§ !• 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
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
(A14) and when a has
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
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
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