Expansions of Modular Forms and Their Interpolation Properties

We define a power series expansion of a 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×-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 interpolation for the square roots of a family of twisted special values of the automorphic L-function associated with the base change of f to K. 2000 Mathematics Subject Classification 11F67