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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
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