Revêtements du demi-plan de Drinfeld et Langlands p-adique catégorique
Yang Pei
TL;DR
This work extends Lue Pan’s exact sequence for the Drinfeld upper half-plane to the entire tower of coverings, providing a geometric construction of the p-adic Langlands correspondence at higher levels. It introduces two Langlands-categorified functors, in Banach and locally analytic settings, and computes the associated sheaves for representations arising in the sequence, notably LL$(M)$, $ ext{O}[M]^*$, and $ ext{Omega}^1[M]^*$. A central outcome is that all proper quotients of the universal unitary completion of the supercuspidal LL$(M)$ have finite length, with a precise description of possible Jordan–Hölder factors in terms of $ ext{Pi}_{M, rak L}$ and its deformations. The paper also develops a categorical Langlands framework, connecting local Galois data to geometric objects on $P^1$ via both analytic and Banach methods, and establishes finiteness results for Ext-groups and endomorphisms that underpin the non-splitting exact sequences. Overall, the results deepen the local–geometric link between p-adic Langlands, Drinfeld towers, and representation theory of $ ext{GL}_2(Q_p)$, with implications for understanding quotients of universal completions and their analytic structure.
Abstract
We generalize to all levels of the tower of coverings of the Drinfeld upper plane an exact sequence established by Lue Pan for the first covering. Furthermore, we introduce two functors, inspired by the categorification of the $p$-adic local Langlands correspondence, in Banach and locally analytic versions respectively. We then compute the sheaves associated with the representations appearing in our sequence. As an application, we show that all proper quotients of the universal unitary completion of a supercuspidal representation have finite length.