  The statistics of peaks in weak gravitational lensing maps is a promising
technique to constrain cosmological parameters in present and future surveys.
Here we investigate its power when using general extreme value statistics which
is very sensitive to the exponential tail of the halo mass function. To this
end, we use an analytic method to quantify the number of weak lensing peaks
caused by galaxy clusters, large-scale structures and observational noise.
Doing so, we further improve the method in the regime of high signal-to-noise
ratios dominated by non-linear structures by accounting for the embedding of
those counts into the surrounding shear caused by large scale structures. We
derive the extreme value and order statistics for both over-densities (positive
peaks) and under-densities (negative peaks) and provide an optimized criterion
to split a wide field survey into sub-fields in order to sample the
distribution of extreme values such that the expected objects causing the
largest signals are mostly due to galaxy clusters. We find good agreement of
our model predictions with a ray-tracing $N$-body simulation. For a Euclid-like
survey, we find tight constraints on $\sigma_8$ and $\Omega_\text{m}$ with
relative uncertainties of $\sim 10^{-3}$. In contrast, the equation of state
parameter $w_0$ can be constrained only with a $10\%$ level, and $w_\text{a}$
is out of reach even if we include redshift information.
