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Primitive submodules for Drinfeld modules

Published online by Cambridge University Press:  30 June 2015

WENTANG KUO  and
DAVID TWEEDLE
WENTANG KUO
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. e-mail: wtkuo@uwaterloo.ca
DAVID TWEEDLE
Affiliation:
Department of Mathematics and Statistics, University of The West Indies, St. Augustine, Trinidad and Tobago, West Indies. e-mail: david.tweedle@sta.uwi.edu
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Abstract

The ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 275 - 302
Copyright
Copyright © Cambridge Philosophical Society 2015 

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