INdAM Program on Serre Conjectures and the p-adic local Langlands program
The notion of cyclotomic multivariable (phi,Gamma)-modules were introduced recently in order to generalize (parts of) Colmez's work on the p-adic Langlands programme from GL_2(Qp) to groups of higher rank. More specifically: there exists a functor with promising exactness- and compatibility properties from the category of smooth mod p^n representations of the group G of Qp-points of a Qp-split reductive group with connected centre to d-variable (phi,Gamma)-modules where d is the number of simple roots of G. Further, there is a Fontaine-style equivalence of categories between these multivariable objects and p-adic representations of d-fold products of local Galois groups. There is a new proof of this fact using Drinfeld's lemma for perfectoid spaces (jt. with Annie Carter and Kiran S. Kedlaya). Usual methods like overconvergence and Herr's complex computing cohomology generalize to this context, too. In part also joint work with Aprameyo Pal.