Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank

We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa's inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.
Comment: 22 pages, LaTeX. Minor improvements in the statement of Theorem 1.1. Added Theorem 1.7 and its proof. To appear in Algebra and Number Theory