Real closed fields with nonstandard and standard analytic structure

We consider the ordered field which is the completion of the Puiseux series field over \bR equipped with a ring of analytic functions on [-1,1]^n which contains the standard subanalytic functions as well as functions given by t-adically convergent power series, thus combining the analytic structures from [DD] and [LR3]. We prove quantifier elimination and o-minimality in the corresponding language. We extend these constructions and results to rank n ordered fields \bR_n (the maximal completions of iterated Puiseux series fields). We generalize the example of Hrushovski and Peterzil [HP] of a sentence which is not true in any o-minimal expansion of \bR (shown in [LR3] to be true in an o-minimal expansion of the Puiseux series field) to a tower of examples of sentences \sigma_n, true in \bR_n, but not true in any o-minimal expansion of any of the fields \bR,\bR_1,...,\bR_{n-1}.Comment: 15 pages, no figure

Uniform approximate functional equation for principal L-functions

We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a number field with unitary central character. We investigate the decay rate of the terms involved using the analytic conductor of Iwaniec and Sarnak as a guideline. Straightforward extensions of the results exist for products of central values. We hope that these formulae will help further understanding of the central values of principal L-functions, such as finding good bounds on their various power means, or establishing subconvexity or nonvanishing results in certain families. A crucial role in the proofs is played by recent progress on the Ramanujan--Selberg conjectures achieved by Luo, Rudnick and Sarnak. The bounds at the non-Archimedean places enter through the work of Molteni.Comment: 8 pages, LaTeX2e; v2: shorter abstract in paper, recent amsart package used; v3: small alterations in the text (most importantly correcting the definition of the analytic conductor (4)), references updated; to appear soon in Internat. Math. Res. Notice

Strictly analytic functions on p-adic analytic open sets

Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of T-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets D, and a class of admissible sets making a covering of such a D, so that we obtain a sheaf on D. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets

The theory of Hahn meromorphic functions, a holomorphic Fredholm theorem and its applications

We introduce a class of functions near zero on the logarithmic cover of the complex plane that have convergent expansions into generalized power series. The construction covers cases where non-integer powers of $z$ and also terms containing $\log z$ can appear. We show that under natural assumptions some important theorems from complex analysis carry over to the class of these functions. In particular it is possible to define a field of functions that generalize meromorphic functions and one can formulate an analytic Fredholm theorem in this class. We show that this modified analytic Fredholm theorem can be applied in spectral theory to prove convergent expansions of the resolvent for Bessel type operators and Laplace-Beltrami operators for manifolds that are Euclidean at infinity. These results are important in scattering theory as they are the key step to establish analyticity of the scattering matrix and the existence of generalized eigenfunctions at points in the spectrum.Comment: 27 page

Sur le th\'eor\`eme de l'indice des \'equations diff\'erentielles p-adiques. III

This paper works out the structure of singular points of p-adic differential equations (i.e. differential modules over the ring of functions analytic in some annulus with external radius 1). Surprisingly results look like in the formal case (differential modules over a one variable power series field) but proofs are much more involved. However, unlike in the Turritin theorem, even after ramification, in the p-adic theory there are irreducible objects of rank >1. The first part is devoted to the definition of p-adic slopes and to a decomposition along p-adic slopes theorem. The case of slope 0 (p-adic analogue of the regular singular case) was already studied in the paper with the same title but number II [Ann. of Math. (2) 146 (1997), 345-410]. The second part states several index existence theorems and index formulas. As a consequence, vertices of the Newton polygon built from p-adic slopes are proved to have integral components (analogue of the Hasse-Arf theorem). After the work of the second author, existence of index implies finitness of p-adic (Monsky-Washnitzer) cohomology for affine varieties over finite fields. The end of the paper outlines the construction of a p-adic-coefficient category over curves (over a finite field) with all needed finitness properties. In the paper with the same title but number IV [Invent. Math. 143 (2001), 629-672], further insights are given.Comment: 73 pages, French, published versio

Analytic cell decomposition and analytic motivic integration

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over \FF_q((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields $K$ with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over $K$. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of \emph{analytic} motivic integration and \emph{analytic} motivic constructible functions in the line of R. Cluckers and F. Loeser [\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]