Good reduction of elliptic curves defined over \({\mathbb Q}(^ 3\sqrt{2})\)
A classical theorem of J. Tate states that there are no elliptic curves over \({\mathbb{Q}}\) with good reduction everywhere. Generalizations to complex quadratic fields are due to Stroeker and Setzer. Stroeker proved that if an elliptic curve on a imaginary quadratic field has a global minimal equation then it has bad reduction somewhere. Setzer proved that there are no elliptic curves with a good reduction everywhere on imaginary quadratic fields with class number prime to 6. In these papers the finiteness of the group of units is crucial. We prove that on \(K={\mathbb{Q}}(^ 3\sqrt{2})\) there are no elliptic curves with good reduction everywhere. In this field the group of units is infinite. We show first that any such curve must have a point of order two K-rational. We show then that the solutions of the diophantine equations we have to consider do not correspond to elliptic curves with good reduction everywhere using Tate's algorithm.
no elliptic curves with good reduction everywhere group of units
scientific article; zbMATH DE number 3995864