poins of an elliptic curve, Monatsh. Math. 168 (2012), 157–181. MR2984145
[5]
A. Bandini and L. Paladino, Fields generated by torsion poins of elliptic curves, J. Number Theory 169 (2016), 103–133. MR3531232
[6]
P. L. Clark, Rational points on Atkin-Lehner quotients of Shimura curves, Ph.D. Thesis, Harvard University, Cambridge, MA, 2003. MR2704676
[7]
P. Clark and X. Xarles, Local bounds for torsion points on abelian varieties, Canad. J. Math. 60 (2008), 532–555. MR2414956
[8]
R. Dvornicich and A. Paladino, Local-global questions for divisibility in commutative algebraic groups, arXiv:1706.03726v4, preprint, 2018.
[9]
R. Dvornicich and U. Zannier, Local-global divisibility of rational points in some commutative algebraic groups, Bull. Soc. Math. France 129 (2001), 317–338. MR1881198
[10]
R. Dvornicich and U. Zannier, On local-global principle for the divisibility of a rational point by a positive integer, Bull. Lond. Math. Soc. 39 (2007), 27–34. MR2303515
[11]
E. González-Jiménez and Á. Lozano-Robledo, Elliptic curves with abelian division fields, Math. Z. 283 (2016), 835–859. MR3519984
[12]
N. M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Ann. of Math. Stud., 108, Princeton Univ. Press, Princeton, NJ, 1985. MR0772569
[13]
A. W. Knapp, Elliptic curves, Math. Notes, 40, Princeton Univ. Press, Princeton, NJ, 1992. MR1193029
[14]
L. Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, (French), Invent. Math. 124 (1996), 437–449. MR1369424
[15]
L. Paladino, Local-global divisibility by $$4$$ in elliptic curves defined over $$\mathbb Q$$, Ann. Mat. Pura Appl. (4) 189 (2010), 17–23. MR2556757
[16]
L. Paladino, Elliptic curves with $$\mathbb Q({\mathcal{E}}[3])=\mathbb Q(\zeta_3)$$ and counterexamples to local-global divisibility by $$9$$, J. Théor. Nombres Bordeaux 22 (2010), 139–160. MR2675877
[17]
L. Paladino, G. Ranieri and E. Viada, On minimal set for counterexamples to the local-global principle, J. Algebra 415 (2014), 290–304. MR3229518
[18]
V. Rotger and C. de Vera-Piquero, Galois representations over fields of moduli and rational points on Shimura curves, Canad. J. Math. 66 (2014), 1167–1200. MR3251768
[19]
J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, (French), J. Reine Angew. Math. 327 (1981), 12–80. MR0631309
[20]
E. F. Schaefer and M. Stoll, How to do a $$p$$-descent on an elliptic curve, Trans. Amer. Math. Soc. 356 (2004), 1209–1231. MR2021618
[21]
G. Shimura, On the real points of an arithmetic quotient of a bounded symmetric domain, Math. Ann. 215 (1975), 135–164. MR0572971
[22]
G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, NJ, 1994. MR1291394
[23]
J. H. Silverman, The arithmetic of elliptic curves, 2nd ed., Grad. Texts in Math., 106, Springer, Dordrecht, 2009. MR2514094
[24]
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts in Math., 151, Springer-Verlag, New York, 1994. MR1312368
[25]
S. Wong, Power residues on Abelian varieties, Manuscripta Math. 102 (2000), 129–137. MR1771232

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