Bounds on the Mordell-Weil rank of elliptic fibrations

We present two proofs for a bound on the rank of the Mordell-Weil group of some elliptic fibrations. The bounds apply to Calabi-Yau varieties, which are also of interest to the physics of string theory. We prove explicit bounds for Calabi-Yau threefolds, as predicted by physics, and give new explicit bounds for fourfolds under mild assumptions. These results motivate a conjecture for bounds in any dimensions.