Number Theory Seminar

A conjecture by R. Greenberg asserts that a modular 2-dimensional p-adic Galois representation of a cusp form of weight larger than or equal to 2 is indecomposable over the p-inertia group unless it is induced from an imaginary quadratic field. I start with a survey of the known results and try to reach a brief description of a new case of indecomposability. Fix a prime p ≥ 3 and an absolutely irreducible odd representation ρ : Gal(Q/Q) → GL2(F) (F/Fp finite) of prime-to-p conductor 0 < N ∈ Z.