Cyclicity and Titchmarsh divisor problem for Drinfeld modules

Abstract

Let $A={\mathbb{F}}_q\left[T\right]$, where ${\mathbb{F}}_q$ is a finite field, let $Q={\mathbb{F}}_q\left(T\right)$, and let $F$ be a finite extension of $Q$. Consider $\varphi$ a Drinfeld $A$-module over $F$ of rank $r$. We write $r=hed$, where $E$ is the center of $D:={End}_{\overline{F}}\left(\varphi \right)\otimes Q$, $e=[E:Q]$, and $d=[D:E{]}^{\frac{1}{2}}$. If $\wp$ is a prime of $F$, we denote by ${\mathbb{F}}_{\wp }$ the residue field at $\wp$. If $\varphi$ has good reduction at $\wp$, let $\overline{\varphi }$ denote the reduction of $\varphi$ at $\wp$. In this article, in particular, when $r\ne d$, we obtain an asymptotic formula for the number of primes $\wp$ of $F$ of degree $x$ for which $\overline{\varphi }\left({\mathbb{F}}_{\wp }\right)$ has at most $(r-1)$ cyclic components. This result answers an old question of Serre on the cyclicity of general Drinfeld $A$-modules. We also prove an analogue of the Titchmarsh divisor problem for Drinfeld modules.