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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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