Heidelberg Lectures on Coleman Integration

Besser, Amnon

doi:10.1007/978-3-642-23905-2_1

Cited by 10 publications

(6 citation statements)

References 61 publications

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“…That is, one requires that 1 INTRODUCTION 2 the extended system satisfies all integrability conditions with respect to the principal and parametric variables. In this paper, we study such isomonodromic systems via the parameterized Picard-Vessiot (PPV) theory [8] and differential Tannakian categories [15,42,43,26,25,4].…”

Section: Introductionmentioning

confidence: 99%

Isomonodromic differential equations and differential categories

Gorchinskiy

Ovchinnikov

2014

Journal de Mathématiques Pures et Appliquées

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We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between Gauss-Manin connection and parameterized differential Galois groups. RésuméOnétudie l'isomonodromie des systèmes d'équations différentielles linéaires paramétrées et les propriétés liéesà la conjugaison des groupes algébriques différentiels linéaires en utilisant les catégories différentielles. On démontre que l'isomonodromie estéquivalenteà l'isomonodromie relativeà chaque paramètre pris séparément, si le corps des fonctions des paramètres est filtré linéairement clos. Ce résultat implique quil nest pas nécessaire de résoudre deséquations différentielles non linéaires pour tester lisomonodromie. Un contre-exemple montre qu'on ne peut pas améliorer ce résultat en affaiblissant la condition sur les paramètres. On démontre, en termes de catégories, que l'isomonodromie estéquivalente,à une conjugaison près, au fait que le groupe de Galois différentiel paramétré associé est constant, généralisant ainsi un résultat de P. Cassidy et M. Singer. On illustre nos résultats fondamentaux par une série d'exemples utilisant, en particulier, un lien entre la connexion de Gauss-Manin et les groupes de Galois différentiels paramétrés.

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Section: Introductionmentioning

confidence: 99%

Isomonodromic differential equations and differential categories

Gorchinskiy

Ovchinnikov

2014

Journal de Mathématiques Pures et Appliquées

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“…Also, we do not choose a basis of the space of derivations, allowing us to give a functorial description of the constructions involved. One reason that this generalization is needed was explained in [5], in the context of Coleman integration. The paper [55] considers the case of several derivations but chooses a basis in the space of derivations and uses a fiber functor to give the axioms of a differential Tannakian category.…”

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confidence: 99%

Parameterized Picard–Vessiot extensions and Atiyah extensions

Gillet

Gorchinskiy

Ovchinnikov

2013

Advances in Mathematics

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Generalizing Atiyah extensions, we introduce and study differential abelian tensor categories over differential rings. By a differential ring, we mean a commutative ring with an action of a Lie ring by derivations. In particular, these derivations act on a differential category. A differential Tannakian theory is developed. The main application is to the Galois theory of linear differential equations with parameters. Namely, we show the existence of a parameterized Picard-Vessiot extension and, therefore, the Galois correspondence for many differential fields with, possibly, non-differentially closed fields of constants, that is, fields of functions of parameters. Other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic, which will appear in a separate paper. This application is based on differential categories developed in the present paper, and not just differential algebraic groups and their representations.Keywords: Differential algebra, Tannakian category, Parameterized differential Galois theory, Atiyah extension 2010 MSC: primary 12H05, secondary 12H20, 13N10, 20G05, 20H20, 34M15 of constants than being differentially closed, Theorem 2.5. This assumption is satisfied by many differential fields used in practice, Theorem 2.8.The importance of the existence of a PPV extension is that it leads to a Galois correspondence, Section 8.1. The Galois group is a differential algebraic group [8,9,44,62,11,55,56,71] defined over the field of constants, which, after passing to the differential closure, coincides with the parameterized differential Galois group from [10], Corollary 8.10. The Galois correspondence, as usual, can be used to analyze how one may build the extension, step-by-step, by adjoining solutions of differential equations of lower order, corresponding to taking intermediate extensions of the base field. For example, consider the special function known as the incomplete Gamma function γ, which is the solution of a second-order parameterized differential equation [10, Example 7.2] over Q(x, t). Knowing the relevant Galois correspondence, one could show how to build the differential field extension of Q(x, t) containing γ without taking the (unnecessary and unnatural) differential closure of Q(t).The general nature of our approach will allow in the future to adapt it to the Galois theory of linear difference equations, which has numerous applications. Differential algebraic dependencies among solutions of difference equations were studied in [31,32,33,18,20,19,21,17,16,26]. Among many applications of the Galois theory, one has an algebraic proof of the differential algebraic independence of the Gamma function over C(x), [33] (the Gamma function satisfies the difference equation Γ(x + 1) = x · Γ(x)). Moreover, such a method leads to algorithms, given in the above papers, that test differential algebraic dependency with applications to solutions of even higher order difference equations (hypergeometric functions, etc.). Gen...

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“…Coleman integration is a p-adic (line) integration theory developed by Robert Coleman in the 1980s [Col82,CdS88,Col85]. Here we briefly summarise the setup for this theory (for more precise details, see, for example, [Bes12]). We also recall the key inputs, which are obtained from Kedlaya's algorithm, for performing explicit Coleman integration on hyperelliptic curves, as described in [BBK10].…”

Section: Coleman Integrationmentioning

confidence: 99%

Explicit Coleman integration in larger characteristic

Best

2019

Open Book Series

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We describe a more efficient algorithm to compute p-adic Coleman integrals on odd degree hyperelliptic curves for large primes p. The improvements come from using fast linear recurrence techniques when reducing differentials in Monsky-Washnitzer cohomology, a technique introduced by Harvey [Har07] when computing zeta functions. The complexity of our algorithm is quasilinear in √ p and is polynomial in the genus and precision. We provide timings comparing our implementation with existing approaches.

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Heidelberg Lectures on Coleman Integration

Cited by 10 publications

References 61 publications

Isomonodromic differential equations and differential categories

Isomonodromic differential equations and differential categories

Parameterized Picard–Vessiot extensions and Atiyah extensions

Explicit Coleman integration in larger characteristic

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