The growth of the discriminant of the endomorphism ring of the reduction of a rank 2 generic Drinfeld module

For q an odd prime power, A = F q [ T ] , and F = F q ( T ) , let ψ : A → F { τ } be a Drinfeld A-module over F of rank 2 and without complex multiplication, and let p = p A be a prime of A of good reduction for ψ, with residue field F p . We study the growth of the absolute value | Δ p | of the discriminant of the F p -endomorphism ring of the reduction of ψ modulo p and prove that, for all p , | Δ p | grows with | p | . Moreover, we prove that, for a density 1 of primes p , | Δ p | is as close as possible to its upper bound | a p 2 − 4 μ p p | , where X 2 + a p X + μ p p ∈ A [ X ] is the characteristic polynomial of τ deg p .