Good reduction of elliptic curves over imaginary quadratic fields

Masanari Kida

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 201-209
  • ISSN: 1246-7405

Abstract

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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.

How to cite

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Kida, Masanari. "Good reduction of elliptic curves over imaginary quadratic fields." Journal de théorie des nombres de Bordeaux 13.1 (2001): 201-209. <http://eudml.org/doc/248704>.

@article{Kida2001,
abstract = {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.},
author = {Kida, Masanari},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {201-209},
publisher = {Université Bordeaux I},
title = {Good reduction of elliptic curves over imaginary quadratic fields},
url = {http://eudml.org/doc/248704},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Kida, Masanari
TI - Good reduction of elliptic curves over imaginary quadratic fields
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 201
EP - 209
AB - 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.
LA - eng
UR - http://eudml.org/doc/248704
ER -

References

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  2. [2] S. Comalada, E. Nart, Modular invariant and good reduction of elliptic curves. Math. Ann.293 (1992), no. 2, 331-342. Zbl0734.14013MR1166124
  3. [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. Zbl0927.11034MR1627777
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  7. [7] M. Kida, Computing elliptic curves having good reduction everywhere over quadratic fields. Preprint (1998). MR1874989
  8. [8] M. Kida, Non-existence of elliptic curves having good reduction everywhere over certain quadratic fields. Preprint (1999). MR1831499
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  10. [10] M. Kida, TECC manual version 2.2. The University of Electro-Communications, November 1999. 
  11. [11] B. Setzer, Elliptic curves over complex quadratic fields. Pacific J. Math.74 (1978), no. 1, 235-250. Zbl0394.14018MR491710
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  13. [13] J.H. Silverman, Advanced topics in the arithmetic of elliptic curves. Springer-Verlag, New York, 1994. Zbl0911.14015MR1312368
  14. [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. Zbl0881.11054MR1443583
  15. [15] N.P. Smart, The algorithmic resolution of Diophantine equations. Cambridge University Press, Cambridge, 1998. Zbl0907.11001MR1689189
  16. [16] R.J. Stroeker, Reduction of elliptic curves over imaginary quadratic number fields. Pacific J. Math.108 (1983), no. 2, 451-463. Zbl0491.14024MR713747
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  18. [18] H.G. Zimmer, Basic algorithms for elliptic curves. Number theory (Eger, 1996), de Gruyter, Berlin, 1998, pp. 541-595. Zbl1066.11514MR1628868

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