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

Abstract:

Let $K = \mathbb {F}_q(T)$. For a Drinfeld $A$-module $\phi$ of rank $2$ defined over $C_\infty$, there are an associated exponential function $e_\phi$ and lattice $\Lambda _\phi$ in $C_\infty$ given by uniformization over $C_\infty$. We explicitly determine the Newton polygons of $e_\phi$ and the successive minima of $\Lambda _{\phi }$. When $\phi$ is defined over $K_\infty$, we give a refinement of Gardeyn’s bounds for the action of wild inertia at $\infty$ on the torsion points of $\phi$ and a criterion for the lattice field to be unramified over $K_\infty$. If $\phi$ 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 $\phi$, using results of Rosen, and this gives an explicit bound on the degree of the different divisor of division fields of $\phi$ over $K$.