  We study the arithmetic self-intersection number of the dualizing sheaf on
arithmetic surfaces with respect to morphisms of a particular kind. We obtain
upper bounds for the arithmetic self-intersection number of the dualizing sheaf
on minimal regular models of the modular curves associated with congruence
subgroups $\Gamma_0(N)$ with square free level, as well as for the modular
curves X(N) and the Fermat curves with prime exponent.
