Commensurabilities among Lattices in PU (1,n). (AM-132), Volume 132 (Annals of Mathematics Studies)

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, __twists__ of hypergeometric functions in __n__-variables. These are treated as an (__n__+1) dimensional vector space of multivalued locally holomorphic functions defined on the space of __n__+3 tuples of distinct points on the projective line __P__ modulo, the diagonal section of Auto __P__=__m__. For __n__=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of __PU__(1,__n__). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in __PU__(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in __n__-variables of the Kummer identities for __n__-1 involving quadratic and cubic changes of the variable. The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is, twists of hypergeometric functions in n -variables. These are treated as an ( n +1) dimensional vector space of multivalued locally holomorphic functions defined on the space of n +3 tuples of distinct points on the projective line P modulo, the diagonal section of Auto P = m . For n =1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms of PU (1, n ). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices in PU (1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations in n -variables of the Kummer identities for n -1 involving quadratic and cubic changes of the variable. CONTENTS §1. Introduction §2. Picard Group and Cohomology §3. Computations for Q and Q+ §4. Lauricella’s Hypergeometric Functions §5. Gelfand’s Description of Lauricella’s Hypergeometric Functions §6. Strict Exponents §7. Characterization o f Hypergeometric-like Local Systems §8. Preliminaries on Monodromy Groups §9. Background Heuristics §10. Some Commensurability Theorems §11. Another Isogeny §12. Commensurability and Discreteness §13. An Example §14. Orbifold §15. Elliptic and Euclidean μ’s, Revisited §16. Livne’s Construction of Lattices in PU(1, 2) §17. Line Arrangements o f Complex Reflection Groups: Questions Bibliography