Elliptic curves over complex quadratic fields. (English) Zbl 0394.14018

Setzer, Bennett

Pac. J. Math. 74, 235-250 (1978).

Let \( \mathrm{k}=\mathrm{Q}[\sqrt{-\mathrm{m}}] \) where \( \mathrm{m} \) is a positive square-free integer. The question considered in this paper is whether there exist elliptic curves defined over \( \mathrm{k} \) having good reduction everywhere. The program followed is to consider separately the cases in which the desired curve does or does not have a 2-division point rational over \( k \). The author proves (theorem 3) that there exist such curves with a rational 2-division point if and only if \( m=65 m_{1} \) where \( m_{1} \) is a square mod 5 and \( \bmod 13 \), and 65 is a square \( \bmod m_{1} \). If this is the case, then the number of isomorphism classes of such curves is \( 2^{r} \), where \( r \) is the number of primes which ramify in the extension \( \mathrm{k} / \mathrm{Q} \). - For curves without a 2 -division point rational over \( k \), the author determines the possible Galois structures of the normal closure \( K \) over \( Q \) of the 2-division field: it turns out (theorem 2) that the Galois group of \( K / Q \) is either the symmetric group \( S_{3} \), or \( S_{3} \times S_{2} \), or \( S_{3} \times S_{3} \). These two theorems make it possible to determine all the elliptic curves with good reduction everywhere, for certain values of \( m \). For example (theorem 5 ), if the class number of \( \mathrm{k} \) is prime to 6, there are no such curves.

Reviewer: I. G. Macdonald

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MSC:

11G05	Elliptic curves over global fields
14H52	Elliptic curves
14G05	Rational points
14G25	Global ground fields in algebraic geometry

Keywords:

Good Reduction; Imaginary Quadratic Field; Elliptic Curve; Rational Point

References:

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