We prove that the value of the critical probability for percolation on an Abelian Cayley graph is determined by its local structure. This is a partial positive answer to a conjecture of Schramm: the function pc defined on the set of Cayley graphs of Abelian groups of rank at least 2 is continuous for the Benjamini.Schramm topology. The proof involves group-theoretic tools and a new block argument.