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=pA 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_p²−4μ_pp|, where X²+a_pX+μ_pp∈A[X] is the characteristic polynomial of τ^degp.