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

 

Class numbers of cyclotomic function fields


Authors: Li Guo and Linghsueh Shu
Journal: Trans. Amer. Math. Soc. 351 (1999), 4445-4467
MSC (1991): Primary 11R29, 11R58; Secondary 11R23
Published electronically: June 10, 1999
MathSciNet review: 1608317
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Abstract: Let $q$ be a prime power and let ${\mathbb F}_q$ be the finite field with $q$ elements. For each polynomial $Q(T)$ in ${\mathbb F}_q [T]$, one could use the Carlitz module to construct an abelian extension of ${\mathbb F}_q (T)$, called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of ${\mathbb F}_q(T)$, similar to the role played by cyclotomic number fields for abelian extensions of ${\mathbb Q}$. We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in ${\mathbb F}_q [T]$. Two types of properties are obtained for the $l$-parts of the class numbers of the fields in this tower, for a fixed prime number $l$. One gives congruence relations between the $l$-parts of these class numbers. The other gives lower bound for the $l$-parts of these class numbers.


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Additional Information

Li Guo
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey 07102
Email: liguo@andromeda.rutgers.edu

Linghsueh Shu
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Address at time of publication: 228 Paseo del Rio, Moraga, California 94556
Email: shul@wellsfargo.com

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02325-9
PII: S 0002-9947(99)02325-9
Keywords: Function fields, class numbers
Received by editor(s): May 15, 1997
Published electronically: June 10, 1999
Additional Notes: The authors were supported in part by NSF Grants #DMS-9301098 and #DMS-9525833.
Article copyright: © Copyright 1999 American Mathematical Society