Ramification of Tate modules for rank $2$ Drinfeld modules

TL;DR

This paper investigates the ramification properties of extensions generated by division points of rank 2 Drinfeld modules and defines conductors analogous to elliptic curves, leading to an analogue of Szpiro's conjecture.

Contribution

It introduces a new analysis of ramification in rank 2 Drinfeld modules and defines conductors similar to elliptic curves, enabling a Szpiro's conjecture analogue.

Findings

Conductors for certain rank 2 Drinfeld modules are explicitly calculated.

An analogue of Szpiro's conjecture is established under specific conditions.

The study enhances understanding of ramification in function field extensions.

Abstract

In this paper, we study the ramification of extensions of a function field generated by division points of rank 2 Drinfeld modules. Also conductors of certain rank 2 Drinfeld modules are defined as analogues of those for elliptic curves. A calculation of these conductors allows us to show an analogue of Szpiro's conjecture under a certain limited situation.