Solid locally analytic representations in mixed characteristic

TL;DR

The article extends solid locally analytic representation theory to mixed-characteristic coefficients by building a robust framework of analytic rings and solid modules. It proves that $h$-analytic $G$-representations over a Banach ring $B$ are equivalent to modules over the distribution algebra $\mathcal{D}_{h\text{-an}}(G,B)$, with cohomology described via Lazard–Serre and Kohlhaase resolutions and a cohomological comparison between $G$-cohomology and locally analytic vectors. The work also handles semilinear actions through twisted rings $B_{\blacksquare}[G]'$ and $\mathcal{D}_{h\text{-an}}(G,B)'$, establishing idempotency of distribution algebras and structural characterizations of locally analytic representations in this mixed-characteristic setting. These results lay groundwork for connections to mixed-characteristic $p$-adic Hodge theory, extended eigenvarieties, and potential Langlands-type correspondences in characteristic $p$ contexts. Overall, the framework provides a unified, homologically robust approach to studying locally analytic representations with mixed-characteristic coefficients.

Abstract

The theory of locally analytic representations of $p$-adic Lie groups with $\mathbf{Q}_p$-coefficients is a powerful tool in $p$-adic Hodge theory and in the $p$-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. Rodríguez Camargo and Rodrigues Jacinto developed in \cite{RJRC22} a solid version of this theory using the language of condensed mathematics. This provides more robust homological tools (comparison theorems, spectral sequences...) for studying these representations. In this article, we extend the solid theory of locally analytic representations to a much broader class of mixed characteristic coefficients, such as $\mathbf{F}_p((X))$ or $\mathbf{Z}_p[[X]]\langle p/X\rangle[1/X]$, as well as to semilinear representations. In the introduction, we explain how these ideas could relate to mixed characteristic phenomena in $p$-adic Hodge theory, extend eigenvarieties, and the Langlands program.

Theorems & Definitions (136)

• Theorem 1.4: Theorem \ref{['thm:full_faithfullness']}
• Theorem 1.5: Theorem \ref{['thm:cohomological_comparison']}
• Corollary 1.6
• Corollary 1.7
• Theorem 1.8: Theorem \ref{['thm:lazard_comparison']}
• Theorem 1.9
• Remark 1.10
• Definition 2.1
• Remark 2.2
• Example 2.3