in [@Coh Theorem 1.2] and [@Zh Theorem 3.2] (cf. also [@Ve Section 1.1]) that generalizes work of Duke [@D]: under the assumption of a subconvex bound as above it is proved that a certain family of Heegner points and certain $d$-dimensional subvarieties are equidistributed on the Hilbert modular surface $\operatorname{SL}_2({\mathfrak{o}}_K)\backslash \mathcal{H}^d$.\
The core of Theorem \[theorem1\], from which it will follow in a fairly straightforward procedure, is the spectral decomposition of smooth shifted convolution sums which implies strong upper bounds for these sums. This is stated as Theorem \[theorem4\] in Section \[section3\] after the necessary notation is developed. We give another application of this result in Theorem \[theorem5\] of Section \[section5\]: we prove the analytic continuation and spectral decomposition of the Dirichlet series associated to shifted convolution sums with polynomial growth on vertical lines. This problem goes back to Selberg [@Se].

Section \[section2\] contains the necessary background on automorphic forms. This section turned out to be very long; although much of the material presented there is essentially known, many of the results and computations in the number field case do not seem to be explicit in the literature. Therefore we felt that it makes the paper more useful (also a 