Rank jumps on elliptic surfaces and the Hilbert property

Loughran, Daniel; Salgado, Cecília

Annales de l'Institut Fourier, Tome 72 (2022) no. 2, pp. 617-638.

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 $1$.

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 $1$.

Classification : 14G05, 14J27, 11G05
Keywords: elliptic surfaces, thin sets, rank jumps
Mot clés : surfaces elliptiques, ensembles minces, saut de rang

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