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Locally Analytic Functions over Completions of Fr[U]

✍ Scribed by Zifeng Yang

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 235 KB
Volume: 73
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1998.2308

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✦ Synopsis

It is well known that a continuous function f : Z p Ä Q p can be expanded by Mahler's basis: f (x)= n=0 a n ( x n ), with a n Ä 0. Amice (Bull. Soc. Math. France 92 (1964), 117 180) has established conditions on the coefficients a n for the function f to be locally analytic, as well as more general results when Z p is replaced by some compact subset of a local field. We study the function field case in this paper. The function field analogue of Mahler's basis is the Carlitz polynomials, and the corresponding result for continuous functions has already been established by Wagner (Acta Arith. 17 (1971), 389 406). We show that the conditions for a continuous function to be locally analytic in the function field case are completely similar to the Q p case. An application to using integral calculus to analytically continue characteristic p-valued L-series is briefly mentioned at the end of the paper.

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