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 | References | Similar Articles | Additional Information

Abstract: Let be a prime power and let be the finite field with elements. For each polynomial in , one could use the Carlitz module to construct an abelian extension of , called a Carlitz cyclotomic extension. Carlitz cyclotomic extensions play a fundamental role in the study of abelian extensions of , similar to the role played by cyclotomic number fields for abelian extensions of . We are interested in the tower of Carlitz cyclotomic extensions corresponding to the powers of a fixed irreducible polynomial in . Two types of properties are obtained for the -parts of the class numbers of the fields in this tower, for a fixed prime number . One gives congruence relations between the -parts of these class numbers. The other gives lower bound for the -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

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