#### 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.

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