Notes on an analogue of the Fontaine-Mazur conjecture

We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.Comment: 12 pages; minor revisions according to referees' comment

On the cyclicity of the rational points group of abelian varieties over finite fields

We propose a simple criterion to know if an abelian variety $A$ defined over a finite field $\mathbb{F}_q$ is cyclic, i.e., it has a cyclic group of rational points; this criterion is based on the endomorphism ring End$_{\mathbb{F}_q}(A)$. We also provide a criterion to know if an isogeny class is cyclic, i.e., all its varieties are cyclic; this criterion is based on the characteristic polynomial of the isogeny class. We find some asymptotic lower bounds on the fraction of cyclic $\mathbb{F}_q$-isogeny classes among certain families of them, when $q$ tends to infinity. Some of these bounds require an additional hypothesis. In the case of surfaces, we prove that this hypothesis is achieved and, over all $\mathbb{F}_q$-isogeny classes with endomorphism algebra being a field and where $q$ is an even power of a prime, we prove that the one with maximal number of rational points is cyclic and ordinary.Comment: 13 pages, this is a preliminary version, comments are welcom

Smale's mean value conjecture for finite Blaschke products

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale's mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by V. Dubinin and T. Sugawa for the dual mean value conjecture for polynomials.Comment: To appear in an issue of Journal of Analysis denoted to the Proceedings of the Conference on Modern Aspects of Complex Geometry (MindaFest)

Analysis of astrometric catalogues with vector spherical harmonics

Comparison of stellar catalogues with position and proper motion components using a decomposition on a set of orthogonal vector spherical harmonics. We show the theoretical and practical advantages of this technique as a result of invariance properties and the independence of the decomposition from a prior model. We describe the mathematical principles used to perform the spectral decomposition, evaluate the level of significance of the multipolar components and examine the transformation properties under space rotation. The principles are illustrated with a characterisation of the systematic effects in the FK5 catalogue compared to Hipparcos and with an application to the extraction of the rotation and dipole acceleration in the astrometric solution of QSOs expected from Gaia.Comment: accepted for publication in Astronomy & Astrophysic

ƿ-adic Fourier analysis

Let Dk be the ring of integers of a finite extension of Q(_p), and let h ɛ Q≥(_0) be in its value group. This thesis considers the space of locally analytic functions of order h on Ok with values in Cp-. that is, functions that are defined on each disc of radius by a convergent power series. A necessary and sufficient condition for a sequence of polynomials, with coefficient in C(_p), to be orthogonal in this space is given, generalising a result of Amice [1] . This condition is used to prove that a particular sequence of polynomials defined in Schneider Teitelbaum [19] is not orthogonal