S. Bloch decomposed the diagonal on complex surfaces with trivial Chow group of 0- cycles in order to show Mumford’s theorem asserting that the corner piece of the Hodge filtration dies. We shall review how we could use this idea, together with Deligne’s integrality theorem, in order to find rational points over finite fields. We shall show what is precisely the obstruction to use this method in order to understand abstract singular varieties, and, most specifically, mod p reduction of varieties defined over local fields. We shall review questions and remarks on this.