Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Families of elliptic curves over cyclic cubic number fields with prescribed torsion


Author: Daeyeol Jeon
Journal: Math. Comp. 85 (2016), 1485-1502
MSC (2010): Primary 11G05; Secondary 11G18, 14H37
DOI: https://doi.org/10.1090/mcom/3012
Published electronically: August 3, 2015
MathSciNet review: 3454372
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we construct infinite families of elliptic curves with given torsion group structures over cyclic cubic number fields.


References [Enhancements On Off] (What's this?)

  • [B] 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 (2011i:11056), https://doi.org/10.1090/S0025-5718-10-02332-X
  • [Ba1] 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 (2009b:20092), https://doi.org/10.1080/00927870801949682
  • [Ba2] F. Bars, On the automorphisms groups of genus 3 curves. Surveys in Math. and Math. Sciences 2 (2012), 83-124.
  • [CIY] 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 (2006k:30042), https://doi.org/10.1007/s00229-005-0593-y
  • [C] Harvey Cohn, The density of abelian cubic fields, Proc. Amer. Math. Soc. 5 (1954), 476-477. MR 0064076 (16,222a)
  • [Cr] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, Cambridge, 1992. MR 1201151 (93m:11053)
  • [DR] Andrei Duma and Wolfgang Radtke, Automorphismen und Modulraum Galoisscher dreiblättriger Überlagerungen, Manuscripta Math. 50 (1985), 215-228 (German, with English summary). MR 784144 (86i:32041), https://doi.org/10.1007/BF01168832
  • [FKK] 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, https://doi.org/10.1112/jlms/jds018
  • [IM] N. Ishii and F. Momose, Hyperelliptic modular curves, Tsukuba J. Math. 15 (1991), no. 2, 413-423. MR 1138196 (93b:14037)
  • [IJK] B. Im, D. Jeon and C. H. Kim, Normalizers of intermediate congruence subgroups of the Hecke subgroups. Preprint.
  • [J] Daeyeol Jeon, Automorphism groups of hyperelliptic modular curves, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 7, 95-100. MR 3365402, https://doi.org/10.3792/pjaa.91.95
  • [JK] Daeyeol Jeon and Chang Heon Kim, On the arithmetic of certain modular curves, Acta Arith. 130 (2007), no. 2, 181-193. MR 2357655 (2008m:11119), https://doi.org/10.4064/aa130-2-7
  • [JKL1] 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 (2011m:11113), https://doi.org/10.1090/S0025-5718-10-02369-0
  • [JKL2] 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, https://doi.org/10.1016/j.jnt.2012.06.014
  • [JKS1] 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 (2005f:11112), https://doi.org/10.4064/aa113-3-6
  • [JKS2] D. Jeon, C. H. Kim and A. Schweizer, Bielliptic intermediate modular curves. Preprint.
  • [KK] Chang Heon Kim and Ja Kyung Koo, The normalizer of $ \Gamma _1(N)$ in $ {\rm PSL}_2({\bf R})$, Comm. Algebra 28 (2000), no. 11, 5303-5310. MR 1785501 (2002d:20075), https://doi.org/10.1080/00927870008827156
  • [KS] 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 (96k:11081)
  • [Kr] D. Krumm, Quadratic Points on Modular Curves. Ph.D. thesis, University of Georgia.
  • [L] Mong-Lung Lang, Normalizer of $ \Gamma _1(m)$, J. Number Theory 86 (2001), no. 1, 50-60. MR 1813529 (2002f:11047), https://doi.org/10.1006/jnth.2000.2555
  • [La] Serge Lang, Elliptic Functions, 2nd ed., with an appendix by J. Tate, Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. MR 890960 (88c:11028)
  • [M] 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 (83i:12006), https://doi.org/10.1016/0022-314X(81)90001-9
  • [Mo] F. Momose, Automorphism groups of the modular curves $ X_1(N)$. Preprint.
  • [N] Filip Najman, Torsion of elliptic curves over cubic fields, J. Number Theory 132 (2012), no. 1, 26-36. MR 2843296 (2012j:11122), https://doi.org/10.1016/j.jnt.2011.06.013
  • [R] 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 (87f:11039), https://doi.org/10.2307/2008003
  • [S] Andrew V. Sutherland, Constructing elliptic curves over finite fields with prescribed torsion, Math. Comp. 81 (2012), no. 278, 1131-1147. MR 2869053 (2012m:11079), https://doi.org/10.1090/S0025-5718-2011-02538-X

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11G05, 11G18, 14H37

Retrieve articles in all journals with MSC (2010): 11G05, 11G18, 14H37


Additional Information

Daeyeol Jeon
Affiliation: Department of Mathematics Education, Kongju National University, Kongju, Chungnam, South Korea
Email: dyjeon@kongju.ac.kr

DOI: https://doi.org/10.1090/mcom/3012
Keywords: Elliptic curve, modular curve, torsion, cyclic cubic number field, cyclic trigonal
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)
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society