Fuchsian structure for differential modules on an ultrametric polyannulus
This paper sets out the theory of regular singularities for \( p\)-adic differential modules in the several variables case. Even if regular singularities seem less basic in the \( p\)-adic case than in the complex case, this work is the first (and until now the only one) significant step toward a \( p\)-adic \( D\)-module theory. Let \( K \) be a complete extension of \( \mathbb{Q}_{p}\), let \( (I_{i})_{1\leq i\leq n} \) be intervals of \( \mathbb{R}^{+}\), let \( I=\prod I_{i} \) and let \( {\mathcal A}_{K}(I) \) be the ring of functions analytic in the (poly)annulus \( {\mathcal C}(I)\)=\(\{(x_{1},\cdots,x_{n});x_{i}\in I_{i}\} \) with coefficients in \( K\). A differential module, namely a free \( {\mathcal A}_{K}(I)\)-module of finite type endowed with an integrable connection, is said to have the Robba property if its local solutions near any point \( t \) in \( {\mathcal C}(I) \) do converge in the maximal polydisk \( (|x_{i}-t_{i}|<|t_{i}|)_{1\leq i<n} \) centered in \( t \) and contained in \( {\mathcal C}(I)\). The basic step in this paper is to give the definition of exponents for differential modules with the Robba property. One of the striking properties of these exponents is their invariance when specializing variables. The main result of the paper is that any differential module with the Robba property whose exponents have non-Liouville differences is Fuchsian, namely has a basis in which the matrices of the \( x_{i} {d \over dx_{i}} \) are constant. Both results and proofs are generalizations of the one variable theory as it is explained in Dwork's paper [\textit{B. Dwork}, J. Reine Angew. Math. 484, 85-126 (1997; Zbl 0870.12008)]. However this generalization is not at all straightforward. It needed, among other, to extend the explicit majoration of Dwork-Robba in the several variables setting and, above all, to prove a \( p\)-adic Hartogs theorem (analyticity with respect to each variable implies global analyticity).
scientific article; zbMATH DE number 1395896
scientific article