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.

We introduce an elementary congruence-based procedure to look for qth power multiples in arbitrary binary recurrence sequences (q ≥ 3). The procedure allows to prove that no such multiples exist in many instances.

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots”of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].