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9.0

Jun 29, 2018
06/18

by
Yves André

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In the mid sixties, A. Grothendieck envisioned a vast generalization of Galois theory to systems of polynomials in several variables, motivic Galois theory, and introduced tannakian categories on this occasion. In characteristic zero, various unconditional approaches were later proposed. The most precise one, due to J. Ayoub, relies on Voevodsky theory of mixed motives and on a new tannakian theory. It sheds new light on periods of algebraic varieties, and shows in particular that polynomial...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1606.03714

9
9.0

Jun 29, 2018
06/18

by
Yves André

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Let $A$ be an abelian scheme over a smooth affine complex variety $S$, $\varOmega_A$ the $\sO_S$-module of $1$-forms of the first kind on $A$, $\sD_S\varOmega_A$ the $\sD_S$-module spanned by $\varOmega_A$ in the first algebraic De Rham cohomology module, and $\theta_\partial: \varOmega_A \to \sD_S\varOmega_A/\varOmega_A$ the Kodaira-Spencer map attached to a tangent vector field $\partial$ on $S$. We compare the rank of $\sD_S\varOmega_A/\varOmega_A$ to the maximal rank of $\theta_\partial$...

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1606.03691

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4.0

Jun 29, 2018
06/18

by
Yves Andre

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We extend Faltings's "almost purity theorem" on finite etale extensions of perfectoid algebras (as generalized by Scholze and Kedlaya-Liu) to the ramified case, without restriction on the discriminant. The key point is a perfectoid version of Riemann's extension theorem. Categorical aspects of uniform Banach algebras and perfectoid algebras are revisited beforehand.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1609.00320

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65

Sep 20, 2013
09/13

by
Yves André

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Classical applications of Galois theory concern algebraic numbers and algebraic functions. Still, the night before his duel, Galois wrote that his last mathematical thoughts had been directed toward applying his "theory of ambiguity to transcendental functions and transcendental quantities". We outline some more or less recent ideas and results in this direction.

Source: http://arxiv.org/abs/1207.3381v1

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Sep 21, 2013
09/13

by
Yves André

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We study the behaviour of semistability under tensor product in various settings: vector bundles, euclidean and hermitian lattices (alias Humbert forms or Arakelov bundles), multifiltered vector spaces. One approach to show that semistable vector bundles in characteristic zero are preserved by tensor product is based on the notion of nef vector bundles. We revisit this approach and show how far it can be transferred to hermitian lattices. J.-B. Bost conjectured that semistable hermitian...

Source: http://arxiv.org/abs/1008.1553v1

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74

Sep 22, 2013
09/13

by
Yves André

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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.

Source: http://arxiv.org/abs/0812.3921v2

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83

Sep 22, 2013
09/13

by
Yves Andre

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This is a introductory survey of some recent developments of "Galois ideas" in Arithmetic, Complex Analysis, Transcendental Number Theory and Quantum Field Theory, and of some of their interrelations.

Source: http://arxiv.org/abs/0805.2568v1

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94

Sep 22, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/math/0003239v1

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78

Sep 21, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/math/0203274v1

4
4.0

Jun 29, 2018
06/18

by
Yves Andre

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Building on his reduction to the case of an unramified complete regular local ring R of mixed characteristic, we propose a proof in the framework of P. Scholze's perfectoid theory. The main ingredients are the perfectoid "Abhyankar lemma" and an analysis of Kummer extensions of R by a thickening technique.

Topics: Algebraic Geometry, Mathematics

Source: http://arxiv.org/abs/1609.00345

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117

Sep 21, 2013
09/13

by
Yves André

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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.

Source: http://arxiv.org/abs/math/0203194v1

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119

Sep 22, 2013
09/13

by
Yves Andre

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From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number...

Source: http://arxiv.org/abs/0805.2569v1

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115

Sep 19, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/math/0701895v1

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37

Sep 19, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/math/0701894v1

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115

Jul 20, 2013
07/13

by
Yves Andre

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We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated by finitely many polynomials in a fundamental set of solutions), and affine quasi-homogeneous varieties (over the constant field) for the differential Galois group of the equation. Solution algebras can be associated to any connection over a smooth affine...

Source: http://arxiv.org/abs/1107.1179v2

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259

Sep 22, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/math/0003238v1

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98

Sep 20, 2013
09/13

by
Yves André

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Real blow-ups and more refined "zooms" play a key role in the analysis of singularities of complex-analytic differential modules. They do not change the underlying topology, but the uniform structure. This suggests to revisit the cohomology theory of differential modules with help of a suitable new notion of uniform sheaves based on the uniformity rather than the topology. We also investigate the $p$-adic situation (in particular, an analog of real blow-ups) from this uniform...

Source: http://arxiv.org/abs/1207.3380v1

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59

Sep 21, 2013
09/13

by
Yves André

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One proves the Crew-Tsuzuki "p-adic local monodromy conjecture" (for local fields of characteristic p>0).

Source: http://arxiv.org/abs/math/0203248v1

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51

Sep 21, 2013
09/13

by
Yves André

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We develop a $p$-adic version of the so-called Grothendieck-Teichm\"uller 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 $\pi_1$-functor in...

Source: http://arxiv.org/abs/math/0203181v2

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72

Sep 22, 2013
09/13

by
Yves André

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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...

Source: http://arxiv.org/abs/0812.3920v1

1
1.0

Apr 19, 2022
04/22

by
Yves André (Agrumos)

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This is a remix of SpongeBob sponge holder by _Fatguy with drain holes at the bottom so it can be sticked on a wall with double side tape. https://www.thingiverse.com/thing:5225957

Topics: Organization, stl, thingiverse

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102

Sep 23, 2013
09/13

by
Yves André; Francesco Baldassarri

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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...

Source: http://arxiv.org/abs/math/0411549v2

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65

Sep 21, 2013
09/13

by
Yves André; Bruno Kahn; avec un appendice de Peter O'Sullivan

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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...

Source: http://arxiv.org/abs/math/0203273v3