Families of elliptic curves over cyclic cubic number fields with prescribed torsion
HTML articles powered by AMS MathViewer
- by Daeyeol Jeon PDF
- Math. Comp. 85 (2016), 1485-1502 Request permission
Abstract:
In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields.References
- Houria Baaziz, Equations for the modular curve $X_1(N)$ and models of elliptic curves with torsion points, Math. Comp. 79 (2010), no. 272, 2371–2386. MR 2684370, DOI 10.1090/S0025-5718-10-02332-X
- Francesc Bars, The group structure of the normalizer of $\Gamma _0(N)$ after Atkin-Lehner, Comm. Algebra 36 (2008), no. 6, 2160–2170. MR 2418382, DOI 10.1080/00927870801949682
- F. Bars, On the automorphisms groups of genus 3 curves. Surveys in Math. and Math. Sciences 2 (2012), 83–124.
- Antonio F. Costa, Milagros Izquierdo, and Daniel Ying, On Riemann surfaces with non-unique cyclic trigonal morphism, Manuscripta Math. 118 (2005), no. 4, 443–453. MR 2190106, DOI 10.1007/s00229-005-0593-y
- Harvey Cohn, The density of abelian cubic fields, Proc. Amer. Math. Soc. 5 (1954), 476–477. MR 64076, DOI 10.1090/S0002-9939-1954-0064076-8
- J. E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1992. MR 1201151
- Andrei Duma and Wolfgang Radtke, Automorphismen und Modulraum Galoisscher dreiblättriger Überlagerungen, Manuscripta Math. 50 (1985), 215–228 (German, with English summary). MR 784144, DOI 10.1007/BF01168832
- Jack Fearnley, Hershy Kisilevsky, and Masato Kuwata, Vanishing and non-vanishing Dirichlet twists of $L$-functions of elliptic curves, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 539–557. MR 2980924, DOI 10.1112/jlms/jds018
- N. Ishii and F. Momose, Hyperelliptic modular curves, Tsukuba J. Math. 15 (1991), no. 2, 413–423. MR 1138196, DOI 10.21099/tkbjm/1496161667
- B. Im, D. Jeon and C. H. Kim, Normalizers of intermediate congruence subgroups of the Hecke subgroups. Preprint.
- Daeyeol Jeon, Automorphism groups of hyperelliptic modular curves, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 7, 95–100. MR 3365402, DOI 10.3792/pjaa.91.95
- Daeyeol Jeon and Chang Heon Kim, On the arithmetic of certain modular curves, Acta Arith. 130 (2007), no. 2, 181–193. MR 2357655, DOI 10.4064/aa130-2-7
- 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, Infinite families of elliptic curves over dihedral quartic number fields, J. Number Theory 133 (2013), no. 1, 115–122. MR 2981403, DOI 10.1016/j.jnt.2012.06.014
- 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
- D. Jeon, C. H. Kim and A. Schweizer, Bielliptic intermediate modular curves. Preprint.
- Chang Heon Kim and Ja Kyung Koo, The normalizer of $\Gamma _1(N)$ in $\textrm {PSL}_2(\textbf {R})$, Comm. Algebra 28 (2000), no. 11, 5303–5310. MR 1785501, DOI 10.1080/00927870008827156
- Matthew J. Klassen and Edward F. Schaefer, Arithmetic and geometry of the curve $y^3+1=x^4$, Acta Arith. 74 (1996), no. 3, 241–257. MR 1373711, DOI 10.4064/aa-74-3-241-257
- D. Krumm, Quadratic Points on Modular Curves. Ph.D. thesis, University of Georgia.
- Mong-Lung Lang, Normalizer of $\Gamma _1(m)$, J. Number Theory 86 (2001), no. 1, 50–60. MR 1813529, DOI 10.1006/jnth.2000.2555
- Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960, DOI 10.1007/978-1-4612-4752-4
- Jean-François Mestre, Corps euclidiens, unités exceptionnelles et courbes élliptiques, J. Number Theory 13 (1981), no. 2, 123–137 (French, with English summary). MR 612679, DOI 10.1016/0022-314X(81)90001-9
- F. Momose, Automorphism groups of the modular curves $X_1(N)$. Preprint.
- Filip Najman, Torsion of elliptic curves over cubic fields, J. Number Theory 132 (2012), no. 1, 26–36. MR 2843296, DOI 10.1016/j.jnt.2011.06.013
- 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
- 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
Additional Information
- Daeyeol Jeon
- Affiliation: Department of Mathematics Education, Kongju National University, Kongju, Chungnam, South Korea
- MR Author ID: 658790
- Email: dyjeon@kongju.ac.kr
- Received by editor(s): July 18, 2014
- Received by editor(s) in revised form: October 22, 2014
- Published electronically: August 3, 2015
- Additional Notes: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2010-0023942)
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp. 85 (2016), 1485-1502
- MSC (2010): Primary 11G05; Secondary 11G18, 14H37
- DOI: https://doi.org/10.1090/mcom/3012
- MathSciNet review: 3454372