Schedule of the Workshop on the Serre conjecture

Denis Benois: Trivial zeros of p-adic L-functions and Iwasawa theory

Abstract: We prove that expected properties of Euler systems imply quite general Mazur-Tate-Teitelbaum type formulas for derivatives of p-adic L-functions. We also discuss the Iwasawa theory of p-adic representations in the trivial zero case.

Tobias Berger: Eisenstein congruences and modularity of Galois representations

Abstract: I will report on joint work with Kris Klosin (CUNY) on congruences of Eisenstein series and cuspforms modulo prime powers and its application in proving the modularity of residually reducible Galois representations.

Tommaso Centeleghe: Computing the number of certain mod p Galois representations

Abstract: We report on computations aimed to obtain for a given prime p the number R(p) of two-dimensional, odd, mod p Galois representations of Q which are irreducible and unramified outside p. Thanks to Serre's Conjecture, this amounts to compute the number of non-Eisenstein systems of Hecke eigenvalues arising from mod p modular forms of level one. Using well-known dimension formulas for modular forms (and the fact that any mod p eigensystem can be Tate-twisted to one arising from a weight ≤ p+1), an explicit upper bound U(p) of R(p) can be drawn. While discussing the reasons which might make the "error term" U(p) - R(p) large, we stress how, in practice, one can control it from above using only one of the first Hecke operators. This gives a lower bound for R(p), which coincides in many cases with R(p) itself.

Fred Diamond: Explicit Serre weights for two-dimensional Galois representations

Abstract: I will discuss joint work with Savitt making the set of Serre weights more explicit for indecomposable two-dimensional mod p representations of Galois groups over ramified extensions of Q_p. In particular the results indicate a structure on the set of such weights.

Luis Dieulefait: Some new cases of Langlands functoriality solved

Abstract: We combine the method of Propagation of Automorphy with recent Automorphy Lifting Theorems (A.L.T.) of Barnet-Lamb, Gee, Geraghty and Taylor to prove some new cases of Langlands functoriality (tensor products and symmetric powers). In particular, we establish automorphy for lots of Galois representations of G_Q of arbitrarily large dimension (and their base changed counterparts). We also prove some variants of the available A.L.T., which are needed at some steps of our proof. Remark: Some technical improvements required to extend some A.L.T. to the case of "small primes" were accomplished with the kind cooperation of R. Guralnick and T. Gee.

Wojciech Gajda: Abelian varieties and l-adic representations

Abstract: We will discuss monodromies for abelian varieties, and independence (in the sense of Serre) for families of some geometric l-adic representations over finitely generated fields.

David Geraghty: Modularity lifting beyond the "numerical coincidence" of the Taylor-Wiles method.

Abstract: Modularity lifting theorems have proven very useful since their invention by Taylor and Wiles. However, as explained in the introduction to Clozel-Harris-Taylor, they only apply in situations where a certain numerical coincidence holds. In this talk, I will describe a method to overcome this restriction. The method is conditional on the existence of Galois representations associated to integral cohomology classes (which can be established in certain cases). This is joint work with Frank Calegari.

Aftab Pande: Deformations of Galois Representations and the Theorems of Sato-Tate, Lang-Trotter and others

Abstract: We construct infinitely ramified Galois representations ρ such that the a_l(ρ)'s have distributions in contrast to the statements of Sato-Tate, Lang-Trotter and others. Using similar methods we deform a residual Galois representation for number fields and obtain an infinitely ramified representation with very large image, generalising a result of Ramakrishna.

Florian Pop: Faithful representations of absolute Galois groups

Abstract:  In his "Esquisse d'un Programme" Grothendieck suggested to studythe absolute Galois group of the rationals via its representations on the algebraic fundamental group of natural categories of varieties, e.g., the Teichmueller modular tower. This lead to the intensive study of the so called Grothendieck-Teichmueller group and its variants, and the I/OM (Ihara/Oda-Matsumoto conjecture). I plan to explain variants of I/OM, and discuss its state of the art.

Dinakar Ramakrishnan: Picard modular surfaces, residual Albanese quotients, and rational points

Abstract: The Picard modular surfaces X are at the crossroads of rich interplay between geometry, Galois representations, and automorphic forms on G=U(2,1) associated to an imaginary quadratic field K. The talk will introduce an ongoing project with M. Dimitrov on the quotients of the albanese variety Alb(X) coming from residual automorphic forms on G, give examples with finite Mordell-Weil group, and investigate possible consequences, inspired by classical arguments of Mazur, for the K-rational points on X.

Mehmet Sengun: Mod p Cohomology of Bianchi Groups and Mod p Galois Representations

Abstract: Given an imaginary quadratic field K with ring of integers R, consider the Bianchi group GL(2,R). It is suspected since the numerical investigations of Fritz Grunewald in the late 1970's that there is a connection between the Hecke eigenclasses in the mod p cohomology of (congruence subgroups of) Bianchi groups and the 2-dimensional continuous mod p representations of the absolute Galois group of K. Most of the basic tools used for establishing this connection (and its surrounding problems) in the classical setting fail to work in the setting of Bianchi groups. The situation has an extra layer of complication by the fact that there are "genuinely mod p" Hecke eigenvalue systems, resulting from the existence of torsion in the integral cohomology.
In this talk I will elaborate on the above paragraph, presenting numerical examples for illustration. Towards the end, I will also talk about how the "even" 2-dimensional continuous mod p representations of the absolute Galois group of Q come into the picture.

Jack Thorne: Symmetric power functoriality for GL(2)

Abstract: We discuss some new automorphy lifting theorems, and their applications to the existence of new cases of Langlands' functoriality for GL(2). This is joint work with L. Clozel.

Jacques Tilouine: Image of Galois and congruence ideals, a program joint work with H. Hida

Abstract: In a recent preprint, H. Hida showed that the image of the Galois representation associated to a non CM Hida family contains a congruence subgroup of GL₂ over Λ, whose level is given in terms of p-adic L-functions. We try to generalize this to Hida families for bigger groups, replacing p-adic L-functions by congruence ideals.

Jeanine Van Order: Critical values of GL₂ Rankin-Selberg L-functions

Abstract: The aim of this talk is to explain the subtle but powerful link between the algebraicity of critical values of automorphic L-functions, the existence of associated p-adic L-functions, and the generic nonvanishing of these values, particularly in the setting of Rankin-Selberg L-functions of GL₂ over a totally real number field. More precisely, the aim is to explain how to extend the conjectures of Mazur to the non self-dual setting, thereby extending the works of Vatsal, Cornut, and Cornut-Vatsal, via a combination of techniques from Iwasawa theory, analytic number theory, and the theory of automorphic forms.
If time permits, then some open problems will also be introduced.