Drinfeld modules, Frobenius endomorphisms, and CM-liftings

TL;DR

This paper provides a comprehensive analysis of Frobenius elements in division fields of rank 2 Drinfeld modules, introduces criteria for polynomial splitting, and extends results to higher ranks, including CM-lifting theorems.

Contribution

It offers a global description of Frobenius elements, new splitting criteria for polynomials, and generalizes results to higher rank Drinfeld modules with CM-lifting theorems.

Findings

Derived a criterion for polynomial splitting modulo primes.

Analyzed the frequency of small endomorphism rings in reductions.

Extended results to higher rank Drinfeld modules and proved CM-lifting theorems.

Abstract

We give a global description of the Frobenius elements in the division fields of Drinfeld modules of rank $2$. We apply this description to derive a criterion for the splitting modulo primes of a class of non-solvable polynomials, and to study the frequency with which the reductions of Drinfeld modules have small endomorphism rings. We also generalize some of these results to higher rank Drinfeld modules and prove CM-lifting theorems for Drinfeld modules.