**Benjamin Schraen (CNRS) **http://schraen.perso.math.cnrs.fr/

**p-adic automorphic forms.** There exist interesting congruences between coefficients of modular forms. An explanation to this phenomenon is the existence of p-adic families of modular forms. The goal for this cours is to introduce the notion of p-adic modular forms and their arithmetic applications, especially in the etale cohomology of locally symmetric spaces.

**Florian Herzig (Toronto) **http://www.math.toronto.edu/~herzig/

**p-modular and locally analytic representation theory of p-adic groups. **This course will give an introduction to the mod p and p-adic representation theory of a p-adic reductive group (like GLn(Qp)). Such representations naturally appear in the p-adic Langlands program. In the first part of the course we focus on smooth representations over Fp, as first studied by Barthel-Livne. In the second part we discuss locally analytic representations over a p-adic field, as initiated by Schneider-Teitelbaum.

**Sug Woo Shin (Berkley) **https://math.berkeley.edu/~swshin/

**The local Langlands correspondence and local-global compatibility for GL(2). **We will review the local Langlands correspondence for GL(n) over p-adic fields, with the n = 2 case as a key example, and describe the statement and proof of the local-global compatibility for GL(2) over Q.

**Stefano Morra (Montpellier) **http://www.math.univ-montp2.fr/~morra/

**The interpolation of the p-adic local Langlands correspondence. **In this course we present the Kisin-Taylor-Wiles patching method, and its further improvement appearing in the work of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin toward applications to the p-adic local Langlands program. In the first part of the course we start from the notions of spaces of algebraic modular forms, Hecke algebras, local deformation rings (introduced in the courses of Schraen and Hellmann) and move further to global deformation problems and deformations over Hecke algebras. In the second part we present the Taylor-Wiles method, with a particular emphasis on the ring-theoretical properties of the modules of ''patched" modular forms with locally algebraic actions of p-adic analytic groups (multiplicity one, Cohen-Macaulayness, etc...)

**Eugen Hellmann (Munster)** https://www.uni-muenster.de/IVV5WS/Arithmetic/hellman/

**p-adic Hodge theory and deformations of Galois representations. **In this course we will introduce the deformation theory of Galois representations as well as p-adic Hodge theory with the aim of constructing and studying deformation spaces of Galois representations with a fixed p-adic Hodge type. In the rest part of the course we will discuss deformation theory with the focus on deformations of continuous representations of a profinite group - the interesting special case being the Galois group of a local or global field. In particular we will link obstruction classes and tangent vectors for deformations with Galois cohomology. In the second part of the course we want to study p-torsion and p-adic representations of the Galois group of a local p-adic field. This includes Fontaines equivalence of categories with so called etale (phi,Gamma)-modules, as well as an introduction to p-adic Hodge theory, which defines and studies certain interesting and important classes of p-adic Galois representation. We will finish this course with a discussion of potentially semi-stable deformation rings, i.e. deformation rings of local Galois representations with a prescribed p-adic Hodge theoretic behavior.