FOREWORD 45As informal is my use of the word "Conjecture": if Conjecture 1 is unimpeach-
able, my formulation of Conjecture 2, stronger than Arthur's, may be too strong
(see the discussion after Conjecture 2A in§ 2.4). The same remark applies to some
of the subsequent Conjectures. I have adopted the Popperian attitude of sticking
one's neck out: counterexamples to some of my surmises, for instance those of§ 3.4,
would lead one to refine the global picture.
Finally, an obvious weakness of these notes is the lack of a detailed descrip-
tion of Arthur's parametrization at the real prime, and its implications for the
eigenvalues of the Laplacian. This would be too long. It should be contained in
a forthcoming volume by Bergeron and myself. Neither have I touched my recent,
related geometric work with Otal and Ullmo [18, 19b, 19c].
I would like to thank James Arthur, Nicolas Bergeron, Joseph Bernstein, Don
Blasius, Jacques Dixmier, Dinakar Ramakrishnan, Peter Sarnak, J.-P. Serre, Frey-
doon Shahidi and J .-L. Waldspurger for useful discussions.