Good reduction of elliptic curves over imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, p. 201-209

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.

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.

@article{JTNB_2001__13_1_201_0,
     author = {Kida, Masanari},
     title = {Good reduction of elliptic curves over imaginary quadratic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {201-209},
     zbl = {02081359},
     mrnumber = {1838081},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_201_0}
}
Kida, Masanari. Good reduction of elliptic curves over imaginary quadratic fields. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 201-209. http://www.numdam.org/item/JTNB_2001__13_1_201_0/

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