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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Carlitz module analogue of a conjecture of Erdős 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
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

$\displaystyle \frac{\omega(f_a(m))-\frac{1}{2}(\log \deg m)^2}{\frac{1}{\sqrt{3}}{(\log \deg m)^{3/2}}}.$

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 $ \vert b\vert>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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04723-0
PII: S 0002-9947(09)04723-0
Keywords: The Carlitz module, Erd\H {o}s-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.