On the surjectivity of $(T)$-adic Galois Representations of Drinfeld $A$-Modules of Rank 2 and 3: Density results

TL;DR

This paper provides explicit valuation criteria for Drinfeld $A$-modules of ranks 2 and 3 to ensure surjectivity of their $(T)$-adic Galois representations, and calculates the densities of such modules.

Contribution

It introduces concrete valuation-based criteria for surjectivity and computes the densities of Drinfeld $A$-modules satisfying these conditions.

Findings

Criteria involving valuations guarantee Galois representation surjectivity.

Calculated densities of Drinfeld $A$-modules with surjective representations.

Explicit conditions for ranks 2 and 3 modules.

Abstract

Let $\mathbb{F}_{q}$ be a finite field, and $A:=\mathbb{F}_{q}[T]$. In this article, we give explicit criteria, involving concrete valuations, on the coefficients of the Drinfeld $A$-modules of rank $r$ for $r=2,3$, which ensure the surjectivity of the associated $(T)$-adic Galois representation. As a result, we shall calculate the densities of such Drinfeld $A$-modules.