Rankin-Eisenstein classes for modular forms

In this paper we make a systematic study of certain motivic cohomology classes (``Rankin-Eisenstein classes'') attached to the Rankin-Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin-Selberg convolutions of cusp forms.