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Determination of elliptic curves with everywhere good reduction over \(\mathbb{Q}(\sqrt{37})\). (English) Zbl 0915.11033
Let \(k\) be a number field. The author considers the problem of determining the elliptic curves with everywhere good reduction over \(k\). For \(k= \mathbb{Q}\) such curves do not exist, and over imaginary quadratic fields \(k\) such curves do not exist either in case the class number of \(k\) is prime to 6 [see R. J. Stroeker, Pac. J. Math. 108, 451-463 (1983; Zbl 0524.14034)]. The author turns to real quadratic fields and briefly describes the relevance Shimura’s elliptic curves have for this problem [see K. Shiota, J. Math. Soc. Japan 38, 649-659 (1986; Zbl 0608.14023)].
The main theorem of this paper states that up to isomorphism the only elliptic curves with everywhere good reduction over \(k= \mathbb{Q}(\sqrt{37})\) are the Shimura curve and its conjugate over this field. The proof is mainly by means of diophantine equations over \(k\) and certain quadratic extensions of \(k\).
The author also points out that in [M. Kida and T. Kagawa, J. Number Theory 66, 201-210 (1997; Zbl 0895.11023)] the case \(k= \mathbb{Q}(\sqrt{41})\) is settled.

MSC:
11G05 Elliptic curves over global fields
14H52 Elliptic curves
11D25 Cubic and quartic Diophantine equations
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