Construction of rigid local systems and integral representations of their sections

Haraoka, Yoshishige; Yokoyama, Toshiaki

doi:10.1002/mana.200310360

Cited by 14 publications

(13 citation statements)

References 11 publications

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“…As a byproduct, one obtains integral expressions for the solutions of these Fuchsian systems. Compare to the work of Haraoka and Yokoyama [7,13] who use a different approach (in the case of semisimple monodromy) to obtain rigid differential systems and integral expression of such solutions.…”

Section: Introductionmentioning

confidence: 91%

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Middle convolution of Fuchsian systems and the construction of rigid differential systems

Dettweiler

Reiter

2007

Journal of Algebra

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In [M. Dettweiler, S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000) 761-798], a purely algebraic analogon of Katz' middle convolution functor (see [N.M. Katz, Rigid Local Systems, Ann. of Math. Stud., vol. 139, Princeton University Press, 1997])is given. In this paper, we find an explicit Riemann-Hilbert correspondence for this functor. This leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the Riemann-Hilbert correspondence.

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Section: Introductionmentioning

confidence: 91%

“…By the construction of mc μ , it is clear that everything can be done in an algorithmic way and is easily implemented on the computer. Moreover, one obtains the sections of the local system F in a concrete way as iterated integrals, compare to [7] and [13].…”

Section: J(b I ) =mentioning

confidence: 99%

Middle convolution of Fuchsian systems and the construction of rigid differential systems

Dettweiler

Reiter

2007

Journal of Algebra

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“…Yokoyama [14] also studied rigid local systems in another viewpoint, and we obtained integral representations of solutions of rigid Fuchsian systems [6,8] in this direction. These integral representations are related to ones in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…[6][7][8]). Although we do not give such compact forms in general, the results in this paper will give a framework for the study of an integral representation for each particular rigid Fuchsian system.…”

Section: Introductionmentioning

confidence: 99%

Topological theory for Selberg type integral associated with rigid Fuchsian systems

Haraoka

Hamaguchi

2011

Math. Ann.

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Using the analytic realization of middle convolution due to Dettweiler and Reiter, we show that any rigid Fuchsian system can be obtained as a subsystem of some generating system which has an integral representation of solutions of Selberg type. Twisted homology groups and twisted cohomology groups associated with such integrals are studied. In particular, contiguity relations and twisted cycles which realize local exponents are obtained.

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“…In another case of arbitrary dimension, but with highly restricted coefficient matrices A 0 , A 1 , one may compute n − 1 linearly independent solutions in terms of generalized hypergeometric functions [1]. More work in this direction has been done in papers by T. Yokoyama [19][20][21], Y. Haraoka [5], Haraoka and Yokoyama [6]. Even earlier, T. Sasai [15] showed in dimension n = 4 that certain cases occur which have solutions related to Appel's function F 3 .…”

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confidence: 97%

Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

Balser

Röscheisen

2009

Journal of Differential Equations

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We introduce one scalar function f of a complex variable and finitely many parameters, which allows to represent all solutions of the so-called hypergeometric system of Okubo type under the assumption that one of the two coefficient matrices has all distinct eigenvalues. In the simplest non-trivial situation, f is equal to the hypergeometric function, while in other more complicated cases it is related, but not equal, to the generalized hypergeometric functions. In general, however, this function appears to be a new higher transcendental one. The coefficients of the power series of f about the origin can be explicitly given in terms of a generalized version of the classical Pochhammer symbol, involving two square matrices that in general do not commute. The function can also be characterized by a Volterra integral equation, whose kernel is expressed in terms of the solutions of another hypergeometric system of lower dimension.

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Construction of rigid local systems and integral representations of their sections

Cited by 14 publications

References 11 publications

Middle convolution of Fuchsian systems and the construction of rigid differential systems

Middle convolution of Fuchsian systems and the construction of rigid differential systems

Topological theory for Selberg type integral associated with rigid Fuchsian systems

Solving the hypergeometric system of Okubo type in terms of a certain generalized hypergeometric function

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