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On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields
Faculty of Mathematics, Kyushu University, Fukuoka, Japan
- 1 Faculty of Mathematics, Kyushu University, Fukuoka, Japan
American Journal of Computational Mathematics·Volume 02 (2012)·Pages 358–366·Published 18 December 2012·DOI10.4236/ajcm.2012.24049
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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.
KeywordsElliptic Curves over Number FieldsMordell-Weil GroupTwo-Descent
- 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
- J. Cremona, “Elliptic Curves with Everywhere Good Reduction over Quadratic Fields.” http://www.warwick.ac.uk/staff/J.E.Cremona//ecegr/ecegrqf.html
- 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.
- T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Q( ),” Acta Arithmetica, Vol. 83, 1998, pp. 253-269.
- 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
- 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
- T. Kagawa, “Determination of Elliptic Curves with Everywhere Good Reduction over Real Quadratic Fields, II”, 2012 (in print).
- M. Kida, “On a Characterization of Shimura’s Elliptic Curve over Q( ),” Acta Arithmetica, Vol. 77, No. 2, 1996, pp. 157-171.
- 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.
- 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
- 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
- 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
- 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.
- R. G. E. Pinch, “Elliptic Curves over Number Fields,” Ph.D. Thesis, Oxford, 1982.