18 STEVEN ZELDITCH
we get (1.8). Forfurther details onthederivation of (1.8), particularly on
the estimation of error terms, werefer to[Z2].
The main problem ingeneralizing (1.8) tofinite area istodeal withthe
new features inthe trace formulae. Theoperators rR, or Op(^(r)h(R) are now
0 2
traceable only onthe space L of cusp forms (A9, A17). Thetrace formula
then takes the preliminary shape:
(1.16) Tr
V
RJT
Q
= (hyp) + (ell) + (III),
where (ell) isa finite sumover elliptic conjugacy classes andwhere (III) is
a combination ofparabolic andcontinuous spectral terms. It ismost
illuminating toview (III) asa spectral term, contributing an integral of the
form
(1.17) (III) = /DDRN0p(^)E(.,^ir), E(- ,^ir)S^(J+ir)dr
GO
to thediscrete spectral term ontheleft of (1.16). Here
0p(r)E(-,w+ir), E(-,j+ir) does notactually make sense if a isan Eisenstein
series, a constant oran eigenfunction oftheresidual spectrum. Thenotation
"RNff in (1.17) stands for"renormalization," or, inother words, for the
finite part of thedivergent integral (the terminology isduetoZagier;the
rigorous version will begiven in§ 2). From this point ofview, (1.17)
really has thefamiliar form (spec) = (hyp), up to some very minor terms that
can beanalyzed as intheclassical case where a = 1.
The subtlest terms arise from thefinite parts of theinner products
0p(r)E(-,^+ir), E(-,j+ir). These give rise tothefunctions Mp((r,^,T)
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