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In this paper, we construct twists of Albanese varieties of abelian and dihedral covers of projective spaces with large Mordell–Weil rank over function fields of certainly defined high dimensional varieties. We achieve this by using the theory of twists, a generalization of the notion of Prym variety to higher dimensions, and a structure theorem for the Mordell–Weil group of abelian varieties over function fields.

Twists of Albanese varieties of abelian and dihedral covers of ℙ ℓ with large ranks over function fields