A pairing on the cuspidal eigenvariety for $\mathrm{GSp}_{2g}$ and the ramification locus

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.