Iwasawa Theory for $p$-adic Representations

Iwasawa invariants of galois deformations

“…The latter group is finite by our last assumption on t. However, H 1 (F Σ /F ∞ , A t ) has no proper Λ-submodules of finite index by [5,Proposition 5]. Thus H 1 (F Σ /F ∞ , A t ) Γ must in fact vanish, so that (ker γ t ) Γ vanishes as well, as desired.…”

“…Since ker γ n,t is finite by (i), it follows that H 1 (F Σ /F n , A t ) has corank [F : Q]c(V )p n and that H 2 (F Σ /F n , A t ) is finite. In fact, H 2 (F Σ /F ∞ , A t ) must therefore vanish since by [5,Proposition 4] it is cofree over Λ. Thus H 2,n vanishes, so that H 1,n = coker γ n,t is finite as well.…”

“…where the latter Selmer group is as in [5,Section 7], A(m) = A ⊗ ε m is the twist of A by m powers of the cyclotomic character and ω is the Teichmüller character.…”

Compatible families of elliptic type

“…For the sake of clarity, let us call a prime p / ∈ S classically ordinary (relative to {ρ }) if p a(p), and ordinary (again relative to {ρ }) if p satisfies the definition on pp. 97-98 of Greenberg [2]. These notions are complementary in the sense that the former is a condition on ρ for = p and the latter a condition on ρ p .…”

On finite $\Lambda $-submodules of Selmer groups of elliptic curves

“…Thus, by (2), it suffices to show that lim ← − n C Γn ∞ has no non-zero finite Λ-submodule to complete the proof. We prove this using the same argument as in the proof of [7,Prop.…”