Kida: Good reduction of elliptic curves over imaginary quadratic fields

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Masanari Kida
Good reduction of elliptic curves over imaginary quadratic fields
Journal de théorie des nombres de Bordeaux, 13 no. 1 (2001), p. 201-209, doi: 10.5802/jtnb.315
Article PDF | Reviews MR 1838081 | Zbl 02081359

Résumé - 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.

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