§ !• Review of [Z.2] on the compact case

At the end of this paper, the reader will find an appendix containing a

long list of notations and definition. Most are relatively standard and

follow [F], [Hej], [L], [Zal-2], [V]. A few, in particular those involving

^DO's or Harish-Chandra type transforms, are non-standard and follow our

earlier articles [Z2-3]. Reference numbers of the form (A10) (e.g) refer to

this appendix.

Our purpose here is to give a brief resume of the results of [Z2]-[Z3] on

generalized trace formulae, Veyl laws and prime geodesic theorems for the

compact case. These results will play a crucial role in § 5. Ve also

indicate the kinds of modifications (many quite substantial) which are

required to extend them to the present cofinite case.

The basic technique of this paper, as in [Z2]-[Z3], is a detailed

analysis of certain trace formulae. These trace formulae generalize the

p

classical formula of Selberg for the trace of a convolution operator R, (A14).

p p

The generalization is just to consider the operators rR,, i.e. Ri followed by

multiplication by an automorphic form r. When I e S

Q

(A14) and when a has

p

weight m, dL acts on the space of forms of weight 0 or, equivalently, on

2 r

L (r\h). The operator rR, can then be identified as a composition Op(r)h(R),

1 2

where A = - (j+ R ),where h(R) is introduced in (A14), and where Op(r) (A12) is

the pseudo-differential operator (jfl)0) with complete symbol a in the sense of

[Z1]-[Z3]. Ve refer to [Z3], § 1(d),for background on the ^DO's Op(r), and

to [Zl], § 1(B) for further discussion of the passage from operators of the

p

form dti to those of the form Op(r)h(R). In general, the matrix elements

(Op((r)ujt, u^) are best studied through the traces tr Op(r)h(R), while the

12