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

Gillibert, Jean; Levin, Aaron

doi:10.2140/ant.2022.16.311

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

Jean Gillibert and Aaron Levin

Vol. 16 (2022), No. 2, 311–333

DOI: 10.2140/ant.2022.16.311

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.

Keywords

elliptic surfaces, Mordell–Weil rank, Igusa's inequality, $p$-descent

Mathematical Subject Classification

Primary: 14D10

Secondary: 14G25, 14K15

Milestones

Received: 30 April 2020

Revised: 23 April 2021

Accepted: 17 June 2021

Published: 27 April 2022

Authors

Jean Gillibert
Institut de Mathématiques de Toulouse France

Aaron Levin
Department of Mathematics Michigan State University East Lansing, MI United States

Vol. 16, No. 2, 2022