Galois representations, automorphic forms and their L-functions – GALF

The GALF project represents an inter-European and transatlantic fundamental research project with the broad objective to study several of the most relevant problems in current Number Theory using p-adic methods. 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 study of p-adic properties of special cycles on algebraic varieties plays a key role in this theory.The GALF project will investigate various facets of this, including applications of special cycles on Shimura varieties to p-adic analogs of the Birch and Swinnerton-
Dyer conjecture, and p-adic aspects of the Plectic Conjecture. Connexions between Plectic Conjecture, Iwasawa theory and Fargues-Fontaine theory 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.

The Langlands programme is a vast international research effort establishing deep links between various mathematical areas and appearing in various different forms. It relates automorphic forms and representations with Galois representations and hence number theory. 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.

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 behavior at p of the attached Galois representation in view of a formulation of a T-adic Main Conjecture. 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 programme.