Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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

Authors: Toshiyuki Mano and Humihiko Watanabe
Journal: Proc. Amer. Math. Soc. 140 (2012), 3867-3881
MSC (2010): Primary 33C05; Secondary 14K25, 55N25, 14F40, 32C35
Published electronically: March 8, 2012
MathSciNet review: 2944728
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study twisted cohomology and homology groups on a one-dimensional 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 non-vanishing 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.

References [Enhancements On Off] (What's this?)

  • 1. Kazuhiko Aomoto, Les équations aux différences linéaires et les intégrales des fonctions multiformes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297 (French). MR 0508176
  • 2. Kazuhiko Aomoto, On the structure of integrals of power product of linear functions, Sci. Papers College Gen. Ed. Univ. Tokyo 27 (1977), no. 2, 49–61. MR 0590052
  • 3. Kazuhiko Aomoto, Une correction et un complément à l’article: “Les équations aux différences linéaires et les intégrales des fonctions multiformes” [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 271–297; MR 58 #22688c], J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 3, 519–523 (French). MR 560011
  • 4. Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections of integrals. I, Tokyo J. Math. 5 (1982), no. 2, 249–287. MR 688126, 10.3836/tjm/1270214894
  • 5. Kazuhiko Aomoto, Configurations and invariant Gauss-Manin connections for integrals. II, Tokyo J. Math. 6 (1983), no. 1, 1–24. MR 707836, 10.3836/tjm/1270214323
  • 6. Kazuhiko Aomoto and Michitake Kita, Theory of hypergeometric functions, Springer Monographs in Mathematics, Springer-Verlag, Tokyo, 2011. With an appendix by Toshitake Kohno; Translated from the Japanese by Kenji Iohara. MR 2799182
  • 7. K. Chandrasekharan, Elliptic functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 281, Springer-Verlag, Berlin, 1985. MR 808396
  • 8. Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
  • 9. Giovanni Felder and Alexander Varchenko, Integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations, Internat. Math. Res. Notices 5 (1995), 221–233. MR 1333749, 10.1155/S1073792895000171
  • 10. Otto Forster, Lectures on Riemann surfaces, Graduate Texts in Mathematics, vol. 81, Springer-Verlag, New York-Berlin, 1981. Translated from the German by Bruce Gilligan. MR 648106
  • 11. R. C. Gunning, Lectures on Riemann surfaces, Princeton Mathematical Notes, Princeton University Press, Princeton, N.J., 1966. MR 0207977
  • 12. Ko-Ki Ito, The elliptic hypergeometric functions associated to the configuration space of points on an elliptic curve. I. Twisted cycles, J. Math. Kyoto Univ. 49 (2009), no. 4, 719–733. MR 2591113
  • 13. Nicholas M. Katz and Tadao Oda, On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ. 8 (1968), 199–213. MR 0237510
  • 14. Toshiyuki Mano, Studies on monodromy preserving deformation of linear differential equations on elliptic curves, J. Math. Phys. 50 (2009), no. 10, 103501, 21. MR 2573133, 10.1063/1.3204973
  • 15. Toshiyuki Mano, The Riemann-Wirtinger integral and monodromy-preserving deformation on elliptic curves, Int. Math. Res. Not. IMRN , posted on (2008), Art. ID rnn110, 19. MR 2448089, 10.1093/imrn/rnn110
  • 16. Riemann, B., Gesammelte mathematische Werke. In Nachträge, edited by M. Noether and W. Wirtinger. Leipzig, Germany: Teubner, 1902.
  • 17. Humihiko Watanabe, Transformation relations of matrix functions associated to the hypergeometric function of Gauss under modular transformations, J. Math. Soc. Japan 59 (2007), no. 1, 113–126. MR 2302665
  • 18. Humihiko Watanabe, Twisted homology and cohomology groups associated to the Wirtinger integral, J. Math. Soc. Japan 59 (2007), no. 4, 1067–1080. MR 2370006, 10.2969/jmsj/05941067
  • 19. Humihiko Watanabe, Linear differential relations satisfied by Wirtinger integrals, Hokkaido Math. J. 38 (2009), no. 1, 83–95. MR 2501895, 10.14492/hokmj/1248787013
  • 20. Watanabe, H., On the general transformation of the Wirtinger integral, preprint.
  • 21. Wirtinger, W., Zur Darstellung der hypergeometrischen Function durch bestimmte Integrale, Akad. Wiss. Wien. S.-B. IIa, 111 (1902), 894-900.
  • 22. Wirtinger, W., Eine neue Verallgemeinerung der hypergeometrischen Integrale, Akad. Wiss. Wien. S.-B. IIa, 112 (1903), 1721-1733.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 33C05, 14K25, 55N25, 14F40, 32C35

Retrieve articles in all journals with MSC (2010): 33C05, 14K25, 55N25, 14F40, 32C35

Additional Information

Toshiyuki Mano
Affiliation: Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan

Humihiko Watanabe
Affiliation: Kitami Institute of Technology, 165, Koencho, Kitami 090-8507, Hokkaido, Japan

Keywords: Theta function, integral representation
Received by editor(s): December 7, 2009
Received by editor(s) in revised form: August 21, 2010, January 31, 2011, April 19, 2011, and April 28, 2011
Published electronically: March 8, 2012
Additional Notes: The first author was supported in part by GCOE, Kyoto University and MEXT Grant-in-Aid for Young Scientists (B) (No. 21740118).
The second author was supported in part by Grant-in-Aid for Scientific Research (C) (No. 19540158), JSPS
Dedicated: Dedicated to Professor Keizo Yamaguchi on his sixtieth birthday
Communicated by: Ted Chinburg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.