Elliptic threefolds with high Mordell-Weil rank

We present the first examples of smooth elliptic Calabi-Yau three- folds with Mordell-Weil rank 10, the highest currently known value. We relate the multi- plicities of the massless spectrum to genus-zero Gopakumar-Vafa invariants and other geometric quantities of the Calabi-Yau. We show that the gravitational and abelian anomaly cancellation con- ditions are satisfied. We prove a Geometric Anomaly Cancellation equation and we deduce birational equivalence for the quantities in the spectrum. We explicitly describe a Weierstrass model over P2 of the Calabi-Yau threefolds as a log canonical model and compare it to a construction by Elkies and classical results of Burkhardt.