Arithmetic and geometry of rational elliptic surfaces
The paper under review deals with how some geometric properties of an rational elliptic surface defined over a number field are related to the minimal model over the base field. Let \(k\) be a number field and \(\mathcal{E}\) is an elliptic rational surface defined over \(k\). An \(k\)-minimal model of \(\mathcal{E}\), \(X\) is a del Pezzo surface of \(k\)-Picard number 1 or a conic bundle of \(k\)-Picard number 2. One main result of this paper is Theorem 3.2 which states that the degree and the \(k\)-Picard number of \(X\) have some realtions with the rank of \(k\)-Mordell-Weil group and the rank of the geometric Mordell-Weil group of the elliptic fibration \(\mathcal{E} \to \mathbb{P}^{1}\). In particular, the \(k\)-Mordell-Weil group is of rank 8 if and only if \(X\) is the blow up of \(\mathbb{P}^{2}\) at 9 \(k\)-rational points in general position. The author also studies the relation between the properties of \(X\) and the singular fibers of \(\mathcal{E} \to \mathcal{P}^{1}\) and proves that if the elliptic fibration has a non-reduced fiber, then \(\mathcal{E}\) is \(k\)-rational, in other words, there is a \(k\)-rational map from \(\mathbb{P}^{2}_{k}\) to \(\mathcal{E}\). (Corollary 4.4) By this result, we can see the universal elliptic modular surface over \(X_{1}(N)\) is \(\mathbb{Q}\)-rational for \(N=4,5\) and 6.
scientific article; zbMATH DE number 6673143