  The generalized Verlinde formulae expressing traces of mapping classes
corresponding to automorphisms of certain Riemann surfaces, and the congruence
relations on allowed modular representations following from them are presented.
The surfaces considered are families of algebraic curves given by suitably
chosen equations, the modular curve $\mathcal{X}(11)$, and a factor curve of
$\mathcal{X}(8)$. The examples of modular curves illustrate how the study of
arithmetic properties of suitable modular representations can be used to gain
information on automorphic properties of Riemann surfaces.
