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

Abstract

In the present paper, we first construct a pairing on the space of analytic distributions associated with ${\text{GSp}}_{2g}$. By considering the overconvergent parabolic cohomology groups and following the work of Johansson-Newton, we construct the cuspidal eigenvariety for ${\text{GSp}}_{2g}$. 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 Bellaïche to study the ramification locus of the cuspidal eigenvariety over the corresponding weight space.