Joint with Moritz Kerz. We shall report on some progress on the structure of special subloci of the space of rank one ℓ-adic local systems. If the variety is defined over the field of complex numbers, special means that the sublocus of the Betti moduli is invariant under the Galois group of the field of definition of the variety. If the variety is defined over the algebraic closure of a finite field, it means the same except that one has to understand the analog of the Betti moduli.