An elliptic Calabi-Yau threefold of Mordell-Weil rank twelve

Abstract

We construct a smooth projective Calabi--Yau threefold with a flat elliptic fibration of Mordell--Weil rank twelve and Hodge numbers $(h^{1,1},h^{2,1})=(25,1)$. The construction starts from the Kummer surfaces associated with the products of a fixed elliptic curve and the level-four modular family. The threefold is birational to an anticanonical quartic in $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1}(3)^{\oplus2}\oplus\mathcal{O}_{\mathbb{P}^1}(4)\oplus\mathcal{O}_{\mathbb{P}^1})$. We give twelve independent sections and construct a projective crepant resolution of the associated Weierstrass model.