Torsion subgroups of elliptic curves over quintic and sextic number fields

Derickx, Maarten; Sutherland, Andrew

doi:10.1090/proc/13605

Torsion subgroups of elliptic curves over quintic and sextic number fields
HTML articles powered by AMS MathViewer

by Maarten Derickx and Andrew V. Sutherland PDF

Proc. Amer. Math. Soc. 145 (2017), 4233-4245 Request permission

Abstract:

Let $\Phi ^\infty (d)$ denote the set of finite abelian groups that occur infinitely often as the torsion subgroup of an elliptic curve over a number field of degree $d$. The sets $\Phi ^\infty (d)$ are known for $d\le 4$. In this article we determine $\Phi ^\infty (5)$ and $\Phi ^\infty (6)$.

References

Dan Abramovich, A linear lower bound on the gonality of modular curves, Internat. Math. Res. Notices 20 (1996), 1005–1011. MR 1422373, DOI 10.1155/S1073792896000621
A. O. L. Atkin and Wen Ch’ing Winnie Li, Twists of newforms and pseudo-eigenvalues of $W$-operators, Invent. Math. 48 (1978), no. 3, 221–243. MR 508986, DOI 10.1007/BF01390245
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
Brian Conrad, Bas Edixhoven, and William Stein, $J_1(p)$ has connected fibers, Doc. Math. 8 (2003), 331–408. MR 2029169
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973, pp. 143–316 (French). MR 0337993
Maarten Derickx and Mark van Hoeij, Gonality of the modular curve $X_1(N)$, J. Algebra 417 (2014), 52–71. MR 3244637, DOI 10.1016/j.jalgebra.2014.06.026
Maarten Derickx and Mark van Hoeij, data related to [6], available at www.math.fsu.edu/~hoeij/files/X1N, 2013.
Maarten Derickx, Sheldon Kamienny, and Barry Mazur, Rational families of $17$-torsion points of elliptic curves over number fields, Contemporary Mathematics, AMS, to appear.
Maarten Derickx and Andrew V. Sutherland, GitHub repository related to Torsion subgroups of elliptic curves over quintic and sextic number fields, available at https://github.com/koffie/mdmagma/, 2016.
Maarten Derickx and Andrew V. Sutherland, Explicit models of $X_1(m,mn)$, available at http://math.mit.edu/~drew/X1mn.html, 2016.
Gerhard Frey, Curves with infinitely many points of fixed degree, Israel J. Math. 85 (1994), no. 1-3, 79–83. MR 1264340, DOI 10.1007/BF02758637
Gerd Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175–182. MR 1307396
Sonal Jain, Points of low height on elliptic surfaces with torsion, LMS J. Comput. Math. 13 (2010), 370–387. MR 2685131, DOI 10.1112/S1461157009000151
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 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
Kazuya Kato, $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), ix, 117–290 (English, with English and French summaries). Cohomologies $p$-adiques et applications arithmétiques. III. MR 2104361
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
Henry H. Kim, Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203, DOI 10.1090/S0894-0347-02-00410-1
V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253. MR 1036843
Daniel Sion Kubert, Universal bounds on the torsion of elliptic curves, Proc. London Math. Soc. (3) 33 (1976), no. 2, 193–237. MR 434947, DOI 10.1112/plms/s3-33.2.193
Abhinav Kumar and Tetsuji Shioda, Multiplicative excellent families of elliptic surfaces of type $E_7$ or $E_8$, Algebra Number Theory 7 (2013), no. 7, 1613–1641. MR 3117502, DOI 10.2140/ant.2013.7.1613
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
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
Markus A. Reichert, Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields, Math. Comp. 46 (1986), no. 174, 637–658. MR 829635, DOI 10.1090/S0025-5718-1986-0829635-X
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, Constructing elliptic curves over finite fields with prescribed torsion, Math. Comp. 81 (2012), no. 278, 1131–1147. MR 2869053, DOI 10.1090/S0025-5718-2011-02538-X
André Weil, L’arithmétique sur les courbes algébriques, Acta Math. 52 (1929), no. 1, 281–315 (French). MR 1555278, DOI 10.1007/BF02547409