Nonexistence of elliptic curves having everywhere good reduction and cubic discriminant. (English) Zbl 0991.11029

The author proves that if \(p\) is a prime number such that \(p\equiv 3(4)\) and \(p\not=3,11\), then there is no elliptic curve defined over \(K={\mathbb Q} (\sqrt{3p})\) with everywhere good reduction over \(K\) and whose discriminant is a cube in \(K\).


11G05 Elliptic curves over global fields
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[1] Fröhlich, A., and Taylor, M. J.: Algebraic number theory. Cambridge Stud. Adv. Math., 27 , Cambridge Univ. Press, Cambridge (1991).
[2] Kagawa, T.: Determination of elliptic curves with everywhere good reduction over real quadratic fields \(\textbf{Q}(\sqrt{3p})\). Acta\hphantom. Arith. (to appear). · Zbl 0977.11024 · doi:10.4064/aa96-3-4
[3] Lang, S.: Algebraic Number Theory. 2nd ed., Grad. Texts in Math., 110 , Springer, Berlin-Heidelberg-New York (1994). · Zbl 0811.11001
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