Elliptic fibrations of a \(K3\) surface (Q449699)

Let \(X_1(7)\) be the modular curve parametrizing stable elliptic curves together with a point of order \(7\). Let \(\pi:X\to X_1(7)\) be the universal family. Then \(X\) is an elliptic \(K3\) surface and \(X\) admits several elliptic fibrations. One can easily show that the Picard group has rank 20 and is generated by curves defined over \(\mathbb Q\).NEWLINENEWLINEIn this paper several other elliptic fibrations on \(X\) are constructed, each of which is defined over \(\mathbb Q\). One can use these fibrations to find a genus zero curves \(C\subset X\) such that \(\pi|_C:C\to X_1(7)\) has degree bigger than one. If this happen then the base changed elliptic surface \(X\times_{X_1(7)} C\) has a larger Mordell-Weil group than the original fibration \(X\to X_1(7)\). The authors identify several such examples. Four of them were known before, and in each of these cases the map \(\pi|_C\) has degree 2. In the new examples the map \(\pi|_C\) has degree 3. By specializing one finds elliptic curves over \(\mathbb Q\) with a point of order \(7\) and positive rank.