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Non-arithmetic monodromy of higher hypergeometric functions

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Abstract

We show that all the currently known non-arithmetic lattices in PU(2, 1) are monodromy groups of higher hypergeometric functions.

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References

  1. F. Beukers and G. Heckman, Monodromy for the hypergeometric functionnFn−1, Invent. Math. 95 (1989), 325–354.

    Article  MathSciNet  Google Scholar 

  2. W. Couwenberg, G. Heckman and E. Looijenga, Geometric structures on the complement of a projective arrangement, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 69–161.

    Article  MathSciNet  Google Scholar 

  3. P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89.

    Article  MathSciNet  Google Scholar 

  4. M. Deraux, Non-arithmetic ball quotients from a configuration of elliptic curves in an abelian surface, Comment. Math. Helv. 93 (2018), 533–554.

    Article  MathSciNet  Google Scholar 

  5. M. Deraux, A new non-arithmetic lattice in PU(3, 1), Algebr. Geom. Topol. 20 (2020), 925–963.

    Article  MathSciNet  Google Scholar 

  6. M. Deraux, J. R. Parker and J. Paupert, New non-arithmetic complex hyperbolic lattices, Invent. Math. 203 (2016), 681–771.

    Article  MathSciNet  Google Scholar 

  7. M. Deraux, J. R. Parker and J. Paupert, New non-arithmetic complex hyperbolic lattices II, Michigan Math. J., to appear.

  8. E. Fuchs, C. Mieri and P. Sarnak, Hyperbolic monodromy groups for the hypergeometric equation and Cartan involutions, J. Eur. Math. Soc. (JEMS) 16 (2014), 1617–1671.

    Article  MathSciNet  Google Scholar 

  9. A.H. M.Levelt, Hypergeometric Functions, Thesis, University of Amsterdam, Amsterdam, 1961.

  10. G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), 171–276.

    Article  MathSciNet  Google Scholar 

  11. G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 91–106.

    Article  MathSciNet  Google Scholar 

  12. J. R. Parker and J. Paupert, Unfaithful complex hyperbolic triangle groups II: Higher order reflections, Pacific J. Math. 239 (2009), 357–389.

    Article  MathSciNet  Google Scholar 

  13. J. R. Parker and L.-J. Sun, Complex hyperbolic triangle groups with 2-fold symmetry, Proc. Int. Geom. Cent. 10 (2017), 1–21.

    MathSciNet  MATH  Google Scholar 

  14. É. Picard, Sur les fonctions hyperfuchsiennes provenant des séries hypergéométriques de deux variables, Ann. Sci. École Norm. Sup. 62 (1885), 357–384.

    Article  Google Scholar 

  15. A. Pratoussevitch, Traces in complex hyperbolic triangle groups, Geom. Dedicata 111 (2005), 159–185.

    Article  MathSciNet  Google Scholar 

  16. H. A. Schwarz, Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt, J. Reine Angew. Math. 75 (1873), 292–335.

    MathSciNet  MATH  Google Scholar 

  17. G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304.

    Article  MathSciNet  Google Scholar 

  18. S. Singh and T. N. Venkataramana, Arithmeticity of certain symplectic hypergeometric groups, Duke Math. J. 163 (2014), 591–617.

    Article  MathSciNet  Google Scholar 

  19. K. Takeuchi, A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27 (1975), 600–612.

    Article  MathSciNet  Google Scholar 

  20. K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan 29 (1977), 91–106.

    Article  MathSciNet  Google Scholar 

  21. J. Thomae, Über die Höheren hypergeometrischen Reihen, Math. Ann. 2 (1870), 427–444.

    Article  MathSciNet  Google Scholar 

  22. J. M. Thompson, Complex Hyperbolic Triangle Groups, PhD thesis, Durham University, Durham, 2010.

  23. E. B. Vinberg, Discrete groups generated by reflections in Lobačevskiĭ spaces, Mat. Sbornik (N.S.) 72(114) (1967) 471–488; correction, ibid. 73 (115) (1967), 303; English translation: Math. USSR Sbornik 1 (1968), 429–444.

    MathSciNet  MATH  Google Scholar 

  24. E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1902.

    MATH  Google Scholar 

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Correspondence to John R. Parker.

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Parker, J.R. Non-arithmetic monodromy of higher hypergeometric functions. JAMA 142, 41–70 (2020). https://doi.org/10.1007/s11854-020-0132-5

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  • DOI: https://doi.org/10.1007/s11854-020-0132-5

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