Confluence of meromorphic solutions of q-difference equations

In this paper, we consider a q-analogue of the Borel-Laplace summation where q>1 is a real parameter. In particular, we show that the Borel-Laplace summation of a divergent power series solution of a linear differential equation can be uniformly approximated on a convenient sector, by a meromorphic solution of a corresponding family of linear q-difference equations. We perform the computations for the basic hypergeometric series. Following J. Sauloy, we prove how a fundamental set of solutions of a linear differential equation can be uniformly approximated on a convenient domain by a fundamental set of solutions of a corresponding family of linear q-difference equations. This leads us to the approximations of Stokes matrices and monodromy matrices of the linear differential equation by matrices with entries that are invariants by the multiplication by q

Notes on the Riemann Hypothesis

These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an introduction to the Riemann Hypothesis and a panoramic view of the world of zeta and L-functions. We first review Riemann's foundational article and discuss the mathematical background of the time and his possible motivations for making his famous conjecture. We discuss some of the most relevant developments after Riemann that have contributed to a better understanding of the conjecture.Comment: 2 sections added, 55 pages, 6 figure

Limits of elliptic hypergeometric biorthogonal functions

The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p->0 of the elliptic hypergeometric biorthogonal functions from Spiridonov, with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational functions or polynomials.Comment: 27 pages. This is a self-contained article which can also be seen as part 1 of a 3 part series on limits of (multivariate) elliptic hypergeometric biorthogonal functions and their measure

Notes on isocrystals

For varieties over a perfect field of characteristic p, etale cohomology with Q_l-coefficients is a Weil cohomology theory only when l is not equal to p; the corresponding role for l = p is played by Berthelot's rigid cohomology. In that theory, the coefficient objects analogous to lisse l-adic sheaves are the overconvergent F-isocrystals. This expository article is a brief user's guide for these objects, including some features shared with l-adic cohomology (purity, weights) and some features exclusive to the p-adic case (Newton polygons, convergence and overconvergence). The relationship between the two cases, via the theory of companions, will be treated in a sequel paper.Comment: 32 pages; v5: Remark 5.14 updated; section 9 split and significantly expande