Congruence Properties of Taylor Coefficients of Modular Forms

In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We examine similar congruence properties, but for the coefficients of a modified Taylor expansion about a CM point τ . These coefficients can be shown to be the product of a power of a constant transcendental factor and an algebraic integer. In our work, we give conditions on τ and a prime number p that, if satisfied, imply that pm divides the algebraic part of all the Taylor coefficients of f of sufficiently high degree. We also give effective bounds on the largest n such that pm does not divide the algebraic part of the nth Taylor coefficient of f at τ that are sharp under certain additional hypotheses.