Congruence properties of Taylor coefficients of modular forms

Hannah Larson, Geoffrey Smith
2014 International Journal of Number Theory  
In their work, Serre and Swinnerton-Dyer study the congruence properties of the Fourier coefficients of modular forms. We will examine similar congruence properties, but for the Taylor series coefficients of modular forms 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 a condition on τ and a prime number p that, if satisfied, implies that p m divides all the Taylor coefficients
more » ... f of sufficiently high degree. We also give effective bounds on the largest n such that p m does not divide the n th Taylor coefficient of f at τ that are sharp under certain additional hypotheses.
doi:10.1142/s1793042114500407 fatcat:o7ugi2p5fjfljn3wm37sq7te6y