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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

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Volume 2016, Issue 718


Monodromy of A-hypergeometric functions

Frits Beukers
Published Online: 2014-07-19 | DOI: https://doi.org/10.1515/crelle-2014-0054


Using Mellin–Barnes integrals we give a method to compute elements of the monodromy group of an A-hypergeometric system of differential equations. The method works under the assumption that the A-hypergeometric system has a basis of solutions consisting of Mellin–Barnes integrals. Hopefully these elements generate the full monodromy group, but this has only been verified in some special cases.


  • [1]

    Adolphson A., Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), 269–290. Google Scholar

  • [2]

    Andrews G. E., Askey R. and Roy R., Special functions, Encyclopedia Math. Appl. 71, Cambridge University Press, Cambridge 1999. Google Scholar

  • [3]

    Antipova I. A., Inversion of multidimensional Mellin transforms, Russian Math. Surveys 62 (2007), 977–979. Google Scholar

  • [4]

    Antipova I. A., Inversion of many-dimensional Mellin transforms and solutions of algebraic equations, Sb. Math. 198 (2007), 474–463. Google Scholar

  • [5]

    Beukers F., Algebraic A-hypergeometric functions, Invent. Math. 180 (2010), 589–610. Google Scholar

  • [6]

    Beukers F., Irreducibility of A-hypergeometric systems, Indag. Math. (N.S.) 21 (2011), 30–39. Google Scholar

  • [7]

    Beukers F., Notes on A-hypergeometric functions, Arithmetic and Galois theories of differential equations, Sémin. Congr. 23, Société Mathématique de France, Paris (2011), 25–61. Google Scholar

  • [8]

    Beukers F. and Heckman G., Monodromy for the hypergeometric function Fn-1n, Invent. Math. 95 (1989), 325–354. Google Scholar

  • [9]

    Chen Y.-H., Yang Y. and Yui N., Monodromy of Picard–Fuchs differential equations for Calabi–Yau threefolds, J. reine angew. Math. 616 (2008), 167–203. Google Scholar

  • [10]

    Deligne P. and Mostow G. D., Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 5–89. Google Scholar

  • [11]

    Gelfand I. M., M. I.Graev , Zelevinsky A. V., Holonomic systems of equations and series of hypergeometric type (in Russian), Dokl. Akad. Nauk SSSR 295 (1987), 14–19. Google Scholar

  • [12]

    Gelfand I. M., Kapranov M. M., Zelevinsky A. V., Generalized Euler integrals and A-hypergeometric functions, Adv. Math 84 (1990), 255–271. Google Scholar

  • [13]

    Gelfand I. M., Kapranov M. M. and Zelevinsky A. V., A correction to the paper “Hypergeometric equations and toral manifolds”, Funct. Anal. Appl. 27 (1993), 295–295. Google Scholar

  • [14]

    Gelfand I. M., Zelevinsky A. V. and Kapranov M. M., Equations of hypergeometric type and Newton polytopes (in Russian), Dokl. Akad. Nauk SSSR 300 (1988), 529–534. Google Scholar

  • [15]

    Gelfand I. M., Zelevinsky A. V. and Kapranov M. M., Hypergeometric functions and toral manifolds, Funct. Anal. Appl. 23 (1989), 94–106. Google Scholar

  • [16]

    Goto Y., The monodromy representation of Lauricella’s hypergeometric function FC, preprint 2014, http://arxiv.org/abs/1403.1654.

  • [17]

    Hanamura M. and Yoshida M., Hodge structure on twisted cohomologies and twisted Riemann inequalities I, Nagoya Math. J. 154 (1999), 123–139. Google Scholar

  • [18]

    Haraoka Y. and Ueno Y., Rigidity for Appell’s hypergeometric series F4, Funkcial. Ekvac. 51 (2008), 149–164. Google Scholar

  • [19]

    Kaneko J., Monodromy group of Appell’s system F4, Tokyo J. Math 4 (1981), 35–54. Google Scholar

  • [20]

    Kato M., Appell’s hypergeometric systems F2 with finite irreducible monodromy groups, Kyushu J. Math. 54 (2000), 279–305. Google Scholar

  • [21]

    Kita M. and Yoshida M., Intersection theory for twisted cycles, Math. Nachr. 166 (1994), 287–304; Intersection theory for twisted cycles II, Math. Nachr. 168 (1994), 171–190. Google Scholar

  • [22]

    Maclachlan N. W., Complex variable theory and transform calculus, 2nd ed., Cambridge University Press, Cambridge 1953. Google Scholar

  • [23]

    Matsumoto K., Sasaki T., Takayama N. and Yoshida M., Monodromy of the hypergeometric equation of type (3,6). I, Duke Math. J. 71 (1993), 403–426. Google Scholar

  • [24]

    Matsumoto K., Sasaki T., Takayama N. and Yoshida M., Monodromy of the hypergeometric equation of type (3,6). II: The unitary reflection group of order 293757, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 20 (1993), 617–631. Google Scholar

  • [25]

    Matsumoto K. and Yoshida M., Monodromy of Lauricella’s hypergeometric FA-system, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 13 (2014), 551–577. Google Scholar

  • [26]

    Mimachi K., Intersection numbers for twisted cycles and the connection problem associated with the generalized hypergeometric function Fnn+1, Int. Math. Res. Not. IMRN 2011 (2011), 1757–1781. Google Scholar

  • [27]

    Nilsson L., Amoebas, discriminants, and hypergeometric functions, PhD dissertation, Stockholm University 2009. Google Scholar

  • [28]

    Nørlund N. E., Hypergeometric functions, Acta Math. 94 (1955), 289–349. Google Scholar

  • [29]

    Picard E., Sur une extension aux fonctions de deux variables du problème de Riemann relatif aux fonctions hypergéométriques, Ann. Éc. Norm. Supér. (2) 10 (1881), 304–322. Google Scholar

  • [30]

    Saito M., Sturmfels B. and Takayama N., Gröbner deformations of hypergeometric differential equations, Algorithms Comput. Math. 6, Springer-Verlag, Berlin 2000. Google Scholar

  • [31]

    Sasaki T., On the finiteness of the monodromy group of the system of hypergeometric differential equations (FD), J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 565–573. Google Scholar

  • [32]

    Schulze M. and Walther U., Resonance equals reducibility for A-hypergeometric systems, Algebra Number Theory 6 (2012), 527–537. Google Scholar

  • [33]

    Smith F. C., Relations among the fundamental solutions of the generalized hypergeometric equation when p=q+1. Non-logarithmic cases, Bull. Amer. Math. Soc. 44 (1938), 429–433. Google Scholar

  • [34]

    Stienstra J., GKZ hypergeometric structures, Arithmetic and geometry around hypergeometric functions, Progr. Math. 260, Birkhäuser-Verlag, Basel (2007), 313–371. Google Scholar

  • [35]

    Takano K., Monodromy of the system for Appell’s F4, Funkcial. Ekvac. 23 (1980), 97–122. Google Scholar

  • [36]

    Terada T., Fonctions hypergéométriques F1 et fonctions automorphes I, J. Math. Soc. Japan 35 (1983), 451–475. Google Scholar

  • [37]

    Yoshida M., Hypergeometric functions, my love. Modular interpretations of configuration spaces, Aspects Math. 32, Vieweg-Verlag, Wiesbaden 1997. Google Scholar

  • [38]

    Zhdanov O. N. and Tsikh A. K., Studying the multiple Mellin–Barnes integrals by means of multidimensional residues, Sib. Math. J. 39 (1998), 245–260. Google Scholar

About the article

Received: 2013-06-26

Revised: 2014-03-11

Published Online: 2014-07-19

Published in Print: 2016-09-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 718, Pages 183–206, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2014-0054.

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