Computing Power Series Expansions of Modular Forms

Abstract

We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.

Darmon points on elliptic curves over number fields of arbitrary signature

“…As for Heegner points on Shimura curves attached to division algebras Elkies [19] performed some computations in certain particular situations. More recently, Voight-Willis [53] using Taylor expansions and Nelson [40] using the Shimizu lift have provided more general algorithms and have performed some computations. (b) Non-archimedean computations.…”

The Taylor coefficients of the Jacobi theta constant $$\theta _3$$

“…The results of this paper can be viewed as part of the much broader theme of considering the Taylor coefficients of general modular forms around complex multiplication points. This point of view seems to have its roots in the work of Shimura [8], and was later considered by Villegas and Zagier [10,11] and others [4,6,12]; see also Sections 5.1 and 6.3 of [15]. Our idea of considering the "centered" version of a modular form, and the connection between the Taylor expansions of the centered and non-centered version, are discussed in [15] in that broader setting.…”

Mumford curves covering 𝑝-adic Shimura curves and their fundamental domains

“…The complex uniformisation of Shimura curves, in comparison to that of modular curves, is more difficult to approach computationally (think about the lack of cusps, which complicates the computation of Fourier expansions of their uniformising functions). Nevertheless, several studies make amenable computations for this complex uniformisation such as [AB04], [Voi06], [BT07] and [VW11]. In addition to the complex uniformisation, Shimura curves also admit nonarchimedean uniformisations.…”