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

In the present article, we study the overconvergent cohomology groups related to $\text{GSp}_{2g}$. We construct a pairing on the cohomology groups. On the other hand, by considering the parabolic cohomology groups and applying the strategy in [JN19], we constructed the cuspidal eigenvariety for $\text{GSp}_{2g}$. The pairing on the cohomology groups then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy in [Bel10, Chapter VI] to study the ramification locus of the cuspidal eigenvariety for $\text{GSp}_{4}$ over the corresponding weight space.