Commensurabilities among lattices in PU(1,n)

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Commensurabilities among lattices in PU(1,n)

by Pierre Deligne and G. Daniel Mostow

（Annals of mathematics studies, no. 132）

Princeton University Press, 1993

  • : pbk

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Includes bibliographical references (p. [182]-183)

Description and Table of Contents

Volume

: pbk ISBN 9780691000961

Description

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.

Table of Contents

*Frontmatter, pg. i*CONTENTS, pg. v*ACKNOWLEDGMENTS, pg. vii* 1. INTRODUCTION, pg. 1* 2. PICARD GROUP AND COHOMOLOGY, pg. 10* 3. COMPUTATIONS FOR Q AND Q+, pg. 17* 4. LAURICELLA'S HYPERGEOMETRIC FUNCTIONS, pg. 27* 5. GELFAND'S DESCRIPTION OF HYPERGEOMETRIC FUNCTIONS, pg. 35* 6. STRICT EXPONENTS, pg. 43* 7. CHARACTERIZATION OF HYPERGEOMETRIC-LIKE LOCAL SYSTEMS, pg. 55* 8. PRELIMINARIES ON MONODROMY GROUPS, pg. 71* 9. BACKGROUND HEURISTICS, pg. 80* 10. SOME COMMENSURABILITY THEOREMS, pg. 84* 11. ANOTHER ISOGENY, pg. 102* 12. COMMENSURABILITY AND DISCRETENESS, pg. 119* 13. AN EXAMPLE, pg. 124* 14. ORBIFOLD, pg. 135* 15. ELLIPTIC AND EUCLIDEAN mu'S, REVISITED, pg. 142* 16. LIVNE'S CONSTRUCTION OF LATTICES IN PU(1,2), pg. 161* 17. LIN E ARRANGEMENTS: QUESTIONS, pg. 169*Bibliography, pg. 182
Volume

ISBN 9780691033853

Description

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.

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