09:00 to 10:00 
Sean Howe (The University of Utah, US) 
padic Automorphic Forms and (big) Igusa Varieties There are two classical constructions of the space of padic modular forms — Serre’s construction via the padic completion of the qexpansions of classical modular forms, and Katz’s construction via functions on the KatzIgusa cover of the ordinary locus. In this talk, we explain how to extend both constructions to give a space of padic *automorphic* forms — Serre’s construction is generalized by completing the Kirillov models of the automorphic representations generated by modular forms, and on the geometric side this amounts to replacing Katz’s cover with the big Igusa formal scheme of CaraianiScholze. This perspective yields new insight on Hida’s finiteness for ordinary padic modular forms, overconvergence, and the differential operator theta, and suggests a common generalization of the archimedean and padic theories of autumorphic forms via functions on (big) Igusa varieties.



10:00 to 10:30 
Break 
Break 


10:30 to 11:30 
Shanwen Wang (Renmin university of China, Beijing) 
Compatibility of the Explicit Reciprocity Laws In Kato's paper on Euler system, he constructed an explicit reciprocity law to compute BlochKato's dual exponential map. In this talk, we give a variant of Kato's explicit reciprocity law and show that it is compatible with BlochKato's dual exponential map. As a consequence, the new map is compatible with Kato's original one. This is a joint work with Pierre Colmez.



11:30 to 12:00 
Break 
Break 


12:00 to 13:00 
Stefano Morra (University Paris 8, France) 
Complete cohomology for Shimura curves (Lecture 3) In this course we introduce the padically completed cohomology for Shimura curves associated to quaternion algebras over totally real fields. In particular, we study the properties of the Banach space representations obtained from them, their relation with the locally algebraic vectors, and their use in the KisinTaylorWiles patching.



13:00 to 14:00 
Break 
Lunch 


14:00 to 15:00 
Guido Kings (University of Regensburg, Germany) 
Equivariant Eisenstein classes, critical values of Hecke Lfunctions and padic interpolation (joint with Johannes Sprang) Let K be a CM field and L/K be an extension of degree n and χ be an algebraic critical Hecke character of L. Then we show that the Lvalue L(χ, 0) divided by carefully normalized ShimuraKatz periods is integral and construct a padic Lfunction for χ. This generalizes results by Damerell, Shimura and Katz for CM fields (L = K). Our method relies on a detailed analysis of new equivariant motivic Eisenstein classes and especially on the study of their de Rham realizations and differs even in the CM case from the classical approach. It turns out that the de Rham realization of these Eisenstein classes can be explicitly described in terms of EisensteinKronecker series and that working in equivariant cohomology allows to connect them with the Lfunction of χ. A further important feature of our work is an integral refinement of these classes in coherent cohomology relying on the completion of the Poincar ́e bundle. Here we were inspired by work of BannaiKobayshi in the imaginary quadratic case. With a technique pioneered in Sprang’s thesis, this approach leads directly to the construction of padic Lfunctions for χ.



15:00 to 15:30 
Break 
Break 


15:30 to 16:30 
Benjamin Schraen (Université ParisSud, France) 
On mod p representations of GL2(K) (Lecture 2) These lectures are about mod p representations of the group GL2(K) for K a finite extension of Qp and their relation with an expected local Langlands correspondence. In a first part, I will recall what is the situation for the group GL2(Qp). I will then discuss the situation for GL2(K). I will conclude by a discussion of recent results concerning the GelfandKirillov dimension and some application to deformation rings



16:30 to 17:00 
Break 
Break 


17:00 to 18:00 
Fred Diamond (King's College London, England) 
Serre's Conjecture for $\mathrm{GL}_2$ over totally real fields (Lecture 3) Serre's Conjecture (over $\mathbf{Q}$) states that every odd irreducible representation $\rho:\mathrm{Gal}(K/\mathbf{Q}) \to \mathrm{GL}_2(\mathbf{F}_{p^r})$ arises from a modular form, with level determined by the local behavior of $\rho$ at primes other than $p$, and weight determined by its local behavior at $p$. Serre's Conjecture is now a theorem of Khare and Wintenberger, but the correctness of the prediction of the level and weight was proved first (through work of Mazur, Ribet, Gross, ColemanVoloch, Gross, Edixhoven\ldots) and is key input for the argument of Khare and Wintenberger as well as for arithmetic applications of modularity results. This``refined part'' of Serre's Conjecture can also be viewed as a localglobal compatibility result in the context of a mod $p$ Langlands Programme, driving the question of what shape it takes in conjectural relations between more general Galois and automorphic representations in characteristic $p$. From this point of view, a natural ``next case'' to consider is that of $\mathrm{GL}_2$ over a totally real field $F$, relating Hilbert modular forms and twodimensional representations of Galois groups over $F$. Generalizing Serre's recipe for the weight to this setting turns out to reveal many new features. The lectures will start with a review of Serre's Conjecture over $\mathbf{Q}$, with particular emphasis on the recipe for the weight and a view to its generalizations. I'll then discuss two formulations of Serre's Conjecture in the context of $\mathrm{GL}_2$ over totally real fields and what is known about them. The first of these (formulated in successively greater generality by BuzzardDJarvis, Schein and Gee) is in terms of Hecke eigenforms in the mod $p$ Betti (or \'etale) cohomology of Shimura varieties, and links more directly to the representation theory of $\mathrm{GL}_2$ of $p$adic fields and to mod $p$ local Langlands correspondences. The second of these (formulated more recently with Sasaki) is in terms of sections of automorphic bundles in characteristic $p$, and relates more to the geometry of Shimura varieties and to noncohomological automorphic representations.


