Blocks for mod p representations of GL2(Qp)

Let π1 and π2 be absolutely irreducible smooth representations of G=GL2(Qp) with a central character, defined over a finite field of characteristic p. We show that if there exists a non-split extension between π1 and π2 then they both appear as subquotients of the reduction modulo p of a unit ball in a crystalline Banach space representation of G. The results of Berger-Breuil describe such reductions and allow us to organize the irreducible representation into blocks. The result is new for p=2, the proof, which works for all p, is new.