Trivial zeros of p-adic L-functions

One says that the $p$-adic $L$-function $L_p(f,s)$ of a modular form $f$ has a trivial zero
if the interpolation property forces $L_p(f,s)$ to vanish at some integer $s=m.$
This phenomenon was first studied at 1980's by Mazur, Tate and Teitelbaum.
For modular forms of even weight they formulated a precise conjecture
about the value of the derivative $L_p'(f,s)$ at $s=m.$
This conjecture was proved by Greenberg-Stevens (using $p$-adic families of modular forms)
and by Kato-Kurihara-Tsuji (using Euler systems) around 1998.

In this  talk we formulate and prove an analog of this result
for modular forms of odd weight. Our key tool is the theory
of $(\varphi,\Gamma)$-modules which allows to define the $\Cal L$-invariant
for a large class of $p$-adic representations.