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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A Carlitz module analogue of a conjecture of Erdos and Pomerance


Authors: Wentang Kuo and Yu-Ru Liu
Journal: Trans. Amer. Math. Soc. 361 (2009), 4519-4539
MSC (2000): Primary 11K36; Secondary 11R58, 14H05
DOI: https://doi.org/10.1090/S0002-9947-09-04723-0
Published electronically: April 6, 2009
MathSciNet review: 2506417
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Abstract: Let $A=\mathbb {F}_q[T]$ be the ring of polynomials over the finite field $\mathbb {F}_q$ and $0 \neq a \in A$. Let $C$ be the $A$-Carlitz module. For a monic polynomial $m\in A$, let $C(A/mA)$ and $\bar {a}$ be the reductions of $C$ and $a$ modulo $mA$ respectively. Let $f_a(m)$ be the monic generator of the ideal $\{f\in A, C_f(\bar {a}) =\bar {0}\}$ on $C(A/mA)$. We denote by $\omega (f_a(m))$ the number of distinct monic irreducible factors of $f_a(m)$. If $q\neq 2$ or $q=2$ and $a\neq 1, T$, or $(1+T)$, we prove that there exists a normal distribution for the quantity \[ \frac {\omega (f_a(m))-\frac {1}{2}(\log \deg m)^2}{\frac {1}{\sqrt {3}}{(\log \deg m)^{3/2}}}.\] This result is analogous to an open conjecture of Erdős and Pomerance concerning the distribution of the number of distinct prime divisors of the multiplicative order of $b$ modulo $n$, where $b$ is an integer with $|b|>1$, and $n$ a positive integer.


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Additional Information

Wentang Kuo
Affiliation: Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
MR Author ID: 698451
Email: wtkuo@math.uwaterloo.ca

Yu-Ru Liu
Affiliation: Department of Pure Mathematics, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: yrliu@math.uwaterloo.ca

Keywords: The Carlitz module, Erdős-Pomerance’s conjecture
Received by editor(s): March 3, 2006
Received by editor(s) in revised form: July 30, 2006
Published electronically: April 6, 2009
Additional Notes: The research of the first author was supported by an NSERC discovery grant.
The research of the second author was supported by an NSERC discovery grant.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.