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Experimental Mathematics Volume 16, 2007 - Issue 3
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Original Article

Finding All Elliptic Curves with Good Reduction Outside a Given Set of Primes

J. E. Cremona Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, UK
&
M. P. Lingham Advantica, Holywell Park, New Ashby Road, Loughborough, Leicestershire, LE11 3GR, UK
Pages 303-312 | Published online: 30 Jan 2011

  • M. A. Bennett and C. M. Skinner. "Ternary Diophantine Equations via Galois Representations and Modular Forms." Canad. J. Math. 56:1 (2004), 23–54.
  • M. A. Bennett, V. Vatsal, and S. Yazdani. "Ternary Diophantine Equations of Signature (p, p, 3)." Compos. Math. 140:6 (2004), 1399–1416.
  • B. J. Birch and W. Kuyk, editors. Modular Functions of One Variable, IV, Lecture Notes in Mathematics, 476. Berlin: Springer-Verlag, 1975.
  • W. Bosma, J. Cannon, and C. Playoust. "The Magma Algebra System. I: The User Language." J. Symbolic Comput. 24:3-4 (1997), 235–265.
  • H. Cohen. Advanced Topics in Computational Number Theory, Graduate Texts in Mathematics, 193. New York: Springer-Verlag, 2000.
  • S. Comalada. "Courbes elliptiques á bonne réduction d'invariant j fixé." C. R. Acad. Sci. Paris Sér. I Math. 311:11 (1990), 667–670.
  • S. Comalada. "Elliptic Curves with Trivial Conductor over Quadratic Fields." Pacific J. Math. 144:2 (1990), 237–258.
  • S. Comalada and E. Nart. "Courbes elliptiques avec bonne réduction partout." C. R. Acad. Sci. Paris Sér. I Math. 305:6 (1987), 223–224.
  • J. E. Cremona. "Hyperbolic Tessellations, Modular Symbols, and Elliptic Curves over Complex Quadratic Fields." Compositio Mathematica 51 (1984), 275–323.
  • J. E. Cremona. "Addendum and errata: 'Hyperbolic Tessellations, Modular Symbols, and Elliptic Curves over Complex Quadratic Fields' [Compositio Math. 51:3 (1984) 275–324; MR0743014 (85j:11063)]." Compositio Math. 63:2 (1987), 271–272.
  • J. E. Cremona. "Abelian Varieties with Extra Twist, Cusp Forms and Elliptic Curves over Imaginary Quadratic Fields." J. London Math. Soc. (2) 45 (1992), 404–416.
  • J. E. Cremona. Algorithms for Modular Elliptic Curves, 2nd edition. Cambridge: Cambridge University Press, 1997.
  • J. E. Cremona and E. Whitley. "Periods of Cusp Forms and Elliptic Curves over Imaginary Quadratic Fields." Math. Comp. 62:205 (1993), 407–429.
  • E. Herrmann. "Bestimmung aller S-ganzen Lösungen auf elliptischen Kurven." PhD thesis, Universit ät des Saarlandes, 2002.
  • E. Herrmann and A. Pethő. "SIntegral Points on Elliptic Curves; Notes on a Paper of B. M. M. de Weger: 'S-Integral Solutions to a Weierstrass Equation.' [J. Theór. Nombres Bordeaux 9:2 (1997), 281–301; MR 99d:11027]." J. Théor. Nombres Bordeaux 13:2 (2001), 443–451.
  • H. Ishii. "The Nonexistence of Elliptic Curves with Everywhere Good Reduction over Certain Quadratic Fields." Japan. J. Math. (N.S.) 12:1 (1986), 45–52.
  • M. Kida. "Reduction of Elliptic Curves over Certain Real Quadratic Number Fields." Math. Comp. 68:228 (1999), 1679–1685.
  • M. Kida. "Computing Elliptic Curves Having Good Reduction Everywhere over Quadratic Fields II." In Algebraic Number Theory and Diophantine Analysis (Graz, 1998), pp. 239–247. Berlin: De Gruyter, 2000.
  • M. Kida. "Computing Elliptic Curves Having Good Reduction Everywhere over Quadratic Fields." Tokyo J. Math. 24:2 (2001), 545–558.
  • M. Kida. "Good Reduction of Elliptic Curves over Imaginary Quadratic Fields." J. Théor. Nombres Bordeaux 13:1 (2001), 201–209.
  • M. Kida. "Nonexistence of Elliptic Curves Having Good Reduction Everywhere over Certain Quadratic Fields." Arch. Math. (Basel) 76:6 (2001), 436–440.
  • M. Kida and T. Kagawa. "Nonexistence of Elliptic Curves with Good Reduction Everywhere over Real Quadratic Fields." J. Number Theory 66:2 (1997), 201–210.
  • M. P. Lingham. "Modular Forms and Elliptic Curves over Imaginary Quadratic Fields." PhD thesis, University of Nottingham, 2005. Available online (http: //etheses.nottingham.ac.uk/archive/00000138/). http://etheses.nottingham.ac.uk/archive/00000138/
  • A. Pethő, H. G. Zimmer, J. Gebel, and E. Herrmann. "Computing All S-Integral Points on Elliptic Curves." Math. Proc. Cambridge Philos. Soc. 127:3 (1999), 383–402.
  • I. R. Šafarevič. "Algebraic Number Fields." In Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 163–176. Djursholm: Inst. Mittag-Leffler, 1963.
  • B. Setzer. "Elliptic Curves over Complex Quadratic Fields." Pacific J. Math. 74:1 (1978), 235–250.
  • B. Setzer. "Elliptic Curves with Good Reduction Everywhere over Quadratic Fields and Having Rational j-Invariant." Illinois J. Math. 25:2 (1981), 233–245.
  • J. H. Silverman. The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. New York: Springer-Verlag, 1986.

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