Covers of P^1 and Number Fields

Let n be an integer. Via the Minkowski embedding, an order O in a degree n number field can be seen as a lattice. Similarly, given a degree n cover of P^1, the pushforward of the structure sheaf is an interesting rank n vector bundle on P^1. This lattice and this vector bundle are analogous, and very little are known about them.

What lattices/vector bundles arise this way? Which lattices/vector bundles arise from maximal orders/smooth curves? How often does a fixed lattice/vector bundle arise? In this talk, we prove new constraints on these lattice and vector bundles, using tools in additive combinatorics. We also discuss joint work with Ravi Vakil towards the Tschirnhausen realization problem.