  We constrain the matter density $\Omega_{\mathrm{m}}$ and the amplitude of
density fluctuations $\sigma_8$ within the $\Lambda$CDM cosmological model with
shear peak statistics and angular convergence power spectra using mass maps
constructed from the first three years of data of the Dark Energy Survey (DES
Y3). We use tomographic shear peak statistics, including cross-peaks: peak
counts calculated on maps created by taking a harmonic space product of the
convergence of two tomographic redshift bins. Our analysis follows a
forward-modelling scheme to create a likelihood of these statistics using
N-body simulations, using a Gaussian process emulator. We include the following
lensing systematics: multiplicative shear bias, photometric redshift
uncertainty, and galaxy intrinsic alignment. Stringent scale cuts are applied
to avoid biases from unmodelled baryonic physics. We find that the additional
non-Gaussian information leads to a tightening of the constraints on the
structure growth parameter yielding
$S_8~\equiv~\sigma_8\sqrt{\Omega_{\mathrm{m}}/0.3}~=~0.797_{-0.013}^{+0.015}$
(68% confidence limits), with a precision of 1.8%, an improvement of ~38%
compared to the angular power spectra only case. The results obtained with the
angular power spectra and peak counts are found to be in agreement with each
other and no significant difference in $S_8$ is recorded. We find a mild
tension of $1.5 \thinspace \sigma$ between our study and the results from
Planck 2018, with our analysis yielding a lower $S_8$. Furthermore, we observe
that the combination of angular power spectra and tomographic peak counts
breaks the degeneracy between galaxy intrinsic alignment $A_{\mathrm{IA}}$ and
$S_8$, improving cosmological constraints. We run a suite of tests concluding
that our results are robust and consistent with the results from other studies
using DES Y3 data.
