Modularity of Some Potentially Barsotti-tate Galois Representations

and its Teichmüller lift will be denoted ω̃n. In particular, ω = ω1 is the mod-l reduction of the cyclotomic character ǫ. Recall that if ρ : GQl or WQl → GL2(K) is continuous and tamely ramified, then ρ |Il is isomorphic either to ω̃ ⊕ ω̃ or to ω̃ 2 ⊕ ω̃ 2 , depending on the absolute reducibility or irreducibility of ρ. If an l-adic representation ρ of GQl is potentially semistable (in the sense of Fontaine), then one associates to ρ a Weil-Deligne representation WD(ρ), for ex-