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Tannakian duality for AndersonDrinfeld motives and algebraic independence of Carlitz logarithms
 Invent. Math
"... Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this resul ..."
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Cited by 26 (6 self)
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Abstract. We develop a theory of Tannakian Galois groups for tmotives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given tmotive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent. Contents
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 22 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Differential Galois Theory of Linear Difference Equations
 MATHEMATISCHE ANNALEN
, 2008
"... We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general r ..."
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Cited by 15 (4 self)
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We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hölder’s theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of
On the definitions of Difference Galois groups
, 2006
"... We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of PicardVessiot extensions over fields with not necessarily al ..."
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Cited by 5 (2 self)
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We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of PicardVessiot extensions over fields with not necessarily algebraically closed subfields of constants. 1
Galois groups of basic hypergeometric equations
, 2007
"... Abstract. In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < q  < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been a ..."
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Cited by 5 (3 self)
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Abstract. In this paper we compute the Galois groups of basic hypergeometric equations. In this paper q is a complex number such that 0 < q  < 1. 1 Basic hypergeometric series and equations The theory of hypergeometric functions and equations dates back at least as far as Gauss. It has long been and is still an integral part of the mathematical literature. In particular, the Galois theory of (generalized) hypergeometric equations attracted the attention of many authors. For this issue, we refer the reader to [2, 3, 13] and to the references therein. We also single out the papers [8, 14], devoted to the calculation of some Galois groups by means of a density theorem (Ramis theorem). In this paper we focus our attention on the Galois theory of the basic hypergeometric equations, the later being natural qanalogues of the hypergeometric equations. The basic hypergeometric series φ(z) = 2φ1 (a,b;c;z) with parameters (a,b,c) ∈ (C ∗ ) 3 defined by:
padic differential equations
, 2008
"... Preface ix 0.1. About this book ix 0.2. Structure of the book x ..."
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Preface ix 0.1. About this book ix 0.2. Structure of the book x
GENERALISED HASSE VARIETIES AND THEIR JET SPACES
"... Abstract. Building on the abstract notion of prolongation developed in [7], the theory of iterative Hasse rings and schemes is introduced, simultaneously generalising difference and (Hasse)differential rings and schemes. This work provides a unified formalism for studying difference and differentia ..."
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Cited by 2 (1 self)
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Abstract. Building on the abstract notion of prolongation developed in [7], the theory of iterative Hasse rings and schemes is introduced, simultaneously generalising difference and (Hasse)differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related geometries. As an application, Hasse jet spaces are constructed generally, allowing the development of the theory for arbitrary systems of algebraic partial difference/differential equations, where constructions by earlier authors applied only to the finite dimensional case. In particular, it is shown that under appropriate separability assumptions a Hasse variety is determined by its jet spaces at a point. 1.
Determinants of elliptic hypergeometric integrals, Funkt. Analiz i ego Pril
 Funct. Analysis and its Appl
"... Abstract. We start from an interpretation of the BC2symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higherdimensional integrals and higherorder hypergeometric functions. ..."
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Cited by 2 (2 self)
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Abstract. We start from an interpretation of the BC2symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higherdimensional integrals and higherorder hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, by proving both sides satisfy the same difference equations, and that the difference equations satisfy a Galoistheoretical condition that ensures uniqueness of simultaneous solution. 1.
Relative invariants, difference equations, and the PicardVessiot theory (revised version 5)
, 2006
"... to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heartwarming encouragement all over ..."
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Cited by 1 (1 self)
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to learn mathematics from the beginning when I just started afresh my life. About four years ago he suggested to study on archimedean local zeta functions of several variables as my first research task for the Master’s thesis. I can not give enough thanks to his heartwarming encouragement all over the period of this program. I thank Professor T. Kogiso and Professor K. Sugiyama for being close advisers on prehomogeneous vector spaces. I also would like to thank Professor M. Takeuchi and Professor A. Masuoka. In June or July 2003, I had a chance to know the existence of Professor Takeuchi’s paper [2] when I was trying to understand [1]. Since I once attended to an introductory part of his lecture on Hopf algebras at 2002, the paper seemed like just what I wanted. Then I was absorbed in it and came to think that the results can be extended to involve [1]. But I could not have developed it in the presented form without the collaboration with Professor Masuoka. He suggested the viewpoint from relative Hopf modules as is described in Section 3.2, added many interesting results, and helped me to complete several proofs. Finally I would like to thank my parents for supporting my decision to study mathematics as the lifework.
Slope filtrations for relative Frobenius
, 2007
"... The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobeniussemilinear endomorphism of a finite free module over the Robba ring (the ring of germs of rigid analytic functions on an unspecified open annulus of outer radius 1) over ..."
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Cited by 1 (0 self)
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The slope filtration theorem gives a partial analogue of the eigenspace decomposition of a linear transformation, for a Frobeniussemilinear endomorphism of a finite free module over the Robba ring (the ring of germs of rigid analytic functions on an unspecified open annulus of outer radius 1) over a discretely valued field. In this paper, we give a thirdgeneration proof of this theorem, which both introduces some new simplifications (particularly the use of faithfully flat descent, to recover the theorem from a classification theorem of DieudonnéManin type) and extends the result to allow an arbitrary action on coefficients (previously the action on coefficients had to itself be a lift of an absolute Frobenius). This extension is relevant to a study of (φ,Γ)modules associated to families of padic Galois representations, as initiated by Berger and Colmez.