Elliptic curves with trivial conductor over quadratic fields
This paper deals with elliptic curves defined over quadratic number fields and having good reduction at all primes. It extends work of \textit{B. Setzer} [Pac. J. Math. 74, 235-250 (1978; Zbl 0394.14018)], and Ill. J. Math. 25, 233-245 (1981; Zbl 0471.14019)] and the reviewer [Pac. J. Math. 108, 451-463 (1983; Zbl 0491.14024)]. The results include the following: Those real quadratic number fields \({\mathbb{K}}\) are characterized which admit elliptic curves having a global minimal model and a 2-division point over \({\mathbb{K}}\), the characterization being achieved through the solvability of certain diophantine equations over \({\mathbb{Z}}\). - An effective procedure is given to determine all curves with the said property for each given real quadratic field \({\mathbb{K}}\). Also the number of isomorphism classes is given. A characterization is given of the real quadratic fields \({\mathbb{K}}\) of odd class number which admit curves with good reduction everywhere and having a \({\mathbb{K}}\)-rational 2-division point. A table is added containing all those curves defined over \({\mathbb{Q}}(\sqrt{d})\) in the range \(2\leq d<100\).
scientific article; zbMATH DE number 4210244