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

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

Annals of Mathematics Studies (Paperback)

Paperback Published: 12th September 1993
ISBN: 9780691000961
Number Of Pages: 218

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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.

Introductionp. 1
Picard Group and Cohomologyp. 10
Computations for Q and Q+p. 17
Lauricella's Hypergeometric Functionsp. 27
Gelfand's Description of Lauricella's Hypergeometric Functionsp. 35
Strict Exponentsp. 43
Characterization of Hypergeometric-like Local Systemsp. 55
Preliminaries on Monodromy Groupsp. 71
Background Heuristicsp. 80
Some Commensurability Theoremsp. 84
Another Isogenyp. 102
Commensurability and Discretenessp. 119
An Examplep. 124
Orbifoldp. 135
Elliptic and Euclidean [mu]'s, Revisitedp. 142
Livne's Construction of Lattices in PU(1,2)p. 161
Line Arrangements of Complex Reflection Groups: Questionsp. 169
Bibliographyp. 182
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780691000961
ISBN-10: 0691000964
Series: Annals of Mathematics Studies (Paperback)
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 218
Published: 12th September 1993
Country of Publication: US
Dimensions (cm): 23.42 x 15.44  x 1.32
Weight (kg): 0.27