Distribution of the trace of Frobenius on average for rank 2 Drinfeld modules

TL;DR

This paper studies the average distribution of Frobenius traces for rank 2 Drinfeld modules over finite fields, providing asymptotic formulas for the number of primes with specified Frobenius characteristic polynomials.

Contribution

It computes the average number of primes with a given Frobenius characteristic polynomial for rank 2 Drinfeld modules, extending understanding of their distribution.

Findings

Asymptotic formulas for prime counts with specified Frobenius polynomials

Average distribution results for Frobenius traces in rank 2 Drinfeld modules

Conditions on q for the formulas to hold

Abstract

Let $q$ be an odd prime power, $a \in \mathbb{F}_q[T]$ and $u \in \mathbb{F}_q^*$. Provided $q \geq 17$, we compute the average number of primes $p$ for which the characteristic polynomial of the Frobenius at $p$ is $X^2 - aX + up$ over a family of rank 2 Drinfeld $ \mathbb{F}_q[T]$-modules. Our results give asymptotic formulas in the $x$-limit.