Bounds on the Mordell-Weil rank of elliptic fibrations

Abstract

We prove that the Mordell-Weil group of a higher dimensional elliptic fibration naturally embeds into that of a suitable elliptic surface. We give sufficient conditions for the existence of such surfaces. We apply our result to obtain explicit and uniform bounds for the Mordell-Weil rank of elliptic threefolds of Kodaira dimension zero, including Calabi-Yau threefolds, confirming predictions from physics. We prove new explicit bounds for a broad class of elliptic fourfolds. These results suggest a general linear bound for the Mordell-Weil rank in terms of the dimension of the elliptic fibration, which we formulate as a conjecture.

36 pages; v2: Improved results, including new bounds for non-Calabi-Yau manifolds and new Section 5