Good reduction of elliptic curves over imaginary quadratic fields

Kida, Masanari

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 201-209.

Nous montrons que l’invariant modulaire $j$ d’une courbe elliptique définie sur un corps quadratique imaginaire ayant par-tout bonne réduction vérifie certaines équations diophantiennes, sous réserve que soient vérifiées certaines hypothèses relatives à l’arithmétique du corps. En résolvant explicitement ces équations dans l’anneau des entiers du corps, nous montrons que de telles courbes n’existent pas sur certains corps quadratiques imaginaires. Nos résultats généralisent des résultats antérieurs de Setzer et Stroeker.

We prove that the $j$-invariant of an elliptic curve defined over an imaginary quadratic number field having good reduction everywhere satisfies certain Diophantine equations under some hypothesis on the arithmetic of the quadratic field. By solving the Diophantine equations explicitly in the rings of quadratic integers, we show the non-existence of such elliptic curve for certain imaginary quadratic fields. This extends the results due to Setzer and Stroeker.

[1] H. Cohen, A course in computational algebraic number theory. Springer-Verlag, Berlin, 1993. | MR | Zbl

[2] S. Comalada, E. Nart, Modular invariant and good reduction of elliptic curves. Math. Ann. 293 (1992), no. 2, 331-342. | MR | Zbl

[3] J.E. Cremona, P. Serf, Computing the rank of elliptic curves over real quadratic number fields of class number 1. Math. Comp. 68 (1999), no. 227, 1187-1200. | MR | Zbl

[4] M. Daberkow, C. Fieker, J. Kluners, M. Pohst, K. Roegner, M. Schörnig, K. Wildanger, KANT V4. J. Symbolic Comput. 24 (1997), no. 3-4, 267-283, Computational algebra and number theory (London, 1993). | MR | Zbl

[5] S. David, Minorations de hauteurs sur les variétés abéliennes. Bull. Soc. Math. France 121 (1993), no. 4, 509-544. | Numdam | MR | Zbl

[6] C. Hollinger, P. Serf, SIMATH-a computer algebra system. Computational number theory (Debrecen, 1989), de Gruyter, Berlin, 1991, pp. 331-342. | MR | Zbl

[7] M. Kida, Computing elliptic curves having good reduction everywhere over quadratic fields. Preprint (1998). | MR

[8] M. Kida, Non-existence of elliptic curves having good reduction everywhere over certain quadratic fields. Preprint (1999). | MR

[9] M. Kida, Reduction of elliptic curves over certain real quadratic fields. Math. Comp. 68 (1999), no. 228, 1679-1685. | MR | Zbl

[10] M. Kida, TECC manual version 2.2. The University of Electro-Communications, November 1999.

[11] B. Setzer, Elliptic curves over complex quadratic fields. Pacific J. Math. 74 (1978), no. 1, 235-250. | MR | Zbl

[12] J.H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), no. 183, 339-358. | MR | Zbl

[13] J.H. Silverman, Advanced topics in the arithmetic of elliptic curves. Springer-Verlag, New York, 1994. | MR | Zbl

[14] N.P. Smart, N.M. Stephens, Integral points on elliptic curves over number fields. Math. Proc. Cambridge Philos. Soc. 122 (1997), no. 1, 9-16. | MR | Zbl

[15] N.P. Smart, The algorithmic resolution of Diophantine equations. Cambridge University Press, Cambridge, 1998. | MR | Zbl

[16] R.J. Stroeker, Reduction of elliptic curves over imaginary quadratic number fields. Pacific J. Math. 108 (1983), no. 2, 451-463. | MR | Zbl

[17] L.C. Washington, Class numbers of the simplest cubic fields. Math. Comp. 48 (1987), no. 177, 371-384. | MR | Zbl

[18] H.G. Zimmer, Basic algorithms for elliptic curves. Number theory (Eger, 1996), de Gruyter, Berlin, 1998, pp. 541-595. | MR | Zbl