Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures

Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full set of rational solutions for almost all finite places. It is equivalent to Katz's conjectural description of the generic Galois group. In this paper we prove an analogous statement for arithmetic q-difference equation.Comment: 45 pages. Defintive versio

An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations

We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since \deg_q and \ord_q define two valuations on {\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition.Comment: 11 pages; many language inaccuracies have been correcte

Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.Comment: The content of this paper was previously included in arXiv:1002.483

Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces

We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cX_{\eta} of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the \emph{diameter} \delta_{\cX}(\xi,U) of U at \xi\in U, the \emph{radius} \rho_{\cX}(\xi) of the point \xi\in\cX_{\eta}, the \emph{radius of convergence} of solutions of (\cE,\na) at \xi, R(\xi) = R_{\cX}(\xi, U,(\cE, \na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that \delta_{\cX}(\xi,U), \rho_{\cX}(\xi) and R_{\xi}(U,\cE,\na) are upper semicontinuous functions of \xi; for Laurent domains in the affine space, \delta_{\cX}(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function.Comment: 19 pages. We have simplified and improved the expositio

On q-summation and confluence

This paper is divided in two parts. In the first part we consider irregular singular analytic q-difference equations, with q\in ]0,1[, and we show how the Borel sum of a divergent solution of a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of such a q-difference equation. In the second part, we work under the assumption q\in ]1,+\infty[. In this case, at least four different q-Borel sums of a divergent solution of an irregular singular analytic q-difference equations are spread in the literature: under convenient assumptions we clarify the relations among them.Comment: 36 pages. Following the referee's comments, we have clarified the exposition of some proofs and corrected some misprint

On the Grothendieck conjecture on p-curvatures for q-difference equations

In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with k perfect. Then we consider the generic (also called intrinsic) Galois group in the sense of N. Katz. The result in the first part of the paper lead to a description of the generic Galois group through the properties of the functional equations obtained specializing q on roots of unity. Although no general Galois correspondence holds in this setting, in the case of positive characteristic, where nonreduced groups appear, we can prove some devissage of the generic Galois group. In the last part of the paper, we give a complete answer to the analogue of Grothendieck conjecture on $p$-curvatures for q-difference equations defined over the field of rational function K(x), where K is any finitely generated extension of \mathbb Q and q\neq 0,1: we prove that the generic Galois group of a q-difference module over K(x) always admits an adelic description in the spirit of the Grothendieck-Katz conjecture. To this purpose, if q is an algebraic number, we prove a generalization of the results by L. Di Vizio, 2002.Comment: The content of this paper was previously included in arXiv:1002.483