Compositio Mathematica

Research Article

Extensions of rank one (φ,Γ)-modules and crystalline representations

Seunghwan Changa1 and Fred Diamonda2

a1 Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, Republic of Korea (email:

a2 Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK (email:


Let K be a finite unramified extension of Qp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional $\overline {\F }_p$-representations of GK and characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

(Received September 10 2009)

(Accepted April 26 2010)

(Online publication December 13 2010)

2000 Mathematics Subject Classification

  • 11S25 (primary)


  • p-adic representations;
  • crystalline representations;
  • (φ,Γ)-modules;
  • Wach modules