The Riemann-Wirtinger Integral and Monodromy-Preserving Deformation on Elliptic Curves

Mano, Toshiyuki

doi:10.1093/imrn/rnn110

Cited by 5 publications

(21 citation statements)

References 6 publications

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“…Using some results of Mano and Watanabe [49] and of Mano [46], we use the explicit differential system satisfied by the two elliptic hypergeometric integrals which are the components of Veech's map in this case to look at Veech's CH 1 -structure of an algebraic leaf Y 1 (N ) α 1 of Veech's foliation on the moduli space M 1,2 in the vicinity of one of its cusps. From an easy analysis, one deduces Theorem 1.11, Corollary 1.12 and Corollary 1.13 stated above.…”

Section: Let Dsmentioning

confidence: 99%

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Moduli spaces of flat tori and elliptic hypergeometric functions

Ghazouani,

Pirio

2016

Preprint

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In the genus one case, we make explicit some constructions of Veech [76] on flat surfaces and generalize some geometric results of Thurston [71] about moduli spaces of flat spheres as well as some equivalent ones but of an analytico-cohomological nature of Deligne and Mostow [11], which concern Appell-Lauricella hypergeometric functions. 2 S. GHAZOUANI AND L. PIRIOTable of Contents 1. Introduction 2 2. Notations and preliminary material 23 3. Twisted (co-)homology and integrals of hypergeometric type 26 4. An explicit expression for Veech's map and some consequences 50 5. Flat tori with two conical points 92 6. Miscellanea 105 Appendix A : 1-dimensional complex hyperbolic conifolds 128 Appendix B : the Gauß-Manin connection associated to Veech's map 135 References 152

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Section: Let Dsmentioning

confidence: 99%

“…1.4.3. Note also that in the papers [46,49], which we use in a crucial way in §6, the authors consider functions defined by integral representations of the form…”

Section: Two Appendices Conclude This Papermentioning

confidence: 99%

Moduli spaces of flat tori and elliptic hypergeometric functions

Ghazouani,

Pirio

2016

Preprint

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“…We show that our solutions can be written by the following integrals: for i =1,2, ͵ ␥ e −2 ͱ −1c 0 w 1 ͑w͒ −c 1 1 ͑w − t͒ c 1 s͑w − t i ;͒dw, where we set t 1 =0, t 2 = t. We call this type of integrals Riemann-Wirtinger integrals. 11,12,18 In the latter part of this section, applying Theorem 3.1 to our special solutions, we prove that the Riemann-Wirtinger integrals converge to the hypergeometric integrals as q → 0.…”

Section: Special Solutions and Their Asymptotic Behaviormentioning

confidence: 87%

“…We call this kind of integrals Riemann-Wirtinger integrals, 11,12,18 which may be regarded as an analog on elliptic curves of the hypergeometric integrals ͑we remark that similar integrals appear in the context of integral representations of solutions to the Knizhnik-Zamolodchikov-Bernard equation in conformal field theory on the elliptic curve 1,9 ͒. As is proven in Theorem 3.1, the hypergeometric integrals on a rational curve asymptotically approximate the Riemann-Wirtinger integrals on elliptic curves.…”

Section: Introductionmentioning

confidence: 91%

Studies on monodromy preserving deformation of linear differential equations on elliptic curves

Mano¹

2009

Journal of Mathematical Physics

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We study a monodromy preserving deformation ͑MPD͒ of linear differential equations on elliptic curves. As the first of our results, we describe asymptotic behaviors of solutions to the MPD system when the elliptic curve degenerates to a rational curve. As the second, we find explicit solutions for special values of parameters where the MPD system is linearizable. Our solutions are written in terms of integrals of theta functions. We also show that they converge to the hypergeometric functions applying the above asymptotic formula when the elliptic curve degenerates to a rational curve.

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“…According to Mano [14], [15], the integral (1.1) appears as a particular solution of a system of partial differential equations of the integrability condition of monodromy-preserving deformation of a Fuchsian differential equation with n singularities t 1 , . .…”

Section: Introductionmentioning

confidence: 99%

Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

Mano

Watanabe

2012

Proc. Amer. Math. Soc.

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Abstract. We study twisted cohomology and homology groups on a onedimensional complex torus minus n distinct points with coefficients in a certain local system of rank one. This local system comes from the integrand of the Riemann-Wirtinger integral introduced by Mano. We construct bases of nonvanishing cohomology and homology groups, give an interpretation as a pairing of a cohomology class and a homology class to the Riemann-Wirtinger integral, and finally describe briefly the Gauss-Manin connection on the cohomology groups.

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The Riemann-Wirtinger Integral and Monodromy-Preserving Deformation on Elliptic Curves

Cited by 5 publications

References 6 publications

Moduli spaces of flat tori and elliptic hypergeometric functions

Moduli spaces of flat tori and elliptic hypergeometric functions

Studies on monodromy preserving deformation of linear differential equations on elliptic curves

Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

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