×

The behaviour of rigid analytic functions around orbits of elements of \(\mathbb C_p\). (English) Zbl 1271.11110

Let \(p\) be a prime number and let \({\mathbb C}_p\) be the completion of an algebraic closure \(\bar{\mathbb Q}_p\) of the field of \(p\)-adic numbers \({\mathbb Q}_p\). Let \(G:=\mathrm{Gal}(\bar{\mathbb Q}_p /{\mathbb Q}_p)\cong \mathrm{Gal}_{\mathrm{cont}}({\mathbb C}_p/ {\mathbb Q}_p)\). For an element \(x\in {\mathbb C}_p\), \({\mathcal O}(x)\) denotes the orbit of \(x\) with respect \(G\). This paper studies the behavior of rigid analytic functions defined on \(E(x)=({\mathbb C}_p\cup\{\infty\}) \setminus {\mathcal O}(x)\). When \(x\) is algebraic, the main result is that if \(S\) is a nonempty finite subset of \({\mathbb C}_p\) and \(f: ({\mathbb C}_p\cup \{ \infty\})\setminus S \to {\mathbb C}_p\) is a rigid analytic function that is not rational, then \(f\) has infinitely many zeros in any neighborhood of at least one point of \(S\). The authors give two different proofs of this result; one using the analogue of the classical result of Picard and the second one by exploiting that certain function has a Weierstrass product uniquely determined by its family of zeros.
For the transcendental case \(x\in{\mathbb C}_p\), let \(\delta\) be a fixed positive real number, \(B(x,\delta):= \{y\in{\mathbb C}_p\mid |x-y|< \delta\}\), \(B[x,\delta]:= \{y\in{\mathbb C}_p\mid |x-y|\leq \delta\}\) and let \(f: E[x,\delta]:=B[x,\delta]\setminus{\mathcal O}(x) \to {\mathbb C}_p\) be a rigid analytic function such that the set of real numbers \(\{\epsilon \|f\|_{E[x,\epsilon, \delta]}\}_{\delta>\epsilon>0}\) is bounded where \(E[x,\epsilon,\delta]=\{y\in B[x,\delta]\mid |y-t|\geq \epsilon, \text{ for all } t\in{\mathcal O}(x)\cap B(x,\delta)\}\). Then there exists a unique \(p\)-adic measure \(\mu_f\) on \({\mathcal O}(x)\cap B[x,\delta]\) such that \(f(z)=g(z)+\int_{{\mathcal O}(x)\cap B[x,\delta]} \frac{1}{z-t} d\mu_f(t)\), \(z\in E[x,\delta]\) where \(g(z)\) is an entire function on \(B[x,\delta]\) and the representation of \(f\) is unique. It is also obtained that if \(f: E[x,\delta]\to {\mathbb C}_p\) is a rigid analytic function such that all the points of \({\mathcal O}(x)\cap B[x,\delta]\) are singular points of \(f\) then \(\liminf_{z\to x} |f(z)|=0\). The authors give examples of rigid analytic functions with and without zeros.

MSC:

11S99 Algebraic number theory: local fields
11J25 Diophantine inequalities
11S20 Galois theory
11S80 Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] V. ALEXANDRU, On the transcendence of the trace function, Proceedings of the Romanian Academy, vol. 6, no. 1 (2005), pp. 11-16. · Zbl 1114.11062
[2] V. ALEXANDRU - E. L. POPESCU - N. POPESCU, On the continuity of the trace, Proceedings of the Romanian Academy, Series A, vol. 5, no. 2 (2004), pp. 117-122. · Zbl 1150.11438
[3] V. ALEXANDRU - N. POPESCU - M. VAJAITU - A. ZAHARESCU, The p-adic measure on the orbit of an elementof Cp, Rend. Sem. Mat. Univ. Padova, Vol. 118 (2007), pp. 197-216. · Zbl 1164.11072
[4] V. ALEXANDRU - N. POPESCU - A. ZAHARESCU, On the closed subfields of Cp, J. Number Theory, 68, 2 (1998), pp. 131-150. · Zbl 0901.11035 · doi:10.1006/jnth.1997.2198
[5] V. ALEXANDRU - N. POPESCU - A. ZAHARESCU, Trace on Cp, J. Number Theory, 88, 1 (2001), pp. 13-48.
[6] V. ALEXANDRU - N. POPESCU - A. ZAHARESCU, A representation theorem for a class of rigid analytic functions, Journal de Theories des Nombres de Bordeaux, 15 (2003), pp. 639-650. · Zbl 1070.11053 · doi:10.5802/jtnb.417
[7] Y. AMICE, Les nombres p-adiques, Presse Univ. de France, Collection Sup., 1975. · Zbl 0313.12104
[8] E. ARTIN, Algebraic Numbers and Algebraic Functions, Gordon and Breach, N. Y. 1967. · Zbl 0194.35301
[9] J. AX, Zeros of polynomials over local fields - The Galois action, J. Algebra, 15 (1970), pp. 417-428. · Zbl 0216.04703 · doi:10.1016/0021-8693(70)90069-4
[10] D. BARSKY, Transformation de Cauchy p-adique et algebre d’Iwasawa, Math. Ann., 232 (1978), pp. 255-266. · Zbl 0352.12014 · doi:10.1007/BF01351430
[11] F. BRUHAT, Integration p-adique, Seminaire Bourbaki, 14e annee, 1961/ 62, no. 229.
[12] J. FRESNEL - M. VAN DER PUT, GeÂomeÂtrie Analytique Rigide et Applications, Birkhauser, Basel, 1981.
[13] N. M. KATZ, p-adic L-functions for CM-fields, Invent. Math., 48 (1978), pp. 199-297. · Zbl 0417.12003 · doi:10.1007/BF01390187
[14] M. LAZARD, Les zeÂros d’une fonction analytique d’une variable sur un corps value complet, Publications MatheÂmatiques de l’IHEÂS, 14 (1962), pp. 47-75.
[15] B. MAZUR - P. SWINNERTON-DYER, Arithmetic of Weil curves, Invent. Math., 25 (1974), pp. 1-61. · Zbl 0281.14016 · doi:10.1007/BF01389997
[16] A. M. ROBERT, A course in p-adic analysis, 2000 Springer-Verlag New- York, Inc. · Zbl 0947.11035
[17] W. RUDIN, Functional Analysis, Mc. Graw Hill Book Company, N. Y., 1973.
[18] W. H. SCHIKHOF, Ultrametric calculus. An introduction to p-adic Analysis, Cambridge Univ. Press, Cambridge, U. K., 1984. · Zbl 0553.26006
[19] M. VAÃJAÃITU - A. ZAHARESCU, Lipschitzian Distributions, Rev. Roumaine Math. Pures Appl., 53, no. 1 (2008), pp. 79-88.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.