On the rank of the fibers of rational elliptic surfaces

Abstract

We consider an elliptic surface $\pi :\mathcal{ℰ}\to {\mathbb{P}}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibers over $k$ with that of the generic fiber over $k\left({\mathbb{P}}^1\right)$. We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibers with rank at least equal to the generic rank plus two.