\centerline{Abstract} \bigskip I'll report on recent results with M.~Bertolini, H.~Darmon and M.~Spiess. Let $f$ be a cusp form for $\Gamma _0(N)$ of even weight $k\ge 2$, which is a new eigenform for all the Hecke operators and such that $N$ is square free. Let $K$ be an imaginary quadratic field whose discriminant is prime to $N$ (and satisfies the so called ``Heegner conditions".) Let $p$ be a prime dividing $N$.
We have constructed a $p$-adic L-function, $L_p(f/K, s)$ attached to $f$ over $K$. This construction is done by interpreting in the anticyclotomic setting some old ideas of Drinfeld and Schneider and is purely $p$-adic (as opposed to other constructions of $p$-adi8c $L$ functions). We prove that our $p$-adic $L$-function has the following properties:
1) {\it Interpolation} The $p$-adic L-function interpolates twists (of square roots) of special values of the complex L-function: $L(f/K, s)$.
2) {\it Special values} We have, ``for trivial reasons", that $L_p(f/K, k/2)=0$, so we look at the arithmetic interpretation of the derivative $L_p'(f/K, k/2)$. The result depends on $p$.
a) Suppose $p$ is split in $K$. Then we prove that: $$ L_p'(f/K, k/2)/\Omega _p={\cal L}_T(f)\dot (L(f/K, k/2))/\Omega)^{1/2} $$ where $\Omega _p$ and $\Omega$ are $p$-adic and respectively real periods and ${\cal L}_T(f)$ is the ${\cal L}$-invariant of $f$ defined by Teitelbaum. The above formula is the anticyclotomic analogue of the conjecture of Mazur-Tate-Teitelbaum.
b) Suppose now that $p$ is inert in $K$. Then we can make sense of and prove th following formula $$ L_p'(f/K, k/2)=AJ_{p,f}({\cal H}_k), $$ where $AJ_{p,f}$ is the $p$-adic Abel-Jacobi map composed with the projection onto the $f$-isotypic component under the action of the Hecke-algebra, and ${\cal H}_k$ is the weight $k$ Heegner cycle. \end{document}