  We show that the modular symbol $(0,\infty)$, considered as an element of the
dual of Emerton's completed cohomology, interpolates Kato's Euler system at
classical points, and we deduce from this a factorisation of Beilinson-Kato's
system as a product of two symbols $(0,\infty)$ (an algebraic analog of
Rankin's method). The proof uses the $p$-adic local Langlands correspondence
for ${\mathbf {GL}}_2({\mathbf Q}_p)$ and Emerton's description of the
completed cohomology which we refine by imposing conditions at classical
points; the existence of such a refinement is a manifestation of an analyticity
property for $p$-adic periods of modular forms.
