  In this article, we prove (many parts of) the rank two case of the Kato's
local epsilon-conjecture using the Colmez's p-adic local Langlands
correspondence for GL_2(Q_p). We show that a Colmez's pairing defined in his
study of locally algebraic vectors gives us the conjectural
epsilon-isomorphisms for (almost) all the families of p-adic representations of
Gal(overline{Q}_p/Q_p) of rank two, which satisfy the desired interpolation
property for the de Rham and trianguline case. For the de Rham and non
trianguline case, we also show this interpolation property for the "critical"
range of Hodge-Tate weights using the Emerton's theorem on the compatibility of
classical and p-adic local Langlands correspondence. As an application, we
prove that the Kato's Euler system associated to any Hecke eigen new form
satisfies a functional equation which has the same form as predicted by the
Kato's global epsilon-conjecture.
