Connectedness of Hecke Algebras and the Rayuela conjecture: a path to   functoriality and modularity
  Let $\rho_1$ and $\rho_2$ be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field $F$. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking $\rho_1$ to $\rho_2$. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality. 
Luis Dieulefait and Ariel Pacetti
math.NT