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

TL;DR

This paper develops p-descent methods for elliptic surfaces over fields of characteristic not 2 or 3, providing new upper bounds for their Mordell-Weil ranks, especially refining known geometric bounds when p=2, with applications to surfaces over rationals.

Contribution

It introduces p-descent techniques for elliptic surfaces and derives new arithmetic bounds for their Mordell-Weil ranks, answering a question by Ulmer.

Findings

Derived upper bounds for the rank of elliptic surfaces using p-descent.

Refined geometric bounds for rank when p=2, based on Igusa's inequality.

Applied the bounds to elliptic surfaces over the rational numbers.

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.