  In the present paper, we first construct a pairing on the space of analytic
distributions associated with $\mathrm{GSp}_{2g}$. By considering the
overconvergent parabolic cohomology groups and following the work of
Johansson--Newton, we construct the cuspidal eigenvariety for
$\mathrm{GSp}_{2g}$. The pairing on the analytic distributions then induces a
pairing on some coherent sheaves of the cuspidal eigenvariety. As an
application, we follow the strategy of Bella\"{i}che to study the ramification
locus of the cuspidal eigenvariety over the corresponding weight space.
