Vol. 16, No. 2, 2022

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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
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