### Abstracts

 Fabrizio Andreatta: The spectral halo I We define a Banach space of characteristic p overconvergent Hilbert modular forms, endowed with an action of Hecke operators and of a compact operator U_p. Its characteristic power series is the mod p reduction of the characteristic power series of the U_p-operator on overconvergent Hilbert modular forms. In the ellptic case this proves a conjecture of R. Coleman. This is joint work of F. Andreatta, A. Iovita and V. Pilloni. Tobias Berger: Totally odd polarizable Galois representations and Bloch-Kato conjecture Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. This strategy can be extended to construct elements in the Bloch-Kato Selmer groups of general ±-Asai representations. I will explain how suitable congruences between automorphic forms over a CM field, whose associated Galois representations are totally odd polarizable, always give rise to elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of Deligne. I will discuss, in particular, a case involving Bianchi and low weight Siegel modular forms. Lassina Dembélé: Theta lifts and applications to modularity In this talk, we will state some conjectures on the existence of certain theta lifts for Hilbert modular forms, and explain their connection with modularity of abelian varieties of higher dimension defined over Q with trivial endomorphism rings. Fred Diamond: Geometric Serre weights for Hilbert modular forms I will discuss conjectures predicting which Galois representations should arise from Hilbert modular forms of partial weight one in characteristic p, and how some cases can be deduced from results of Andreatta and Goren. (This is joint work with S. Sasaki.) Luis Dieulefait: Automorphy for GL(2) \otimes GL(n) in the self-dual case (This is joint work with S. Arias de Reyna and Toby Gee, with the collaboration of R. Guralnick at several critical points). We will explain the main ingredients in our proof of Langlands functoriality for the tensor product of a classical modular form and an n-dimensional automorphic representations which is RAESDC (in the notation of Harris and Taylor) assuming regularity and irreducibility. The proof is only unconditional in the case of a level 1 newform, the proof for higher levels depends on assuming a strengthening of an Automorphy Lifting Theorem (A.L.T.) of Thorne for p=2. Ingredients include A.L.T., "safe" chains of congruences between newforms, good dihedral and micro good dihedral primes, criteria for adequacy of residual images (for small p, and for tensor products), level raising in GL_n, criteria for "maximal" residual image for GSp_n and GL_n, existence of lifts with prescribed local properties, including some recent results about existence of weight 0 potentially diagonalizable lifts obtained by Savitt, Herzig, Gee and Liu (see Savitt's talk). Remark: some of the tools just mentioned are taken from the paper "Potential Automorphy and change of weight" (Barnet-Lamb, Gee, Geraghty and Taylor), in particular we use the main three A.L.T. in that paper, together with three variants of them that we prove with similar arguments. Mladen Dimitrov: On Galois representations attached to weight one Hilbert modular forms in positive characteristic In this talk I will present a joint work with Gabor Wiese, in which we prove that the Galois representation attached to a parallel weight one Hilbert modular eigenform in positive characteristic is unramified outside the level of the form. Bas Edixhoven: Some elliptic curves from the real world (talk in the Mathematics colloquium) Elliptic curves are very important in my work in number theory and arithmetic geometry, and so it makes me happy to encounter them as well in other areas of mathematics, and even outside mathematics. In this non-technical lecture I will give a few examples of elliptic curves showing up in plane geometry (Poncelet), in Escher's Print Gallery'' (de Smit and Lenstra), in classical mechanics (Euler), and in the Guggenheim museum in Bilbao (minimal art by Richard Serra). The first three examples are well known, but the last one appears to be new. Hui Gao: Crystalline liftings and Serre weight conjectures We prove some new cases of weight part of Serre's conjecutures, for mod p Galois representations associated to automorphic representations on unitary groups. The approach is a generalization of the work of Gee-Liu-Savitt, namely, we study reductions of certain crystalline representations. Florian Herzig: Explicit Serre weight conjectures We discuss Serre weight conjectures for GL_n and some more general reductive groups, with a particular focus on explicit versions for Galois representations that are semisimple locally at p. This is joint work with M. Emerton, T. Gee, and D. Savitt. Kai-Wen Lan: Vanishing theorems for coherent automorphic cohomology I will review the notion of coherent cohomology of canonical and subcanonical extensions of automorphic bundles over the toroidal compactifications of locally symmetric varieties (such as Shimura varieties) and over their mixed characteristics models when they make sense, and report on some results by Junecue Suh and me on the vanishing of such cohomology, with examples emphasizing the low weight'' cases. If time permits, I will also explain how to algorithmically determine the vanishing degrees, which might be of some independent interest for users of our results. Vincent Pilloni: The spectral halo II David Savitt: Potentially crystalline lifts of certain prescribed types. We discuss several results concerning the existence of potentially crystalline lifts of prescribed Hodge-Tate weights and inertial types of a given mod p representation of the absolute Galois group of a finite extension of the p-adics. Some of these results are proved by purely local methods, and are expected to be useful in the application of automorphy lifting theorems. The proofs of other results are global, making use of automorphy lifting theorems. This is joint work with Toby Gee, Florian Herzig, and Tong Liu. Shu Sasaki: Weight one forms and conjectures of Artin, Langlands, and Fontaine-Mazur In this talk, I will explain how to prove modularity of continuous irreducible, totally odd, two-dimensional representations of the absolute Galois group of a totally real field in the case Hodge-Tate weights are all equal. I will also explain its applications to the strong Artin conjecture and a conjecture of Fontaine-Mazur. Haluk Sengun: Computations with mod p modular forms for GL(2) We will survey current methods to compute with cohomological modular forms for GL(2) over number fields that are not totally real, focusing on the mod p case. I will discuss numerical data (some collected via collaborations with A.Page and with A.Jones) supporting several related conjectures and predictions. Jack Thorne: Derived Hecke algebras and torsion Scholze has constructed Galois representations associated to torsion classes in the cohomology of GL(n) symmetric spaces. More precisely, he constructs a pseudorepresentation valued in a quotient of a Hecke algebra by a nilpotent ideal. The degree of nilpotence is bounded in terms of n and the degree of the base field. We will explain how these results can be improved by using derived categories. This is joint work with James Newton. Jacques Tilouine: A survey on the modularity of certain rank 4 motives The Fontaine-Mazur conjecture predicts the modularity of certain Galois representations. In the case of symplectic rank 4 motives over the rationals, the cases of regular and non-regular Hodge Tate weights are completely different. We'll discuss these differences and give some evidence towards the conjecture. Konstantinos Tsaltas: On congruences of modular forms over imaginary quadratic fields In this talk, I will discuss joint work with Frazer Jarvis on congruences between Galois representations associated to automorphic representations for GL(2) over imaginary quadratic fields. This will be done subject to the existence of congruences for automorphic representations for GSp(4) over the rationals, which arise as global theta lifts.

Last modification: 15 June 2015.