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A representation theorem for a class of rigid analytic functions. (English) Zbl 1070.11053

For a prime number \(p\), let \({\mathbb Q}_ p\) denote the field of \(p\)-adic numbers and \({\mathbb C} _ p\) the completion of a fixed algebraic closure \(\overline{\mathbb Q}_ p\) of \({\mathbb Q}_ p\). Let \(G\) be the absolute Galois group of \({\mathbb Q}_ p\): \(G=\text{Gal}(\overline{\mathbb Q}_ p/{\mathbb Q}_ p)\). The group \(G\) is canonically isomorphic with the group of all continuous automorphisms of \({\mathbb C} _ p\) over \({\mathbb Q}_ p\). The purpose of the paper under review is to obtain a representation theorem for rigid analytic functions on \(E(t,\varepsilon)= ({\mathbb C}_ p \cup \infty)\setminus C(t,\varepsilon)\) which are equivariant with respect \(G\), where \(t\) is a {Lipschitzian} element of \({\mathbb C}_ p\) and \(C(t,\varepsilon)\) denotes the \(\varepsilon\)-neighborhood of the orbit of \(t\) under the action of \(G\). When \(t\) is algebraic over \({\mathbb Q}_ p\), these functions can be described easily: sending \(t\) to the point at infinity, they correspond to the power series \(\sum_{n=0}^{\infty} a_ n z^ n\) with \(a_ n\in {\mathbb Q}_ p\) and \(\lim_{n\to\infty} \root n \of {| a_ n| }=0\). It \(t\) is transcendental over \({\mathbb Q}_ p\) it is not obvious that there exist nonconstant equivariant rigid analytic functions on \(E(t,\varepsilon)\). However, for Lipschitzian elements, the authors [J. Number Theory 88, No. 1, 13–38 (2001; Zbl 0965.11049)] construct such a function \(z\mapsto F(t,z)\). In this paper rigid analytic functions \(F_ {m,n} (t,z)\) are defined on \(E(t,\varepsilon)\) for any Lipschitzian element \(t\) and any nonnegative integer numbers \(m,n\). Then any equivariant rigid analytic function on \(E(t,\varepsilon)\) is expressed in terms of these functions \(F_{m,n}(t,z)\).

MSC:

11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
12F20 Transcendental field extensions
12J10 Valued fields
14G22 Rigid analytic geometry

Citations:

Zbl 0965.11049

References:

[1] Alexandru, V., Popescu, N., Zaharescu, A., On closed subfields of Cp. J. Number Theory68 (1998), 131-150. · Zbl 0901.11035
[2] Alexandru, V., Popescu, N., Zaharescu, A., Trace on Cp. J. Number Theory88 (2001), 13-48. · Zbl 0965.11049
[3] Amice, Y., Les nombres p-adiques, Presse Univ. de France, Collection Sup. 1975. · Zbl 0313.12104
[4] Ax, J., Zeros of Polynomials over Local Fields - The Galois Action, J. Algebra15 (1970), 417-428. · Zbl 0216.04703
[5] Popescu, N., Zaharescu, A., On the main invariant of an element over a local field, Portugaliae Mathematica54 (1997), 73-83. · Zbl 0894.11045
[6] Artin, E., Algebraic Numbers and Algebraic Functions, Gordon and Breach, New York/London/ Paris, 1967. · Zbl 0194.35301
[7] Fresnel, J., Van Der Put, M., Géometrie Analytique Rigide et Applications, Birkhäuser. Boston. Basel. Stuttgart, 1981. · Zbl 0479.14015
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