Power Series Expansions of Modular Forms and Their Interpolation Properties

Abstract

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.

The Birch and Swinnerton–Dyer formula for elliptic curves of analytic rank one

“…For the above, the only hypotheses on the real quadratic field M that is needed is that B ⊗ M is split and p splits in M . No additional hypotheses are needed on the idempotent e. The extra hypotheses imposed in Section 4.2 are assumed in the construction of Brooks [Bro14] where they are used to obtain a certain explicit formula for the Maass-Shimura operators; they follow closely the choices made by Mori [Mor11] and Hashimoto [Has95]. For the comparison between periods of CM elliptic curves and false elliptic curves in the next section we do not need to make these extra assumptions.…”

p-Adic and analytic properties of period integrals and values of L-functions

“…Factorizations of p-adic L-functions as in question (3) have been obtained in a number of situations in low dimension; see for example [3,18,27,33]. Whenever there is a factorization of the central value of the complex L-function -references [33] and [27] Returning to the situation of Question 4.8, we arrive at a p-adic analogue of Question 2.1.…”

“…Whenever there is a factorization of the central value of the complex L-function -references [33] and [27] Returning to the situation of Question 4.8, we arrive at a p-adic analogue of Question 2.1. Because the role of the local zeta integrals is unclear, we are not even confident enough to formulate an Expectation.…”

The ring of differential Fourier expansions

“…In all the discussion above, we considered the modular curves X 1 (N ) parameterizing elliptic curves with Γ 1 (N ) level structures; nevertheless a substantial part of the theory will be developed also in the case of Shimura curves X D (U) parameterizing false elliptic curves with level U structures [12]. The role of Fourier expansions in the Shimura curve case will be played by Serre-Tate expansions, in the sense of [20,5,6]. We will also consider the theory over modular curves with respect to Serre-Tate expansions.…”