The Zeta Functions of Picard Modular Surfaces


PM013
492 pages
ISBN 2-921120-08-9
1992

Robert P. Langlands and Dinakar Ramakrishnan, editors

Although they are central objects in the theory of diophantine equations, the zeta-functions of Hasse–Weil are not well understood. One large class of varieties whose zeta-functions are perhaps within reach are those attached to discrete groups, and called generically Shimura varieties. The techniques involved are difficult: representation theory and harmonic analysis; the trace formula and endoscopy; intersection cohomology and L2-cohomology; and abelian varieties with complex multiplication. The simplest Shimura varieties for which all attendant problems occur are those attached to unitary groups in three variables over imaginary quadratic fields, referred to in the present volume as Picard modular surfaces. In an attempt to render this very new domain accessible to mathematicians in related fields and to students, the contributors have provided a coherent and thorough account of the necessary ideas and techniques, many of them novel and not previously published, and of some applications.

Contents

  • R. P. Langlands and D. Ramakrishnan, Foreword
  • B. Gordon, Canonical models of Picard modular surfaces
  • M. J. Larsen, Arithmetic compactification of some Shimura surfaces
  • M. Goresky, L2 cohomology is intersection cohomology
  • J. Rogawski, Analytic expression for the number of points mod p
  • R. E. Kottwitz and M. Rapoport, Contribution of the points at the boundary
  • J. S. Milne, The points on a Shimura variety modulo a prime of good reduction
  • R. P. Langlands and D. Ramakrishnan, The description of the theorem
  • T. C. Hales, Orbital integrals on U(3)
  • R. P. Langlands, Remarks on Igusa theory and real orbital integrals
  • R. E. Kottwitz, Calculation of some orbital integrals
  • D. Blasius and J. Rogawski, Fundamental lemmas for U(3) and related groups
  • J. Rogawski, The multiplicity formula for A-packets
  • D. Blasius and J. Rogawski, Tate classes and arithmetic quotients of the two-ball
  • V. K. Murty and D. Ramakrishnan, The Albanese of unitary Shimura varieties
  • M. Goresky and R. MacPherson, Lefschetz numbers of Hecke correspondences
  • M. Rapoport, On the shape of the contribution of a fixed point on the boundary: The case of Q-rank one

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