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 $\mathfrak{p}=pA$ 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 ${\tau }^{\text{deg}p}$.

arXiv comment19 pages