Blocks for mod p representations of GL₂(Qₚ)

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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.