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We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06]for certain crystalline families of G_Qp-representations. This conjecture canbe regarded as a local analogue of the Iwasawa main conjecture for families.Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), ofBenois and Berger for crystalline G_Qp-representations with respect to thecyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes(cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelianp-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also formulated aversion of the epsilon-conjecture for affinoid families of (phi,Gamma)-modulesover the Robba ring, and proved his conjecture in the rank-1 case. He used thiscase to construct an epsilon-isomorphism for families of trianguline(phi,Gamma)-modules, depending on a fixed triangulation. Our results imply thatthis epsilon-isomorphism is independent of the chosen triangulation for certaincrystalline families. The main ingredient of our proof consists of theconstruction of families of Wach modules generalizing work of Wach and Berger[Ber04] and following the approach of Kisin to the construction of potentiallysemi-stable deformation rings [Kisa].

Wach modules, regulator maps, and epsilon-isomorphisms in families