Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global

Abstract : The subject of this thesis is in the p-adic Langlands programme. Let L be a finite extension of \Q_p, \rho_L a 2-dimensional p-adic representation of the Galois group \Gal(\overline{\Q_p}/L) of L, if \rho_L is the restriction of a global modular Galois representation \rho (i.e. \rho appears in the étale cohomology of Shimura curves), one can associate to \rho an admissible Banach representation \widehat{\Pi}(\rho) of \GL_2(L) by using Emerton's completed cohomology theory. Locally, if \rho_L is crystalline (and sufficiently generic), following Breuil, one can associate to \rho_L a locally analytic representation \Pi(\rho_L) of \GL_2(L). In this thesis, we prove results on the compatibility of \widehat{\Pi}(\rho) and \Pi(\rho_L), called local-global compatibility, in the unitary Shimura curves case. By locally analytic representations theory (for \GL_2(L)), the problem of local-global compatibility can be reduced to the study of eigenvarieties X constructed from the completed H^1 of unitary Shimura curves. We prove results on local-global compatibility in non-critical case by using global triangulation theory. We also study the p-adic modular forms over unitary Shimura curves, from which we construct some closed rigid subspaces of X by Coleman-Mazur's method. We prove the existence of overconvergent companion forms (over unitary Shimura curves) by using p-adic comparison theorems, from which we deduce some results on local-global compatibility in critical case.
Complete list of metadatas

Cited literature [79 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-01141624
Contributor : Abes Star <>
Submitted on : Monday, April 13, 2015 - 1:52:12 PM
Last modification on : Friday, May 17, 2019 - 10:50:24 AM
Long-term archiving on : Tuesday, April 18, 2017 - 5:30:31 PM

File

VD2_DING_YIWEN_19032015.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01141624, version 1

Collections

Citation

Yiwen Ding. Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global. Théorie des nombres [math.NT]. Université Paris Sud - Paris XI, 2015. Français. ⟨NNT : 2015PA112035⟩. ⟨tel-01141624⟩

Share

Metrics

Record views

756

Files downloads

367