## Cyclicity and Titchmarsh divisor problem for Drinfeld modules.(English)Zbl 1427.11056

Summary: Let $$A=\mathbb{F}_{q}[T]$$, where $$\mathbb{F}_{q}$$ is a finite field, let $$Q=\mathbb{F}_{q}(T)$$, and let $$F$$ be a finite extension of $$Q$$. Consider $$\phi$$ a Drinfeld $$A$$-module over $$F$$ of rank $$r$$. We write $$r=hed$$, where $$E$$ is the center of $$D:=\operatorname{End}_{\overline{F}}(\phi)\otimes Q$$, $$e=[E:Q]$$, 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 $$\phi$$ has good reduction at $$\wp$$, let $$\bar{\phi}$$ denote the reduction of $$\phi$$ at $$\wp$$. In this article, in particular, when $$r\neq d$$, we obtain an asymptotic formula for the number of primes $$\wp$$ of $$F$$ of degree $$x$$ for which $$\bar{\phi}(\mathbb{F}_{\wp})$$ 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.

### MSC:

 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11G15 Complex multiplication and moduli of abelian varieties

### Keywords:

Drinfeld modules; cyclicity; Titchmarsh divisor problem
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