Submitted Abstract
The GALF project represents an inter-European and transatlantic fundamental research project targeting several of the most relevant problems in current Number Theory. The team consists of internationally renowned and leading experts from Paris, Lille and Bordeaux in France, from Montreal in Canada and from Luxembourg, joining their complementary expertise in the project.A central problem in Number Theory is the relationship between special values of L-functions and fundamental arithmetic invariants such as regulators or Tate-Shafarevich groups. The GALF project will investigate various facets of this, including extra zeros of p-adic L-functions in the non-ordinary setting and the application of p-adic families of special cycles on Shimura varieties to the Birch and Swinnerton-Dyer conjecture. The Langlands programme is a vast international research effort combining various mathematical areas and appearing in various different forms. It relates automorphic forms (and generalisations) and representations with Galois representations and hence number theory. In particular, holomorphic weight one modular forms correspond to two dimensional Artin representations. The GALF research will target p-adic and mod p aspects of those via Galois deformation techniques.Geometrically defined Hilbert modular forms mod p of parallel and partial weight one play a very special role. In the GALF project, their attached Hecke algebras shall be related to universal deformation rings with a particular focus on the ramification properties at p.The theory of deformations of Galois representations has a geometric counter part in so-called eigenvarieties. The GALF efforts in this context especially target the local structure of the eigenvariety at classical weight one points. Moreover, properties of companion forms attached to specific weight 1 modular forms, in particular their Fourier coefficients, will be investigated both in the archimedean and the p-adic settings, with applications to explicit Class Field Theory and Kudla’s program.A part of the GALF project is to attach an L-function to an overconvergent eigenform mod p of finite slope and to examine the local behaviour at p of the attached Galois representation in view of a formulation of a T-adic Main Conjecture. The GALF efforts will also go into the study of non-critical trivial zeros of p-adic L-functions using the methods of p-adic Hodge theory. Connections with Iwasawa theory and p-adic aspects of the Plectic Conjecture shall be examined. Other important GALF research themes to be examined are the role of p-adic and plectic cohomology methods in extending the theory of complex multiplication to other settings like that of real quadratic fields.