On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

Shun’ichi Yokoyama

Faculty of Mathematics, Kyushu University, Fukuoka, Japan.

DOI: 10.4236/ajcm.2012.24049                

Abstract

We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. 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.

Keywords

Elliptic Curves over Number Fields; Mordell-Weil Group; Two-Descent

Conflicts of Interest

The authors declare no conflicts of interest.

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.