This volume is the first in a series of three dedicated to the $p$-adic Langlands correspondence for $\mathrm{GL}_2(\mathbf{Q}_p)$. The correspondence itself is the subject of the second volume (local aspects) and the third volume (global and geometric aspects). This book begins with a general introduction by Breuil to the three volumes.

The articles in this first volume have three broad themes : the study of classical $p$-adic representations and $(\varphi,\Gamma)$-modules, the study of families of $p$-adic representations, and the study of relative $p$-adic representations.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Readership

Graduate students and research mathematicians interested in $p$-adic representations.

Table of Contents

• C. Breuil -- Introduction générale
• L. Berger -- Equations différentielles $p$-adiques et $(\varphi, N)$-modules filtrés
• F. Andreatta and O. Brinion -- Surconvergence des représentations $p$-adiques: le cas relatif
• P. Colmez -- Espaces vectoriels de dimension finie et représentations de de Rham
• P. Colmez -- Conducteur d'Artin d'une représentation de de Rham
• P. Colmez -- Représentations triangulines de dimension 2
• K. Kedlaya -- Slope filtrations for relative Frobenius
• L. Berger and P. Colmez -- Familles de représentations de de Rham et monodromie $p$-adique
• F. Andreatta and A. Iovita -- Global applications of relative $(\phi, \Gamma)$-modules I
• Appendix A. Galois cohomology via the Tate-Sen method
• Appendix B. Artin-Schreier theory
• References