Torsion subgroups of rational elliptic curves over the compositum of all cubic fields

Daniels, Harris; Lozano-Robledo, Álvaro; Najman, Filip; Sutherland, Andrew

doi:10.1090/mcom/3213

Torsion subgroups of rational elliptic curves over the compositum of all cubic fields
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by Harris B. Daniels, Álvaro Lozano-Robledo, Filip Najman and Andrew V. Sutherland;

Math. Comp. 87 (2018), 425-458

DOI: https://doi.org/10.1090/mcom/3213

Published electronically: May 5, 2017

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Abstract:

Let $E/\mathbb {Q}$ be an elliptic curve and let $\mathbb {Q}(3^\infty )$ be the compositum of all cubic extensions of $\mathbb {Q}$. In this article we show that the torsion subgroup of $E(\mathbb {Q}(3^\infty ))$ is finite and we determine 20 possibilities for its structure, along with a complete description of the $\overline {\mathbb {Q}}$-isomorphism classes of elliptic curves that fall into each case. We provide rational parameterizations for each of the 16 torsion structures that occur for infinitely many $\overline {\mathbb {Q}}$-isomorphism classes of elliptic curves, and a complete list of $j$-invariants for each of the 4 that do not.

References

Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Michael Chou, Torsion of rational elliptic curves over quartic Galois number fields, J. Number Theory 160 (2016), 603–628. MR 3425225, DOI 10.1016/j.jnt.2015.09.013
J. E. Cremona, Elliptic curve data, database available at http://homepages.warwick.ac.uk/ ~masgaj/ftp/data/.
C. J. Cummins and S. Pauli, Congruence subgroups of $\textrm {PSL}(2,{\Bbb Z})$ of genus less than or equal to 24, Experiment. Math. 12 (2003), no. 2, 243–255. MR 2016709
H. Daniels, Á. Lozano-Robledo, J. Morrow, F. Najman, and A.V. Sutherland, Magma scripts related to Torsion subgroups of rational elliptic curves over the compositum of all cubic fields, available at http://math.mit.edu/ drew.
G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366 (German). MR 718935, DOI 10.1007/BF01388432
Gerhard Frey and Moshe Jarden, Approximation theory and the rank of abelian varieties over large algebraic fields, Proc. London Math. Soc. (3) 28 (1974), 112–128. MR 337997, DOI 10.1112/plms/s3-28.1.112
Yasutsugu Fujita, Torsion subgroups of elliptic curves with non-cyclic torsion over $\Bbb Q$ in elementary abelian 2-extensions of $\Bbb Q$, Acta Arith. 115 (2004), no. 1, 29–45. MR 2102804, DOI 10.4064/aa115-1-3
Yasutsugu Fujita, Torsion subgroups of elliptic curves in elementary abelian 2-extensions of $\Bbb Q$, J. Number Theory 114 (2005), no. 1, 124–134. MR 2163908, DOI 10.1016/j.jnt.2005.03.005
Itamar Gal and Robert Grizzard, On the compositum of all degree $d$ extensions of a number field, J. Théor. Nombres Bordeaux 26 (2014), no. 3, 655–673 (English, with English and French summaries). MR 3320497
Bertram Huppert, Normalteiler und maximale Untergruppen endlicher Gruppen, Math. Z. 60 (1954), 409–434 (German). MR 64771, DOI 10.1007/BF01187387
Daeyeol Jeon, Chang Heon Kim, and Yoonjin Lee, Families of elliptic curves over cubic number fields with prescribed torsion subgroups, Math. Comp. 80 (2011), no. 273, 579–591. MR 2728995, DOI 10.1090/S0025-5718-10-02369-0
Daeyeol Jeon, Chang Heon Kim, and Yoonjin Lee, Families of elliptic curves over quartic number fields with prescribed torsion subgroups, Math. Comp. 80 (2011), no. 276, 2395–2410. MR 2813367, DOI 10.1090/S0025-5718-2011-02493-2
Daeyeol Jeon, Chang Heon Kim, and Euisung Park, On the torsion of elliptic curves over quartic number fields, J. London Math. Soc. (2) 74 (2006), no. 1, 1–12. MR 2254548, DOI 10.1112/S0024610706022940
Daeyeol Jeon, Chang Heon Kim, and Andreas Schweizer, On the torsion of elliptic curves over cubic number fields, Acta Arith. 113 (2004), no. 3, 291–301. MR 2069117, DOI 10.4064/aa113-3-6
S. Kamienny, Torsion points on elliptic curves and $q$-coefficients of modular forms, Invent. Math. 109 (1992), no. 2, 221–229. MR 1172689, DOI 10.1007/BF01232025
M. A. Kenku, On the number of $\textbf {Q}$-isomorphism classes of elliptic curves in each $\textbf {Q}$-isogeny class, J. Number Theory 15 (1982), no. 2, 199–202. MR 675184, DOI 10.1016/0022-314X(82)90025-7
M. A. Kenku, The modular curve $X_{0}(39)$ and rational isogeny, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 21–23. MR 510395, DOI 10.1017/S0305004100055444
M. A. Kenku, The modular curves $X_{0}(65)$ and $X_{0}(91)$ and rational isogeny, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 1, 15–20. MR 549292, DOI 10.1017/S0305004100056462
M. A. Kenku, The modular curve $X_{0}(169)$ and rational isogeny, J. London Math. Soc. (2) 22 (1980), no. 2, 239–244. MR 588271, DOI 10.1112/jlms/s2-22.2.239
M. A. Kenku, On the modular curves $X_{0}(125)$, $X_{1}(25)$ and $X_{1}(49)$, J. London Math. Soc. (2) 23 (1981), no. 3, 415–427. MR 616546, DOI 10.1112/jlms/s2-23.3.415
M. A. Kenku and F. Momose, Torsion points on elliptic curves defined over quadratic fields, Nagoya Math. J. 109 (1988), 125–149. MR 931956, DOI 10.1017/S0027763000002816
Michael Laska and Martin Lorenz, Rational points on elliptic curves over $\textbf {Q}$ in elementary abelian $2$-extensions of $\textbf {Q}$, J. Reine Angew. Math. 355 (1985), 163–172. MR 772489, DOI 10.1515/crll.1985.355.163
LMFDB Collaboration, The L-functions and modular forms database, available at http://www.lmfdb.org.
Álvaro Lozano-Robledo, On the field of definition of $p$-torsion points on elliptic curves over the rationals, Math. Ann. 357 (2013), no. 1, 279–305. MR 3084348, DOI 10.1007/s00208-013-0906-5
Álvaro Lozano-Robledo, Division fields of elliptic curves with minimal ramification, Rev. Mat. Iberoam. 31 (2015), no. 4, 1311–1332. MR 3438391, DOI 10.4171/RMI/870
Á. Lozano-Robledo, Uniform boundedness in terms of ramification, preprint, available at http://alozano.clas.uconn.edu/research-articles/.
B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287
B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129–162. MR 482230, DOI 10.1007/BF01390348
James McKee, Computing division polynomials, Math. Comp. 63 (1994), no. 208, 767–771. MR 1248973, DOI 10.1090/S0025-5718-1994-1248973-7
Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437–449 (French). MR 1369424, DOI 10.1007/s002220050059
L. J. Mordell, On the rational solutions of the indeterminate equations of the third and fourth degrees, Proc. Cambridge Philos. Soc. 21 (1922) 179–192.
Filip Najman, Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$, Math. Res. Lett. 23 (2016), no. 1, 245–272. MR 3512885, DOI 10.4310/MRL.2016.v23.n1.a12
Pierre Parent, Bornes effectives pour la torsion des courbes elliptiques sur les corps de nombres, J. Reine Angew. Math. 506 (1999), 85–116 (French, with French summary). MR 1665681, DOI 10.1515/crll.1999.009
H. Poincaré Sur les propriétés arithmétiques des courbes algébriques, J. Math. Pures Appl. Ser 5. 7 (1901), 161–233.
Nicholas M. Katz and Serge Lang, Finiteness theorems in geometric classfield theory, Enseign. Math. (2) 27 (1981), no. 3-4, 285–319 (1982). With an appendix by Kenneth A. Ribet. MR 659153
Derek J. S. Robinson, A course in the theory of groups, 2nd ed., Graduate Texts in Mathematics, vol. 80, Springer-Verlag, New York, 1996. MR 1357169, DOI 10.1007/978-1-4419-8594-1
Jeremy Rouse and David Zureick-Brown, Elliptic curves over $\Bbb Q$ and 2-adic images of Galois, Res. Number Theory 1 (2015), Paper No. 12, 34. MR 3500996, DOI 10.1007/s40993-015-0013-7
Jean-Pierre Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331 (French). MR 387283, DOI 10.1007/BF01405086
Joseph H. Silverman, The arithmetic of elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009. MR 2514094, DOI 10.1007/978-0-387-09494-6
Andrew V. Sutherland, A local-global principle for rational isogenies of prime degree, J. Théor. Nombres Bordeaux 24 (2012), no. 2, 475–485 (English, with English and French summaries). MR 2950703
Andrew V. Sutherland, Computing images of Galois representations attached to elliptic curves, Forum Math. Sigma 4 (2016), Paper No. e4, 79. MR 3482279, DOI 10.1017/fms.2015.33
A. V. Sutherland and D. Zywina, Modular curves of prime power level with infinitely many rational points, to appear in Algebra and Number Theory, available at https://arxiv.org/abs/1605.03988.
André Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), no. 1, 281–315 (French). MR 1555278, DOI 10.1007/BF02547409
Guido Zappa, Remark on a recent paper of O. Ore, Duke Math. J. 6 (1940), 511–512. MR 2118
D. Zywina, On the possible images of mod $\ell$ representations associated to elliptic curves over $\mathbb {Q}$, preprint, available at http://arxiv.org/abs/1508.07660.
D. Zywina, Possible indices for the Galois image of elliptic curves over $\mathbb {Q}$, preprint, available at http://arxiv.org/abs/1508.07663.