Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity

Abstract

Let ρ1 and ρ2 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 ρ1 to ρ2. 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.