# Commensurabilities among Lattices in PU (1,n). (AM-132), Volume 132

**Pierre Deligne & G. Daniel Mostow **

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.

First published in 1993.

*Pierre Deligne*is a Permanent Member of the Department of Mathematics at the Institute for Advanced Study in Princeton.

*G. Daniel Mostow*is Henry Ford II Professor of Mathematics at Yale University.