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Jacques Sauloy

Publications and source records attributed to Jacques Sauloy.

12 recordsLinked to original sources
arXiv2023

Geometry of the space of monodromy data

In a paper published by the Annales de la Faculté de Sciences de Toulouse, with Yousuke Ohyama, we defined and studied a space of monodromy data underlying the well known derivation of q-Painlevé VI equation from q-isomonodromy conditions by Jimbo and Sakai. In a recent ArXiv preprint, Nalini Joshi and Pieter Roffelsen pursued our work. However, both our article and their preprint are ambiguous on some foundational algebro-geometric matters. We proceed here to provide sound bases.

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arXiv2020

On the vanishing of coefficients of the powers of a theta function

A result on the Galois theory of $q$-difference equations \cite{JSTALPAEN} leads to the following question: if $q \in \Cs$, $\lmod q \rmod < 1$ and if one sets $\thq(z) := \sum\limits_{m \in \Z} q^{m(m-1)/2} z^m$, can some coefficients of the Laurent series expansion of $θ_q^k(z)$, $k \in \N^*$, vanish ? We give a partial answer.

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arXiv2012

The q-analogue of the wild fundamental group and the inverse problem of the Galois theory of q-difference equations

In previous papers, we defined $q$-analogues of alien derivations for linear analytic $q$-difference equations with integral slopes and proved a density theorem (in the Galois group) and a freeness theorem. In this paper, we completely describe the wild fundamental group and apply this result to the inverse problem in $q$-difference Galois theory. The new version contains an appendix on pronilpotent completion and the main result on the direct problem is made more precise. (Submitted for publication)

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arXiv2008

Classification de modules aux différences filtrés isogradués

The local analytic classification of irregular linear q-difference equations (Ramis-Sauloy-Zhang) involves the classfication of filtered q-difference modules with a prescribed associated graded module. We prove in a more general setting the existence for this problem of a moduli scheme which is an affine space. ----- La classification analytique locale des equations aux q-differences irregulieres se ramene a la classification de modules aux q-differences filtres a gradue fixe. Nous degageons ici des hypotheses generales qui assurent l'existence d'un schema de modules pour ce probleme, qui soit de plus un espace affine.

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arXiv2004

Algebraic construction of the Stokes sheaf for irregular linear q-difference equations

The local analytic classification of irregular linear q-difference equations has recently been obtained by J.-P. Ramis, J. Sauloy and C. Zhang. Their description involves a q-analog of the Stokes sheaf and theorems of Malgrange-Sibuya type and is based on a discrete summation process due to C. Zhang. We show here another road to some of these results by algebraic means and we describe the q-Gevrey devissage of the q-Stokes sheaf by holomorphic vector bundles over an elliptic curve.

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arXiv2002

Systèmes aux q-différences singuliers réguliers: solutions canoniques, classification, matrice de connexion et monodromie

G.D. Birkhoff extended the classical Riemann-Hilbert problem for differential equations to the case of ``fuchsian'' linear $q$-difference systems with rational coefficients. He solved it in the generic case: the classifying object which he introduces is made up of the connection matrix $P$, together with the exponents at 0 and $\infty$. We follow his method in the general case, but treat symetrically 0 and $\infty$ and use no ``wildly'' growing solutions. When $q$ tends to 1, $P$ tends to a locally constant matrix $\tilde{P}$ such that the (finitely many) values $\tilde{P}(a)^{-1}\tilde{P}(b)$ are the monodromy matrices of the limiting differential system (assumed to be non resonant at 0 and $\infty$) at the singularities on $\mathbf{C}^{*}$. This text is that of preprint 148 of the Laboratoire Emile Picard (february 1999). A shorter version was published by the Annales de l'Institut Fourier, 50, 4, (2000).

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arXiv2002

La filtration canonique par les pentes d'un module aux q-différences et le gradué associé

We show that the Newton polygon of a linear q-difference equation depends only on the corresponding q-difference module. We interpret the classical results of convergent factorisation of Adams-Birkhoff-Guenther in terms of the existence of a canonical filtration. Moreover, the associated graded module has excellent functorial (resp. tensorial) properties, whence its interest for classification (resp. for Galois theory).

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arXiv2002

Galois theory of fuchsian q-difference equations

We propose an analytical approach to the Galois theory of singular regular linear q-difference systems. We use Tannaka duality along with Birkhoff's classification scheme with the connection matrix to define and describe their Galois groups. Then we describe \emph{fundamental subgroups} that give rise to a Riemann-Hilbert correspondence and to a density theorem of Schlesinger's type.

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