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 .
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 .
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3457
Keywords: elliptic surfaces, thin sets, rank jumps
Mot clés : surfaces elliptiques, ensembles minces, saut de rang
Loughran, Daniel 1; Salgado, Cecília 2
@article{AIF_2022__72_2_617_0, author = {Loughran, Daniel and Salgado, Cec{\'\i}lia}, title = {Rank jumps on elliptic surfaces and the {Hilbert} property}, journal = {Annales de l'Institut Fourier}, pages = {617--638}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {72}, number = {2}, year = {2022}, doi = {10.5802/aif.3457}, zbl = {07554665}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3457/} }
TY - JOUR AU - Loughran, Daniel AU - Salgado, Cecília TI - Rank jumps on elliptic surfaces and the Hilbert property JO - Annales de l'Institut Fourier PY - 2022 SP - 617 EP - 638 VL - 72 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3457/ DO - 10.5802/aif.3457 LA - en ID - AIF_2022__72_2_617_0 ER -
%0 Journal Article %A Loughran, Daniel %A Salgado, Cecília %T Rank jumps on elliptic surfaces and the Hilbert property %J Annales de l'Institut Fourier %D 2022 %P 617-638 %V 72 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3457/ %R 10.5802/aif.3457 %G en %F AIF_2022__72_2_617_0
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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