Newton polygons, successive minima, and different bounds for drinfeld modules of rank 2

Abstract

Let K = F q(T). For a Drinfeld A-module φ of rank 2 defined over C ∞, there are an associated exponential function e φ and lattice Λ φ in C∞ given by uniformization over C ∞. We explicitly determine the Newton polygons of e φ and the successive minima of Λ φ. When φ is defined over K ∞, we give a refinement of Gardeyn's bounds for the action of wild inertia at ∞ on the torsion points of φ and a criterion for the lattice field to be unramified over K ∞. If φ is in addition defined over K, we make explicit Gardeyn's bounds for the action of wild inertia at finite primes on the torsion points of φ, using results of Rosen, and this gives an explicit bound on the degree of the different divisor of division fields of φ over K. © 2012 American Mathematical Society.