The normal order of of the divisor-counting function for invariants of rank 2 Drinfeld modules

TL;DR

This paper analyzes the distribution of prime divisors of invariants related to rank 2 Drinfeld modules, providing statistical insights into their behavior over varying primes.

Contribution

It computes the first and second moments of the divisor-counting function for specific invariants of rank 2 Drinfeld modules, revealing their normal order.

Findings

Derived the normal order of the divisor-counting function for invariants.

Established the statistical distribution of prime divisors for these invariants.

Provided moments that characterize the divisor behavior over primes.

Abstract

We compute the first and second moments of the divisor-counting function for the Euler-Poincar\'{e} characteristic and the trace of Frobenius for the reductions modulo $p$ of a rank 2 Drinfeld module with nontrivial endomorphism ring, as the prime $p$ varies over the primes of ordinary reduction of the Drinfeld module. From these moments we derive the normal order of the number of prime divisors of these invariants.