An Algorithm for Determining Torsion Growth of Elliptic Curves

References

• Bourdon, A., Clark, P. (2020). Torsion points and Galois representations on CM elliptic curves. Pacific J. Math. 305(1): 43–88. doi:10.2140/pjm.2020.305.43
   Web of Science ®Google Scholar
• Bourdon, A., Clark, P., Stankewicz, J. (2017). Torsion points on CM elliptic curves over real number fields. Trans. Amer. Math. Soc. 369(12): 8457–8496. doi:10.1090/tran/6905
   Web of Science ®Google Scholar
• Bosma, W., Cannon, J. J., Fieker, C., and Steel, A. (eds.). (2019). Handbook of Magma Functions, Edition 2.24.
   Google Scholar
• Chou, M. (2016). Torsion of rational elliptic curves over quartic Galois number fields. J. Number Theory 160: 603–628. doi:10.1016/j.jnt.2015.09.013
   Web of Science ®Google Scholar
• Clark, P. L., Corn, P., Rice, A, Stankewicz, J. (2014). Computation on elliptic curves with complex multiplication. LMS J. Comput. Math. 17(1): 509–539. doi:10.1112/S1461157014000072
   Google Scholar
• Cremona, J. E. ecdata: 2016-10-17 (Elliptic curve data for conductors up to 400.000), available on http://johncremona.github.io/ecdata/.
   Google Scholar
• Daniels, H. B, González–Jiménez, E. (2020). On the torsion of rational elliptic curves over sextic fields. Math. Comp. 89(321): 411–439. doi:10.1090/mcom/3440
   Web of Science ®Google Scholar
• Daniels, H. B., Lozano-Robledo, A., Najman, F, Sutherland, A. V. (2018). Torsion subgroups of rational elliptic curves over the compositum of all cubic fields. Math. Comp. 87(309): 425–458. doi:10.1090/mcom/3213
   Web of Science ®Google Scholar
• Derickx, M., Sutherland, A. V. (2017). Torsion subgroups of elliptic curves over quintic and sextic number fields. Proc. Amer. Math. Soc. 145(10): 4233–4245. doi:10.1090/proc/13605
   Web of Science ®Google Scholar
• Derickx, M., van Hoeij, M. (2014). Gonality of the modular curve X1(N). J. Algebra 417: 52–71.
   Web of Science ®Google Scholar
• Elkies, N. D. Elliptic curves with 3-adic Galois representation surjective mod 3 but not mod 9, arXiv/math: 0612734.
   Google Scholar
• González–Jiménez, E. (2017). Complete classification of the torsion structures of rational elliptic curves over quintic number fields. J. Algebra 478: 484–505. doi:10.1016/j.jalgebra.2017.01.012
   Web of Science ®Google Scholar
• González-Jiménez, E., Lozano-Robledo, Á. (2017). On the minimal degree of definition of p-primary torsion subgroups of elliptic curves. Math. Res. Lett. 24(4): 1067–1096. Data file 2primary_Ss.txt available at http://matematicas.uam.es/∼enrique.gonzalez.jimenez/). doi:10.4310/MRL.2017.v24.n4.a7
   Web of Science ®Google Scholar
• González–Jiménez, E., Lozano–Robledo, Á. (2018). On torsion of rational elliptic curves over quartic fields. Math. Comp. 87(311): 1457–1478. doi:10.1090/mcom/3235
   Web of Science ®Google Scholar
• González–Jiménez, E., Najman, F., Tornero, J. M. (2016). Torsion of rational elliptic curves over cubic fields. Rocky Mountain J. Math. 46(6): 1899–1917. doi:10.1216/RMJ-2016-46-6-1899
   Web of Science ®Google Scholar
• González–Jiménez, E., Najman, F. (2020). Growth of torsion of elliptic curves upon base change. Math. Comp. 89(323): 1457–1485. doi:10.1090/mcom/3478
   Web of Science ®Google Scholar
• González–Jiménez, E., Najman, F. Magma scripts and electronic transcript of computations for the paper. An algorithm for determining torsion growth of elliptic curves. http://matematicas.uam.es/∼enrique.gonzalez.jimenez/
   Google Scholar
• González–Jiménez, E., Tornero, J. M. (2014). Torsion of rational elliptic curves over quadratic fields. RACSAM 108(2): 923–934. doi:10.1007/s13398-013-0152-4
   Google Scholar
• González–Jiménez, E., Tornero, J. M. (2016). Torsion of rational elliptic curves over quadratic fields II. RACSAM 110(1): 121–143. doi:10.1007/s13398-015-0223-9
   Google Scholar
• Greenberg, R. (2012). The image of Galois representations attached to elliptic curves with an isogeny. Amer. J. Math. 134(5): 1167–1196. doi:10.1353/ajm.2012.0040
   Web of Science ®Google Scholar
• Greenberg, R., Rubin, K., Silverberg, A., Stoll, M. (2014). On elliptic curves with an isogeny of degree 7. Amer. J. Math. 136(1): 77–109. doi:10.1353/ajm.2014.0005
   Web of Science ®Google Scholar
• van Hoeij, M. Low Degree Places on the Modular Curve X1(N), preprint. http://arxiv.org/abs/1202.4355.
   Google Scholar
• Jeon, D., Kim, C. H., Schweizer, A. (2004). On the torsion of elliptic curves over cubic number fields. Acta Arith. 113(3): 291–301. doi:10.4064/aa113-3-6
   Web of Science ®Google Scholar
• Jeon, D., Kim, C. H., Park, E. (2006). On the torsion of elliptic curves over quartic number fields. J. London Math. Soc. 74(1): 1–12. doi:10.1112/S0024610706022940
   Google Scholar
• Kamienny, S. (1992). Torsion points on elliptic curves and q-coefficients of modular forms. Invent. Math. 109(1): 221–229. doi:10.1007/BF01232025
   Google Scholar
• Kenku, M. A., Momose, F. (1988). Torsion points on elliptic curves defined over quadratic fields. Nagoya Math. J. 109: 125–149. doi:10.1017/S0027763000002816
   Web of Science ®Google Scholar
• Landau, S. (1985). Factoring polynomials over algebraic number fields. SIAM J. Comput. 14(1): 184–195. doi:10.1137/0214015
   Web of Science ®Google Scholar
• Laska, M., Lorenz, M. (1985). Rational points on elliptic curves over Q in elementary abelian 2-extensions of Q. J. Reine Angew. Math. 355: 163–172.
   Web of Science ®Google Scholar
• The LMFDB Collaboration. (2019). The L-functions and Modular Forms Database. http://www.lmfdb.org
   Google Scholar
• Lozano-Robledo, A. (2018). Uniform bounds in terms of ramification. Res. number Theory. 4(1): 6. doi:10.1007/s40993-018-0095-0
   Google Scholar
• Mazur, B. (1977). Modular curves and the Eisenstein ideal. Publications Mathématiques de L’institut. Des. Hautes. Scientifiques. 47(1): 33–186. doi:10.1007/BF02684339
   Google Scholar
• Merel, L. (1996). Bornes pour la torsion des courbes elliptiques sur les corps de nombres. Invent. Math. 124(1–3): 437–449. doi:10.1007/s002220050059
   Google Scholar
• Najman, F. (2016). Torsion of rational elliptic curves over cubic fields and sporadic points on X1(n). Math. Res. Letters 23(1): 245–272. doi:10.4310/MRL.2016.v23.n1.a12
   Web of Science ®Google Scholar
• Najman, F. (2015). The number of twists with large torsion of an elliptic curve. Racsam. 109(2): 535–547. doi:10.1007/s13398-014-0199-x
   Google Scholar
• Rouse, J, Zureick-Brown, D. (2015). Elliptic curves over Q and 2-adic images of Galois. Res. Number Theory 1(1): 12. (Data files and subgroup descriptions available at http://users.wfu.edu/rouseja/2adic/). doi:10.1007/s40993-015-0013-7
   Google Scholar
• Serre, J.-P. (1998). Abelian ℓ-Adic Representations and Elliptic Curves, Research Notes in Mathematics, vol. 7. Wellesy, MA: A. K. Peters Ltd.
   Google Scholar
• Sutherland, A. V. (2016). Computing images of Galois representations attached to elliptic curves. Forum Math. Sigma 4: 79.e4.
   Web of Science ®Google Scholar
• Washington, L. (2008). Elliptic curves: Number theory and cryptography. Boca Raton, FL: Chapman & Hall/CRC.
   Google Scholar
• Zywina, D. On the possible images of the mod ℓ representations associated to elliptic curves over Q, arXiv:1508.07660.
   Google Scholar