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Let p be an odd prime and let us fix the embeddings of the algebraic closure Q̄ of the field of rationals into C and the algebraic closure Q̄p of the field of p-adic numbers. Let T be a p-adic family of rank two Galois representations, that is, T is a free module of rank two over a complete local noetherian ring R of characteristic zero with finite residue field of characteristic p on which the absolute Galois group GF of a number field F acts continuously. We fix a Zariski-dense subset S of Homcont(R, Q̄p). We assume that the triple (T , R, S) satisfies the following properties: (1) The action of GF on T is unramified outside a finite set of places Σ. (2) For each κ ∈ S, the GF -representation Tκ = T ⊗R κ(R) is a lattice of the p-adic étale realization Vκ is of a pure motive Mκ which is critical in the sense of [De79]. (3) For every κ ∈ S, the complex L-function L(Mκ, s) is meromorphically continued to the whole complex plane.

Control theorems for Selmer groups of nearly ordinary deformations