Groupe de travail d analyse ultramétrique

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1 Groupe de travail d analyse ultramétrique DANIEL BARSKY On Morita s p-adic Γ-function Groupe de travail d analyse ultramétrique, tome 5 ( ), exp. n o 3, p. 1-6 < 5 A1_0> Groupe de travail d analyse ultramétrique (Secrétariat mathématique, Paris), , tous droits réservés. L accès aux archives de la collection «Groupe de travail d analyse ultramétrique» implique l accord avec les conditions générales d utilisation ( Toute utilisation commerciale ou impression systématique est constitutive d une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

2 Groupe d étude d Analyse ultramétrique (Y. G. CHRISTOL, P. ROBBA) e année, 1977/78, n 3, 6 p. 7 novembre 1977 ON MORITA S p-adic 0393-FUNCTION Daniel BAESKY (*) Y, MORITA proved that, for each prime number p, one can define a p-adic continuous function r (x) from Z to Z, interpolating the sequence where m runs through the integers m prime to p with 1 $ m n. Our aim is to show how this result is related to DWORK s result on the radius of convergence of exp(x + (X~/p)).. 1. Introduction and notations. Let p be a prime, N, JR y Zp, Qp, C as usual [2]. Thé absolute value (resp thé valuation) on Zp, Qp, C is denoted by ). ) (resp, v(.) ), and normalized by )p) = p" (resp, v(p) 1 ). Thé = séquence n ~ n! t can not be thé restriction to N, of a p-adic continuous function from Z to C. However, KORITA n 2014~ (- l)~ ~~) [10] proved that thé = séquence ~ ~ prime to p, 1 ~ m n ~ can be.interpolated by a continuous p-adic function from Z to C denoted by r (x), We shall prove, by means of thé formai Laplace transform ([5] or [3]) y that this result can be deduced from DWORK s estimate of thé p-adic absolute value of thé coefficients b of thé série /n! to the existence of a ie actually prove that the result of DWORK 18 equivalent locally analytic function x - r (px) on Z, with local analyticity radius p p = p-(l/p)-(l/(p-l)), bounded b 1 on interpolating the sequence =.s;- n - (- 1)" «pn)l/pn(n!)). p r (pn). Let us denote Recall [3] that the formal Laplace transform E(f(x)) of is (*) Visting Research Associate of Princeton Université 1977/78. L exposé, au groupe d étude, a été prononcé par Philippe ROBBA.

3 3-02 The following properties of 2 are obvious from the definition : is a C -linear map from C [[x]] onto C [[x]], T -p E i3 continuous for the (p, x)-adic topology on C [[x]]. ---p Let 2. Morita s p-adic DEFINITION 1 [10]). - Let 0393 p(n) be defined, for n ~ N, by LEMMA 1. - For 1 1 $ p, i,çe have Proof. - Obvions from définition 1. The generating function of the séquence is the formal Laplace transform of Proof. the conclusion follows from lemma 1. Q. E. D. Let b p ~ (9) 1 ~ -~- X be thé formai séries at thé of -~-). It is clear thé Z (b /n!)x~ ~ jg[[x]]. LEMMA 3 (DWORK [6 J). - The coefficients b exp(x -!- (xp /p)) satisffy the following p-adic estimate n of the Taylor series of For the proof see [9 ]

4 3-03 LEMMA 4. - If L((x)) =f(x), then Proof. - Let (x) = 03A3n0 an(xn/n!), then This implies THEOREM 1. - The following identity holds Proof. - F(X) = E(exp(- X)(Ii=l exp(x + (xp/p»)). Applying lemma 4 and 2, we obtain COROLLARY 1. - For n EN, we have Proof. THEOREM 2. - The sequence n - r (n) is the restriction to N of a unique locally analytic function on Z, y with local analyticity radius Proof. - Recall that a function from Z to C is locally analytic an Z, if, for each point Z, there exist --p p E R -p such that, on n Z, ---p -p a ~ f. a is the restriction of a function f (x~ _ ~ a ( at~ (x - a~n, analytic on a n>(3 n B(a, p~. The local analyticity radius of f is Take the series By lemma 3, we get

5 is the intégral part of R, that is [a] E N and a - 1 This (17) implies. By theorem [1], 9 this implies that (l6) défines thé unique continuous functions from Z to C (actually to Q ) interpolating thé séquence n 2014~ --p -p -~p Actually (l7) gives us a stronger resuit. Let 1/p p = p-03b1 1, a ~ R, From (20) and (17), we obtain if, and only if, But, we can get more thé following theorem. THEOREM 3. - Thé following two propositions are équivalents (i) Thé séquence (pn) is thé restriction unique locally analytic function on Zp, r (px), local analyticity radius r== p1-(1/p)-(1/p-1), and bounded by one on? (z ). r? "T,-, Proof. From (22), it is clear that obtain, as in corollary 1, Consider the series

6 -3-05 We obtain, as in ( 19), Assuming (ii), we obtain (i) is true. Assuming (1 ), vie obtain so (ii) is true. This theorem shows the close relations that exist between exp(x + Q. E. Do r p (x). (24) (25) 18 that the C fficieflts of the Mahler expansion [ 1] of the function x ~ are given by the coefficients bn of the Taylor expansion of f(exp(x + (xp7p))). This suggests the possibility of a direct proof of Koblitz s formula ([4J, [7J, [8J) between Gauss surns and the p-adic f-function of Marita. and BIBLIOGRAPHY [ 1 ] AMICE ( Y. ). - Interpolation p-adique, Bull. S oc, math. France, t. 9 2, 1964, p (Thèse Sc. math. Paris, 1964). [2] AMICE (Y.). - Les nombres p-adiques. - Paris, Presses universitaires de France, 1975 (Collection Sup, "Le Mathématicien", 14). [3J BARSKY (D.). - Transformation de Cauchy p-adique et algèbre d Iwasawa, Math. Annalen, t. 232, 1978, p [4J BOYARSKY (B. ). - p-adic gamma function and Dwork cohomology (Preprint). [5J DIAMOND (J. ). - On the value of p-adic L functions at positive integers (to appear). [6J DWORK (B. ). - On the rationnality of the zeta function of an algebraic variety, Amer. J. of Math., t. 82, 1960, p [7J KATZ (N.). - Graduate course, at Princeton University, Spring [8J KOBLITZ (N.). - Communication unpublished.

7 [9] MONSKY (P.). - p-adic analysis and zeta functions. - Tokyo, Kinokuniya Book- Store, 1970 (Lectures in Mathematics ; Kyoto University, 4). [10] MORITA (Y.). - A p-adic analogue of the 0393-function, J. Fac. Sc., University Tokyo, Section 1, t. 22, 1975, p Daniel BARSKY Mathématiques, Tour Université Paris-7 2 place Jussieu PARIS CEDEX (Texte reçu le 24 juin 1978)