Extensions of de Rham Galois representations

Abstract

We construct the parabolic version and the reductive version of the integral de Rham moduli stacks of Langlands parameters ($p>3$). We allow the group to be arbitrarily ramified. We propose that the top Chow group of the reduced Emerton-Gee stack $\mathcal{X}_{^L\!G}$ is isomorphic to that of the moduli of Weil-Deligne representations valued in $^L\!B$, where $^L\!B$ is a Borel of $^L\!G$. The latter bears a concrete description by Serre weights corrected by the Kottwitz homomorphism. We explicitly define such a map using parabolic de Rham moduli stacks as the composition of a chain of tautological maps, and confirm it is an isomorphism for (1) algebraic tori, (2) unitary, orthogonal and symplectic groups, (3) tame groups when restricted to the cyclotomic-free part of the Chow groups.