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

Abstract

Let $\psi: A \to F\{\tau\}$ be a Drinfeld $A$-module over $F$ of rank 2 and without complex multiplication, where $A = {\mathbb{F}}_q[T]$, $F = {\mathbb{F}}_q(T)$, and $q$ is an odd prime power. For a prime $\mathfrak{p} = p A$ of $A$ of good reduction for $\psi$ and with residue field ${\mathbb{F}}_{\mathfrak{p}}$, we study the growth of the absolute value $|\Delta_{\mathfrak{p}}|$ of the discriminant of the ${\mathbb{F}}_{\mathfrak{p}}$-endomorphism ring of the reduction of $\psi$ modulo $\mathfrak{p}$. We prove that for all $\mathfrak{p}$, $|\Delta_{\mathfrak{p}}|$ grows with $|p|$. Moreover, we prove that for a density 1 of primes $\mathfrak{p}$, $|\Delta_{\mathfrak{p}}|$ is as close as possible to its upper bound $|a_{\mathfrak{p}}^2 - 4 \mu_{\mathfrak{p}}p|$, where $X^2+a_{\mathfrak{p}}X+\mu_{\mathfrak{p}} p \in A[X]$ is the characteristic polynomial of $\tau^{\text{deg} \ p}$.