Extensions of de Rham Galois representations

TL;DR

This paper extends the theory of de Rham Galois representations by constructing new moduli stacks for arbitrary ramified groups, proposing a Chow group isomorphism with Weil-Deligne representations, and confirming it for specific groups.

Contribution

It introduces parabolic and reductive integral de Rham moduli stacks for ramified groups and establishes a Chow group isomorphism with Weil-Deligne representations in certain cases.

Findings

Constructed parabolic and reductive de Rham moduli stacks for ramified groups.

Proposed and confirmed a Chow group isomorphism for specific groups.

Explicitly defined the isomorphism map using tautological maps and verified it in key cases.

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.