Harvard-MIT Algebraic Geometry Seminar: Bounds on the Mordell-Weil rank of certain elliptic fibrations

It is known that the rank of the Mordell-Weil group of an elliptic K3 surface can take any integer value between 0 and 18.  For elliptic Calabi-Yau varieties,  boundedness results imply the existence of a bound on the rank of the Mordell-Weil group for every fixed dimension. Such bounds play an important role in physics. I will discuss a method that paves the way to determining effective bounds on the rank of the Mordell-Weil group of certain higher dimensional elliptic fibrations,  namely the reduction to particular elliptic surfaces with a scheme morphism to the original variety. I will present applications to threefolds of Kodaira dimension zero and a broad class of Calabi-Yau fourfolds, proving and extending explicit bounds  in the physics literature.

The talk is based on joint work with Miranda, Paranjape, Srinivas and Weigand.