[Paper Review] Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique

Gabriel Dospinescu, Arthur-César Le Bras|arXiv (Cornell University)|Sep 2, 2015

Advanced Algebra and Geometry37 references3 citations

TL;DR

This paper provides a geometric realization of the $p$-adic local Langlands correspondence for certain 2-dimensional de Rham representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$ by computing the de Rham cohomology of étale coverings of Drinfeld's $p$-adic upper half-plane for $\mathrm{GL}_2(\mathbf{Q}_p)$. The result confirms a conjecture by Breuil and Strauch, linking $p$-adic automorphic forms to $p$-adic Galois representations via the cohomology of these coverings.

ABSTRACT

We describe the de Rham complex of the étale coverings of Drinfeld's p-adic upper half-plane for GL_2(Q_p). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of the absolute Galois group of Q_p.

Motivation & Objective

• To provide a geometric realization of the $p$-adic local Langlands correspondence for certain 2-dimensional de Rham Galois representations.
• To compute the de Rham cohomology of étale coverings of Drinfeld's $p$-adic upper half-plane for $\mathrm{GL}_2(\mathbf{Q}_p)$, as conjectured by Breuil and Strauch.
• To establish a link between $p$-adic automorphic forms and $p$-adic Galois representations through the cohomology of the Drinfeld upper half-plane.
• To extend the framework of $p$-adic Hodge theory and $(\varphi,\Gamma)$-modules to the context of $p$-adic automorphic forms.

Proposed method

• Use of the $p$-adic uniformization of Drinfeld's upper half-plane to relate its cohomology to automorphic representations.
• Application of Emerton's compatibility between local and global Langlands correspondences in the $p$-adic setting.
• Construction of a $G$-equivariant morphism between the cohomology of the coverings and the representation space via Kirillov model and duality.
• Employment of $(\varphi,\Gamma)$-modules over the Robba ring to relate the cohomology to Galois representations.
• Use of functional analytic techniques to prove surjectivity of the key morphism $\Phi$ in the cohomological correspondence.
• Analysis of locally analytic vectors and $P$-finite vectors to construct the dual of the representation and relate it to the cohomology.

Experimental results

Research questions

• RQ1 How can the de Rham cohomology of étale coverings of Drinfeld's $p$-adic upper half-plane realize the $p$-adic local Langlands correspondence for de Rham representations of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q}_p)$?
• RQ2 What is the precise structure of the $\mathrm{GL}_2(\mathbf{Q}_p)$-representation realized in the cohomology of these coverings?
• RQ3 How does the cohomology of the Drinfeld upper half-plane relate to the $(\varphi,\Gamma)$-module associated to a de Rham Galois representation?
• RQ4 Can the $p$-adic local Langlands correspondence be geometrically realized via the cohomology of the Drinfeld tower, as conjectured by Breuil and Strauch?
• RQ5 What role do locally analytic vectors and the Kirillov model play in connecting the cohomology to the representation theory of $\mathrm{GL}_2(\mathbf{Q}_p)$?

Key findings

• The de Rham cohomology of the $n$-th covering $\Sigma_n$ of Drinfeld's upper half-plane is computed, and its structure is shown to be compatible with the $p$-adic local Langlands correspondence.
• The first cohomology group $H^1_{\mathrm{dR,c}}(\Sigma_n)$ is shown to carry a $G$-action isomorphic to the representation $\Pi(\pi,0)$, which is the $p$-adic local Langlands lift of a de Rham Galois representation $\rho$.
• The morphism $\Phi$ from the cohomology to the representation space is shown to be surjective, confirming the geometric realization of the correspondence.
• The construction establishes a $G$-equivariant isomorphism between the cohomology of the coverings and the locally analytic dual of the $p$-adic Galois representation via the Kirillov model.
• The paper confirms the Breuil-Strauch conjecture: the de Rham complex of the coverings realizes the $p$-adic local Langlands correspondence for non-trianguline de Rham representations.
• The result provides a $p$-adic analogue of Carayol's non-abelian Lubin-Tate theory, extending the classical geometric realization to the $p$-adic setting.