| Résume | We define a power series expansion of an holomorphic modular form $f$ in the $p$-adic neighborhood of a CM point $x$ of type $K$ for a split good prime $p$. The modularity group can be either a classical conguence group or a group of norm $1$ elements in an order of an indefinite quaternion algebra. The expansion coefficients are shown to be closely related to the classical Maass operators and give $p$-adic information on the ring of definition of $f$. By letting the CM point $x$ vary in its Galois orbit, the $r$-th coefficients define a $p$-adic $K^{\times}$-modular form in the sense of Hida. By coupling this form with the $p$-adic avatars of algebraic Hecke characters belonging to a suitable family and using a Rankin-Selberg type formula due to Harris and Kudla along with some explicit computations of Watson and of Prasanna, we obtain in the even weight case a $p$-adic measure whose moments are essentially the square roots of a family of twisted special values of the automorphic $L$-function associated with the base change of $f$ to $K$. |
