Elliptic threefolds with high Mordell-Weil rank.

We present the first examples of smooth elliptic Calabi-Yau threefolds with Mordell-Weil rank 10, the highest currently known value. They are given by the Schoen threefolds introduced by Namikawa. We explicitly compute the Shioda homomorphism for the generators of the Mordell-Weil group and their induced height pairing. There are six isolated fibers of Kodaira Type IV. Compactification of F-theory on these threefolds gives an effective theory in six dimensions which contains ten abelian gauge group factors. We compute the charges and multiplicities of the massless hypermultiplets as Gopakumar-Vafa invariants and show that they satisfy the gravitational and abelian anomaly cancellation conditions. We explicitly describe a Weierstrass model over $\mathbb P^2$ of the Calabi-Yau threefolds as a log canonical model and relate it to a construction by Elkies.