Rank jumps on elliptic surfaces and the Hilbert property
Annales de l'Institut Fourier, Volume 72 (2022) no. 2, pp. 617-638.

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell–Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree 1.

Pour une surface elliptique sur un corps de nombres, nous étudions la famille des fibres dont le rang de Mordell–Weil est strictement plus grand que le rang générique. Sous des hypothèses appropriées, nous démontrons que cette famille n’est pas un ensemble mince. Nos résultats s’appliquent, par exemple, aux familles quadratiques tordues et aux surfaces de del Pezzo de degré 1.

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DOI: 10.5802/aif.3457
Classification: 14G05,  14J27,  11G05
Keywords: elliptic surfaces, thin sets, rank jumps
Loughran, Daniel 1; Salgado, Cecília 2

1 Department of Mathematical Sciences University of Bath Claverton Down Bath BA2 7AY, (UK)
2 Instituto de Matemática, Univ. Federal do Rio de Janeiro, Rio de Janeiro, (Brazil)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Loughran, Daniel; Salgado, Cecília. Rank jumps on elliptic surfaces and the Hilbert property. Annales de l'Institut Fourier, Volume 72 (2022) no. 2, pp. 617-638. doi : 10.5802/aif.3457. https://aif.centre-mersenne.org/articles/10.5802/aif.3457/

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