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A DETERMINANT FORMULA FOR RELATIVE CONGRUENCE ZETA FUNCTIONS FOR CYCLOTOMIC FUNCTION FIELDS

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  15 July 2010

DAISUKE SHIOMI
DAISUKE SHIOMI*
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan (email: m05019e@math.nagoya-u.ac.jp)
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Abstract

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Rosen gave a determinant formula for relative class numbers for cyclotomic function fields, which may be regarded as an analogue of the classical Maillet determinant. In this paper, we give a determinant formula for relative congruence zeta functions for cyclotomic function fields. Our formula may be regarded as a generalization of the determinant formula for the relative class number.

Keywords

congruence zeta functioncyclotomic function fields

MSC classification

Secondary: 11M38: Zeta and $L$-functions in characteristic $p$
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 133 - 144
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Ahn, J., Choi, S. and Jung, H., ‘Class number formulae in the form of a product of determinants in function fields’, J. Aust. Math. Soc. 78(2) (2005), 227–238.Google Scholar
[2]Bae, S. and Kang, P.-L., ‘Class numbers of cyclotomic function fields’, Acta Arith. 102(3) (2002), 251–259.Google Scholar
[3]Carlitz, L. and Olson, F. R., ‘Maillet’s determinant’, Proc. Amer. Math. Soc. 6 (1955), 265–269.Google Scholar
[4]Galovich, S. and Rosen, M., ‘The class number of cyclotomic function fields’, J. Number Theory 13(3) (1981), 363–375.Google Scholar
[5]Hayes, D. R., ‘Explicit class field theory for rational function fields’, Trans. Amer. Math. Soc. 189 (1974), 77–91.Google Scholar
[6]Rosen, M., ‘A note on the relative class number in function fields’, Proc. Amer. Math. Soc. 125(5) (1997), 1299–1303.CrossRefGoogle Scholar
[7]Rosen, M., Number Theory in Function Fields (Springer, Berlin, 2002).Google Scholar
[8]Shiomi, D., ‘A determinant formula of congruence zeta functions for maximal real cyclotomic function fields’, Acta Arith. 138(3) (2009), 259–268.Google Scholar
[9]Washington, L. C., Introduction to Cyclotomic Fields (Springer, New York, 1982).CrossRefGoogle Scholar