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 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.

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

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• Elliptic surfaces over ${\mathbb{P}}^1$ and large class groups of number fields