I will present a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the family of complex analytic functions and of the Eisenstein series. I will provide the definition of Eisenstein series, and will discuss how the result allows us to give new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set X is a subanalytic set, whose complement has codimension two in X. This is a work in collaboration with Octave Curmi and Guillaume Rond.