arXiv subjects
Publications and source records attributed to Yves André.
Many slope filtrations occur in algebraic geometry, asymptotic analysis, ramification theory, p-adic theories, geometry of numbers... These functorial filtrations, which are indexed by rational (or sometimes real) numbers, have a lot of common properties. We propose a unified abstract treatment of slope filtrations, and survey how new ties between different domains have been woven by dint of deep correspondences between different concrete slope filtrations.
It oftens occurs that Taylor coefficients of (dimensionally regularized) Feynman amplitudes $I$ with rational parameters, expanded at an integral dimension $D= D_0$, are not only periods (Belkale, Brosnan, Bogner, Weinzierl) but actually multiple zeta values (Broadhurst, Kreimer). In order to determine, at least heuristically, whether this is the case in concrete instances, the philosophy of motives - more specifically, the theory of mixed Tate motives - suggests an arithmetic approach (Kontsevich): counting points of algebraic varieties related to $I$ modulo sufficiently many primes $p$ and checking that the number of points varies polynomially in $p$. On the other hand, Kapranov has introduced a new "zeta function", the role of which is precisely to "interpolate" between zeta functions of reductions modulo different primes $p$. In this survey, we outline this circle of ideas and some of their recent developments.
Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only if the restriction of $\nabla$ to every smooth curve on $X$ is fuchsian (i.e. has only regular singularities at infinity). The "only if" part is the difficult implication. Deligne's proof is transcendental and uses Hironaka's resolution of singularities. We present a purely algebraic proof of this implication which does not use resolution beyond the case of plane curves. It relies upon a study of the formal structure of integrable connections on surfaces with (possibly irregular) singularities along a divisor with normal crossings.
We prove Malgrange's conjecture on the absence of confluence phenomena for integrable meromorphic connections. More precisely, if $ Y\to X$ is a complex-analytic fibration by smooth curves, $Z$ a hypersurface of $Y$ finite over $X$, and $\nabla$ an integrable meromorphic connection on $Y$ with poles along $Z$, then the function which attaches to {\smit x} $\in X$ the sum of the irregularities of the fiber $\nabla_{(x)}$ at the points of $Z_x$ is lower semicontinuous. The proof relies upon a study of the formal structure of integrable meromorphic connections in several variables.
Deligne's regularity criterion for an integrable connection $\nabla$ on a smooth complex algebraic variety $X$ says that $\nabla$ is regular along the irreducible divisors at infinity in some fixed normal compactification of $X$ if and only if the restriction of $\nabla$ to every smooth curve on $X$ is regular ({\it i. e.} has only regular singularities at infinity). The ``only if" part is the difficult implication. Deligne's proof is transcendental, and uses Hironaka's resolution of singularities. We give here an elementary and purely algebraic proof of this implication: it is, as far as we know, the first algebraic proof of Deligne's regularity criterion.
We develop a $p$-adic version of the so-called Grothendieck-Teichmüller theory (which studies $Gal(\bar{\bf Q}/{\bf Q})$ by means of its action on profinite braid groups or mapping class groups). For every place $v$ of $\bar{\bf Q}$, we give some geometrico-combinatorial descriptions of the local Galois group $Gal(\bar{\bf Q}_v/{\bf Q}_v)$ inside $Gal(\bar{\bf Q}/{\bf Q})$. We also show that $Gal(\bar{\bf Q}_p/{\bf Q}_p)$ is the automorphism group of an appropriate $π_1$-functor in $p$-adic geometry.
For $K$ a field, a Wedderburn $K$-linear category is a $K$-linear category $\sA$ whose radical $\sR$ is locally nilpotent and such that $\bar \sA:=\sA/\sR$ is semi-simple and remains so after any extension of scalars. We prove existence and uniqueness results for sections of the projection $\sA\to \bar\sA$, in the vein of the theorems of Wedderburn. There are two such results: one in the general case and one when $\sA$ has a monoidal structure for which $\sR$ is a monoidal ideal. The latter applies notably to Tannakian categories over a field of characteristic zero, and we get a generalisation of the Jacobson-Morozov theorem: the existence of a pro-reductive envelope $\Pred(G)$ associated to any affine group scheme $G$ over $K$. Other applications are given in this paper as well as in a forthcoming one on motives.
We show how the Galois-Picard_Vessiot theory of differential equations and difference equations, and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the construction and study of the tensor product of non commutative connections over a commutative base, without any curvature assumption. This theory provides an algebraic frame for the study of the confluence arising when the increment of a difference equation tends to 0.
One proves the Crew-Tsuzuki "p-adic local monodromy conjecture" (for local fields of characteristic p>0).
This is an introduction to $p$-adic geometry and $p$-adic analysis focusing on the theme of $p$-adic period mappings. We follow as closely as possible the development of the classical theory of complex period mappings, blending differential equations, group theory and geometry. The text ends with a detailed discussion of $p$-adic triangle groups.
Gevrey series are ubiquitous in analysis; any series satisfying some (possibly non-linear) analytic differential equation is Gevrey of some rational order. The present work stems from two observations: 1) the classical Gevrey series, e.g. generalized hypergeometric series with rational parameters, enjoy arithmetic counterparts of the Archimedean Gevrey condition; 2) the differential operators which occur in classical treatises on special functions have a rather simple structure: they are either Fuchsian, or have only two singularities, 0 and infinity, one of them regular, the other irregular with a single slope... The main idea of the paper is that the arithmetic property 1) accounts for the global analytic property 2): the existence of an injective arithmetic Gevrey solution at one point determines to a large extent the global behaviour of a differential operator with polynomial coefficients. Proofs use both p-adic and complex analysis, and a detailed arithmetic study of the Laplace transform.
In this second part, we study the Diophantine properties of values of arithmetic Gevrey series of non-zero order at algebraic points. We rely on the fact, proved in the first part, that the minimal differential operator (with polynomial coefficients) which annihilates such a series has no non-trivial singularity outside the origin and infinity. We show how to draw from this fact some transcendence properties, and recover in particular the fundamental theorem of the Siegel-Shidlovsky theory on algebraic independence of values of E-functions. The paradox of the title points out the contrast between the qualitative aspect of this new argument and the essentially quantitative aspect of the traditional approach. At last, we discuss q-analogues of the theory (theta-functions, q-exponential,...).