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

Abstract

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 nonconstant 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.