On successive minimal bases of division points of Drinfeld modules

Abstract

We define successive minimal bases (SMBs) for the space of $u^{n}$-division points of a Drinfeld $\mathbb{F}_{q}[t]$-module over a local field, where $u$ is a finite prime of $\mathbb{F}_{q}[t]$ and $n$ is a positive integer. These SMBs share similar properties to those of SMBs of the lattices associated to Drinfeld modules. We study the relations between these SMBs and those of the lattices. Finally, we apply the relations to study the explicit wild ramification subgroup action on an SMB of the space of $u^{n}$-division points and show the function field analogue of Szpiro's conjecture for rank $2$ Drinfeld modules under a certain limited situation.