On the universal supersingular mod p representations of GL2(F)GL2(F)

The irreducible supersingular mod p   representations of G=GL2(F)G=GL2(F), where F   is a finite extension of QpQp, are the building blocks of the mod p representation theory of G  . They all arise as irreducible quotients of certain universal supersingular representations. We investigate the structure of these universal modules in the case when F/QpF/Qp is totally ramified.