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.

**[Ca]**L. Carlitz,*On certain functions connected with polynomials in the Galois field*, Duke Math. J.**1**(1935), 137-168.**[F-W]**Bruce Ferrero and Lawrence C. Washington,*The Iwasawa invariant 𝜇_{𝑝} vanishes for abelian number fields*, Ann. of Math. (2)**109**(1979), no. 2, 377–395. MR**528968**, 10.2307/1971116**[G-R]**Steven Galovich and Michael Rosen,*Units and class groups in cyclotomic function fields*, J. Number Theory**14**(1982), no. 2, 156–184. MR**655724**, 10.1016/0022-314X(82)90045-2**[G-K]**R. Gold and H. Kisilevsky,*On geometric 𝑍_{𝑝}-extensions of function fields*, Manuscripta Math.**62**(1988), no. 2, 145–161. MR**963002**, 10.1007/BF01278975**[Ha]**David R. Hayes,*Explicit class field theory in global function fields*, Studies in algebra and number theory, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 173–217. MR**535766****[Iw]**Kenkichi Iwasawa,*A class number formula for cyclotomic fields*, Ann. of Math. (2)**76**(1962), 171–179. MR**0154862****[Iw2]**Kenkichi Iwasawa,*On 𝑝-adic 𝐿-functions*, Ann. of Math. (2)**89**(1969), 198–205. MR**0269627****[Sh]**L. Shu,*Narrow ray class fields and partial zeta-functions*, preprint, 1994.**[Th]**Dinesh S. Thakur,*Iwasawa theory and cyclotomic function fields*, Arithmetic geometry (Tempe, AZ, 1993) Contemp. Math., vol. 174, Amer. Math. Soc., Providence, RI, 1994, pp. 157–165. MR**1299741**, 10.1090/conm/174/01858

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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:
https://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