## Torsion of rational elliptic curves over cubic fields.(English)Zbl 1358.11068

The aim of this article is to determine the torsion groups over cubic fields of an elliptic curve $$E$$ defined over $$\mathbb Q$$ with the torsion group over $$\mathbb Q$$ isomorphic to a given group $$G$$. The set $$\Phi_{\mathbb Q}(d)$$ of possible structures of torsion groups over number fields of degree $$1,2$$ and $$3$$ of elliptic curves rational over $$\mathbb Q$$ are determined by B. Mazur [Publ. Math., Inst. Hautes Étud. Sci. 47, 33–186 (1977; Zbl 0394.14008); Invent. Math. 44, 129–162 (1978; Zbl 0386.14009)] in the case $$d=1$$ and the second author [Math. Res. Lett. 23, No. 1, 245–272 (2016; Zbl 1416.11084)] in other cases. The authors determine the set $$\Phi_{\mathbb Q}(3,G)$$ of possible structures of torsion groups over a cubic number field of an elliptic curve $$E$$ defined over $$\mathbb Q$$ such that $$E(\mathbb Q)_{\text{tors}}\simeq G\in \Phi_{\mathbb Q}(1)$$. Further, they determine explicitly all elliptic curves $$E$$ and all cubic fields $$K$$ up to isomorphisms such that $$E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}$$. In particular, for an elliptic curve $$E$$, they show that there is at most one cubic field $$K$$, up to isomorphisms, such that $$E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}\simeq H$$ for a fixed $$H\in\Phi_{\mathbb Q}(3)$$ and that there are at most three non-isomorphic cubic fields $$K$$ such that $$E(\mathbb Q)_{\text{tors}}\neq E(K)_{\text{tors}}$$.

### MSC:

 11G05 Elliptic curves over global fields 11R16 Cubic and quartic extensions 14G05 Rational points 14H52 Elliptic curves

### Keywords:

elliptic curves; torsion subgroup; rationals; cubic fields

### Citations:

Zbl 0394.14008; Zbl 0386.14009; Zbl 1416.11084

Magma
Full Text:

### References:

 [1] B.J. Birch and W. Kuyk, eds., Modular functions of one variable , IV, Lect. Notes Math. 476 , Springer, New York, 1975. [2] W. Bosma, J. Cannon and C. Fieker, et al., eds., Handbook of Magma functions , edition 2.20, http://magma.maths.usyd.edu.au/magma, 2013. URL: [3] J.E. Cremona, Elliptic curve data for conductors up to 320.000, available at http://www.warwick.ac.uk/ masgaj/ftp/data/, 2014. URL: [4] T. Dokchitser and V. Dokchitser, Surjectivity of $$\mbox{\mathrm mod\,}2^n$$ representations of elliptic curves , Math. Z. 272 (2012), 961-964. · Zbl 1315.11046 [5] E. González-Jiménez and J.M. Tornero, Torsion of rational elliptic curves over quadratic fields Rev. Roy. Acad. Cienc. Exactas 110 (2016), 121-143. · Zbl 1366.11080 [6] —-, Torsion of rational elliptic curves over quadratic fields , II, submitted. [7] P. Ingram, Diophantine analysis and torsion on elliptic curves , Proc. Lond. Math. Soc. 94 (2007) 137-154. · Zbl 1117.11033 [8] D. Jeon, C.H. Kim and A. Schweizer, On the torsion of elliptic curves over cubic number fields , Acta Arith. 113 (2004), 291-301. · Zbl 1083.11038 [9] S. Kamienny, Torsion points on elliptic curves and $$q$$-coefficients of modular forms , Invent. Math. 109 (1992), 129-133. · Zbl 0773.14016 [10] M.A. Kenku, The modular curves $$X_0(65)$$ and $$X_0(91)$$ and rational isogeny , Math. Proc. Cambr. Philos. Soc. 87 (1980), 15-20. · Zbl 0479.14014 [11] —-, The modular curve $$X_0(169)$$ and rational isogeny , J. Lond. Math. Soc. 22 (1980), 239-244. · Zbl 0437.14022 [12] —-, The modular curves $$X_0(125)$$, $$X_1(25)$$ and $$X_1(49)$$ , J. Lond. Math. Soc. 23 (1981), 415-427. · Zbl 0425.14006 [13] M.A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields , Nagoya Math. J. 109 (1988), 125-149. · Zbl 0647.14020 [14] A.W. Knapp, Elliptic curves , in Mathematical notes 40 , Princeton University Press, Princeton, 1992. · Zbl 0804.14013 [15] B. Mazur, Modular curves and the Eisenstein ideal , Publ. Math. Inst. Hautes Etud. Sci. 47 (1977), 33-186. · Zbl 0394.14008 [16] —-, Rational isogenies of prime degree , Invent. Math. 44 (1978), 129-162. · Zbl 0386.14009 [17] F. Najman, Torsion of elliptic curves over cubic fields , J. Num. Th. 132 (2012), 26-36. · Zbl 1268.11080 [18] —-, Torsion of rational elliptic curves over cubic fields and sporadic points on $$X_1(n)$$ , Math. Res. Lett. 23 (2016), 245–272. · Zbl 1416.11084 [19] —-, The number of twists with large torsion of an elliptic curve , Rev. Roy. Acad. Cienc. Exactas 109 (2015), 535–547. · Zbl 1341.11031 [20] L. Paladino, Elliptic curves with $$\Q({\mathcal E}[3]) = \Q (\zeta_3)$$ and counterexamples to local-global divisibility by $$9$$ , J. Th. Nombr. Bordeaux 22 (2010), 139-160. · Zbl 1216.11064 [21] J.H. Silverman, The arithmetic of elliptic curves , Springer, New York, 1986. · Zbl 0585.14026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.