Share This Article:

On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

Abstract Full-Text HTML Download Download as PDF (Size:259KB) PP. 358-366
DOI: 10.4236/ajcm.2012.24049    2,923 Downloads   5,301 Views   Citations

ABSTRACT

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. Yokoyama, "On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields," American Journal of Computational Mathematics, Vol. 2 No. 4, 2012, pp. 358-366. doi: 10.4236/ajcm.2012.24049.

References

[1] J. Cremona and M. Lingham, “Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes,” Experimental Mathematics, Vol. 16, No. 3, 2007, pp. 303-312. doi:10.1080/10586458.2007.10129002
[2] J. Cremona, “Elliptic Curves with Everywhere Good Reduction over Quadratic Fields.” http://www.warwick.ac.uk/staff/J.E.Cremona//ecegr/ecegrqf.html
[3] H. Ishii, “The Non-Existence of Elliptic Curves with Everywhere Good Reduction over Certain Quadratic Fields,” Japanese Journal of Mathematics, Vol. 12, 1986, pp. 45-52.
[4] T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Q(),” Acta Arithmetica, Vol. 83, 1998, pp. 253-269.
[5] T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Real Quadratic Fields,” Acta Arithmetica, Vol. 73, No. 1, 1999, pp. 25-32. doi:10.1007/s000130050016
[6] T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Real Quadratic Fields Q(),” Acta Arithmetica, Vol. 96, 2001, pp. 231-245. doi:10.4064/aa96-3-4
[7] T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Real Quadratic Fields, II”, 2012 (in print).
[8] M. Kida, “On a Characterization of Shimura’s Elliptic Curve over Q(),” Acta Arithmetica, Vol. 77, No. 2, 1996, pp. 157-171.
[9] M. Kida and T. Kagawa, “Nonexistence of Elliptic Curves with Good Reduction Everywhere over Real Quadratic Fields,” Journal of Number Theory, Vol. 66, No. 2, 1997, pp. 201-210.
[10] M. Kida, “Reduction of Elliptic Curves over Certain Real Quadratic Number Fields,” Mathematics Computation, Vol. 68, 1999, pp. 1679-1685. doi:10.1090/S0025-5718-99-01129-1
[11] M. Kida, “Nonexistence of Elliptic Curves Having Good Reduction Everywhere over Certain Quadratic Fields,” Acta Arithmetica, Vol. 76, No. 6, 2001, pp. 436-440. doi:10.1007/PL00000454
[12] M. Kida and T. Kagawa, “Nonexistence of Elliptic Curves with Good Reduction Everywhere over Real Quadratic Fields,” Journal of Number Theory, Vol. 66, No. 2, 1997, pp. 201-210. doi:10.1006/jnth.1997.2177
[13] H. Muller, H. Stroher and H. Zimmer, “Torsion Groups of Elliptic Curves with Integral J-Invariant over Quadratic Fields”, Journal Für Die Reine und Angewandte Mathematik, Vol. 1989, No. 397, 2009, 1989, pp. 100-161.
[14] R. G. E. Pinch, “Elliptic Curves over Number Fields,” Ph.D. Thesis, Oxford, 1982.
[15] T. Thongjunthug, “Heights on Elliptic Curves over Number Fields, Period Lattices, and Complex Elliptic Logarithms,” Ph.D. Thesis, The University of Warwick, Coventry, 2011.
[16] A. Umegaki, “A Construction of Everywhere Good Q-Curves with P-Isogeny,” Tokyo Journal of Mathematics, Vol. 21, No. 1, 1998, pp. 183-200.
[17] M. Bertolini and G. Canuto, “Good Reduction of Elliptic Curves Defined over,” Acta Arithmetica, Vol. 50, No. 1, 1988, pp. 42-50. doi:10.1007/BF01313493
[18] N. Takeshi, “On Elliptic Curves Having Everywhere Good Reduction over Cubic Fields,” Master’s Thesis, Tsuda College, Tokyo, 2012.
[19] S. Yokoyama and Y. Shimasaki, “Non-Existence of Elliptic Curves with Everywhere Good Reduction over Some Real Quadratic Fields,” Journal of Math-for-Industry, Vol. 3, 2011, pp. 113-117.
[20] S. Comalada, “Elliptic Curves with Trivial Conductor over Quadratic Fields,” Pacific Journal of Mathematics, Vol. 144, No. 2, 1990, pp. 233-258. doi:10.2140/pjm.1990.144.237
[21] B. Setzer, “Elliptic Curves over Complex Quadratic Fields,” Pacific Journal of Mathematics, Vol. 74, No. 1, 1978, pp. 235-250.
[22] J. H. Silverman, “The Arithmetic of Elliptic Curves,” 2nd Edition, Graduate Texts in Mathematics 106, Springer-Verlag, Berlib, 2009.
[23] T. Kagawa, “Computing Integral Points of Elliptic Curves over Real Quadratic Fields, and Determination of Elliptic Curves Having Trivial Conductor.” http://www.ritsumei.ac.jp/se/~kagawa/waseda.pdf.
[24] G. P. Pari, “A Computer Algebra System Designed for Fast Computations in Number Theory.” http://pari.math.u-bordeaux.fr/.
[25] T. Kagawa, “Elliptic Curves with Everywhere Good Reduction over Real Quadratic Fields,” Ph.D. Thesis, Waseda University, Tokyo, 1998.
[26] KANT/KASH, “Computational Algebraic Number Theory.” http://www.math.tu-berlin.de/~kant/kash.html.
[27] D. Simon, “Computing the Rank of Elliptic Curves over Number Fields,” LMS Journal of Computation and Mathematics, Vol. 5, 2002, pp. 7-17.
[28] Sage, “Open Source Mathematics Software.” http://www.sagemath.org/
[29] W. Bosma, J. Cannon and C. Playoust, “The Magma Algebra System. I. The User Language,” Journal of Symbolic Computation, Vol. 24 No. 3-4, 1997, pp. 235-265.
[30] S. Siksek, “Infinite Descent on Elliptic Curves,” Rocky Mountain Journal of Mathematics, Vol. 25, No. 4, 1995, pp. 1501-1538. doi:10.1216/rmjm/1181072159
[31] N. P. Smart, “The Algorithmic Resolution of Diophantine Equations,” London Mathematical Society Student Text 41, Cambridge University Press, Cambridge, 1998.

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.