Zeta functions of Picard Modular surfaces : based on lectures delivered at a CRM Workshop in the spring of 1988.

Includes Bibliography (p. 488) and references.

1. Canonical models of Picard modular surfaces;
2. Arithmetic compactification of some Shimura surfaces;
3. 2-cohomology is intersection cohomology;
4. Analytic expression for the number of points mod p;
5. Contribution of the points at the boundary;
6. The points on a Shimura variety modulo a prime of good reduction;
7. The description of the theorem;
8. Orbital integrals of U(3);
9. Remarks on Igusa theory and real orbital integrals;
10. Calculation of some orbital integrals;
11. Fundamental lemmas for U(3) and related groups;
12. The multiplicity formula for A-packets;
13. Tate classes and arithmetic quotients of the two-ball;
14. The Albanese of unitary Shimura varieties;
15. Lefschetz numbers of Hecke correspondences;
16. On the shape of the contribution of a fixed point on the boundary:
16a. The case of Q-rank one;
Appendix

Although they are central objects in the theory of diophantine equations, the zeta-functions of Hasse-Weil are not well understood. The AMS are pleased to offer this highly praised and integrated account. The contributors have provided a coherent and thorough account of necessary ideas and techniques.